text:
\begin{document}

\title{The space of ergodic measures for Lorenz attractors}

\author{Yi Shi\footnote{Y. Shi was supported by NSFC (11701015, 11831001)}, Xueting Tian	\footnote{X. Tian was supported by NSFC (11671093)} and Xiaodong Wang
\footnote{X. Wang was supported by NSFC (11701366).}}

\maketitle

\begin{abstract}
	For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we show that the space of ergodic measures is path connected for $C^r$-generic Lorenz attractors while  it is not connected for $C^r$-dense Lorenz attractors.
	Various properties of the ergodic measure space for Lorenz attractors have been showed.
	In particular, a $C^r$-connecting lemma ($r\geq2$) for Lorenz attractors also has been proved.
	In $C^1$-topology,  we obtain similar properties for singular hyperbolic attractors in higher dimensions.
\end{abstract}

\section{Introduction}

In 1963, E. Lorenz~\cite{Lorenz} introduced the following differential equations in the study of meteorology:
\begin{align}\label{Equa:Lorenz}
 \left\{ \begin{array}{ll}
\dot x=\alpha(y-x),\\
\dot y=\beta x-y-xz, \\
\dot z=xy-\gamma z,
\end{array} \right.
\end{align}
which has arisen much interest for dynaminists since then.
For an open neighborhood of the chosen parameters $(\alpha,\beta,\gamma)=(10,28,8/3)$, numerical simulations suggest that the system generated by the solution of~(\ref{Equa:Lorenz})  admits a strange attractor called the {\it Lorenz attractor} which attracts almost all points in the phase space. Dynamics of Lorenz attractors behave in a chaotic way in the sense that it is transitive with dense periodic orbits and thus it is sensitively dependent on initial conditions. Although expressed in a simplified model, Lorenz equations (\ref{Equa:Lorenz}) have been proved to be resistant and  present much difficulty for rigorous mathematical analysis. Thanks to the independent pioneering works of J. Guckenheimer~\cite{guck} and V. Afraĭ\u{\i}movi\v{c}-V. Bykov-L. Sil'nikov~\cite{abs}, a geometric model of Lorenz attractor was introduced whose dynamics admits the behaviors observed by Lorenz. The {\it geometric Lorenz attractor} (see Definition~\ref{def:Lorenz}) is defined through a three dimensional smooth vector field and is proved to possess a singularity (or an equilibrium) where the vector field vanishes and which is accumulated by regular orbits. On the other hand, the geometric Lorenz attractor is not  structurally stable~\cite{guck}.  Works of J. Guckenheimer and R. Williams~\cite{gw,williams} gave  well descriptions of the structure of geometric Lorenz attractors which led more studies, for instance~\cite{am2,c1lorenz,LMP,tucher1,tucher2} and references therein.

Due to the existence of the singularity, the geometric Lorenz attractor is not  hyperbolic (see Definition~\ref{Def:hyp}). To describe its geometric structure, C. Morales-M. Pacifico-E. Pujals~\cite{mpp} introduced the notion of {\it singular hyperbolicity} (or {\it sectional hyperbolicity}).
One can refer to~\cite{lgw,Me-Mo} for higher dimensions.
Let $M^d$ (or $M$ for simplicity) be a $d$-dimensional ($d\geq 3$) $C^{\infty}$  smooth closed Riemannian manifold.
For every $r\in\mathbb{N}\cup\{\infty\}$, we denote by $\mathscr{X}^r(M)$ the set of $C^r$-smooth vector fields on $M$ endowed with $C^r$-topology. Given a vector field $X\in\mathscr{X}^r(M)$. We denote by $\phi_t^X\colon M\rightarrow M$ the $C^r$-flow generated by $X$ and by $\Phi_t^X=D\phi_t^X$ the tangent map of $\phi_t^X$.
When there is no confusion, we also denote them by $\phi_t$ and $\Phi_t$ for simplicity.

An invariant compact set $\Lambda$ of $\phi_t$ is {\it singular hyperbolic} if its tangent bundle admits a dominated splitting $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ with $E^{ss}$ being $\Phi_t$-contracting and $E^{cu}$ being $\Phi_t$-sectional expanding (see details in Definition~\ref{Def:singular-hyp}). It is proved in~\cite{mpp}  that any robustly transitive set of a three dimensional vector field is singular hyperbolic and similar conclusion holds for higher dimensions~\cite{lgw,zgw}.  We refer to \cite{3-dim flow} for a comprehensive introduction of singular hyperbolic vector fields.
\medskip

In this paper, we study the space of invariant measures supported on geometric Lo\-renz attractors as well as on singular hyperbolic attractors of higher dimensions. For $C^r(r\geq 2)$-generic geometric Lorenz attractors, we show that every invariant measure supported on the attractor can be approximated by periodic measures.
This is motivated by works of K. Sigmund~\cite{Sigmund-Axiom A,Sigmund-flow} on basic sets of Axiom A systems~\cite{smale} together with R. Ma\~n\'e's Ergodic Closing Lemma~\cite{mane-ergodic closing}.
Given $X\in\mathscr{X}^r(M)$ and an invariant compact set $\Lambda$ of $\phi_t^X$, denote by $\mathcal{M}_{inv}(\Lambda)$ and $\mathcal{M}_{erg}(\Lambda)$ the spaces of $\phi_t^X$-invariant measures and $\phi_t^X$-ergodic measures  supported on $\Lambda$  respectively endowed with the weak*-topology.
Denote by $\mathcal{M}_{per}(\Lambda)$ the set of periodic measures (the invariant measure equidistributed on a single periodic orbit) supported on $\Lambda$. When $\Lambda$ is a basic set of an Axiom A system (diffeomorphism or vector field), K. Sigmund~\cite{Sigmund-Axiom A,Sigmund-flow} proved that  $\mathcal{M}_{per}(\Lambda)$ is dense in $\mathcal{M}_{inv}(\Lambda)$ and the set of ergodic measures with zero entropy forms a residual subset (i.e. a dense $G_\delta$ subset) of $\mathcal{M}_{inv}(\Lambda)$.
More statistical properties of periodic measures on basic sets are obtained by R. Bowen~\cite{Bowen-Axiom A,Bowen-Flow}.
During the exploration of the stability conjecture, R. Ma\~n\'e proved the famous Ergodic Closing Lemma in 1982~\cite{mane-ergodic closing} which is a milestone work in dynamical systems and ergodic theory.
The Ergodic Closing Lemma plays a key role in solving the stability conjecture.
It states that for $C^1$ generic diffeomorphisms or vector fields (i.e. systems in a residual subset of the whole space of systems), the set of periodic measures and singular measures are dense in the space of ergodic measures.  In this sense, works of K. Sigmund~\cite{Sigmund-Axiom A,Sigmund-flow} can be seen as the Ergodic Closing Lemma restricted to basic sets without perturbation.
However, things become more complicated to verify the Ergodic Closing Lemma restricted to a non-hyperbolic invariant compact  set $\Lambda$. To be precise, considering the problem whether periodic measures together with singular ones are dense in $\mathcal{M}_{erg}(\Lambda)$ if $\Lambda$ is not hyperbolic, one has to guarantee that  the periodic measures (maybe obtained by perturbations) are contained in $\Lambda$.

For a hyperbolic periodic point $p$, its homoclinic class $H(p)$  is the closure of the set of periodic points $q$ homoclinically related to $p$, i.e. $W^s(\orb(q))$ has non-empty transverse intersections with $W^u(\orb(p))$ and vice verse. In his survey~\cite[Conjecture 2]{b-survey}, C. Bonatti conjectured that Ergodic Closing Lemma restricted on a homoclinic classes holds true for $C^1$-generic diffeomorphisms. Progresses and some important criterions can be found in the work of F. Abdenur-C. Bonatti-S. Crovisier~\cite{abc}.
Other works on the approximation of invariant measures by periodic ones can be found in~\cite{btv,bonatti-zhang,dgmr,hirayama,lls,Yang-Zhang}.

\medskip

Recall that a basic set of an Axiom A diffeomorphism is a hyperbolic horseshoe which is topological conjugate to a full shift. By another work of K. Sigmund~\cite{Sigmund}, we know that the space of ergodic measures over a basic set is path connected (or arcwise connected). In the case of Axiom A  vector fields, a basic set $\Lambda$ is the suspension of a hyperbolic horseshoe $\Lambda'$ with a continuous roof function, thus by the one-to-one correspondence between invariant measures over $\Lambda$ and $\Lambda'$ (see for instance~\cite{parry-pollicott}), we know that $\mathcal{M}_{erg(\Lambda)}$ is also path connected.
In the non-hyperbolic case,  a recent work of A. Gorodetski-Y. Pesin~\cite{Go-Pe} proves that if an isolated homoclinic class  $H(p)$ satisfies that all hyperbolic periodic points in $H(p)$ of the same index with $p$ are homoclinically related with each other, then $\mathcal{M}_{erg}(H(p))$ is path connected. This gives a criterion for path connectedness of the ergodic measure space restricted on homoclinic classes. Other results on connectedness and topological properties of the  invariant measure space for certain systems can be found in~\cite{bbg,GK}.

We verify the path connectedness of the space of ergodic measures on geometric Lorenz attractors for $C^r(r\geq 2)$-generic vector fields. Moreover, we prove that for $C^r(r\geq 2)$-dense vector fields admitting a geometric Lorenz attractor, the singular measure is isolated in the space of ergodic measures which breaks the connectedness of the ergodic measure space. This gives a sharp contrast.

\begin{theoremalph}\label{Thm:A}
	For every $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$, there exist a $C^r$-residual subset $\mathscr{R}^r\subset\mathscr{X}^r(M^3)$ and a $C^r$-dense subset $\mathscr{D}^r\subset\mathscr{X}^r(M^3)$, such that
	\begin{itemize}
		\item if $X\in\mathscr{R}^r$ has a geometric Lorenz attractor $\Lambda$, then
		$\overline{\mathcal{M}_{per}(\Lambda)}=
		\overline{\mathcal{M}_{erg}(\Lambda)}=		\mathcal{M}_{inv}(\Lambda)$.
		Moreover, the space $\mathcal{M}_{erg}(\Lambda)$ is path connected.
		\item if $X\in\mathscr{D}^r$ has a geometric Lorenz attractor $\Lambda$, then
		$\overline{\mathcal{M}_{per}(\Lambda)}\subsetneqq \overline{\mathcal{M}_{erg}(\Lambda)}\subsetneqq \mathcal{M}_{inv}(\Lambda)$.
		Moreover, the space $\mathcal{M}_{erg}(\Lambda)$ is not connected.
	\end{itemize}
\end{theoremalph}

Here the definition of geometric Lorenz attractor follows J. Guckenheimer and R. Williams \cite{guck,gw,williams}, see Definition \ref{def:Lorenz}. Following this definition, the vector fields exhibiting a geometric Lorenz attractor is an open subset of $\mathscr{X}^r(M^3)$ for every $r\geq2$, see Proposition~\ref{prop:Lorenz}.

\begin{remark*}
	In our construction for the dense part of Theorem~\ref{Thm:A}, the atomic measure $\delta_{\sigma}$ of the singularity $\sigma$ is isolated in the ergodic measure space. Actually, for every geometric Lorenz attractor $\Lambda$,  let $\mathcal{M}_0=\mathcal{M}_{erg}(\Lambda)\setminus \{\delta_{\sigma}\}$ and $\mathcal{M}_1=\{\mu\in \mathcal{M}_{inv}(\Lambda): \mu(\sigma)=0 \}$, then $\mathcal{M}_0$ is path connected and moreover, $\overline{\mathcal{M}_{per}(\Lambda)}=\overline{\emph{Convex}(\mathcal{M}_0)}=\overline{\mathcal{M}_1}$. Here $\emph{Convex}(\mathcal{M}_0)$ consists all elements that are convex combinations of measures in $\mathcal{M}_0$.
\end{remark*}

As an application of Theorem~\ref{Thm:A}, we have the following corollary concerning the support and entropy of invariant measures for $C^r(r\geq 2)$-generic geometric Lorenz attractors.

\begin{corollary}\label{Cor:Cr-entropy-support}
	For every $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$, there exists a residual subset $\mathscr{R}^r\subset\mathscr{X}^r(M^3)$, such that for every $X\in\mathscr{R}^r$, if $\Lambda$ is a geometric Lorenz attractor of $X$, then
	\begin{itemize}
		\item there exists a residual subset $\mathcal{M}_{res}$ in $\mathcal{M}_{inv}(\Lambda)$ such that every $\mu\in\mathcal{M}_{res}$ is ergodic with $\supp(\mu)=\Lambda$ and $h_{\mu}=0$.
		\item there exists a dense subset $\mathcal{M}_{den}$ in $\mathcal{M}_{inv}(\Lambda)$ such that every $\mu\in\mathcal{M}_{den}$ is ergodic with $h_{\mu}>0$.
	\end{itemize}
\end{corollary}

\begin{remark*}
	For every geometric Lorenz attractor $\Lambda$ of every $X\in \mathscr{X}^r(M^3)$, the measures with zero entropy and full support  form a residual subset in $\mathcal{M}_{inv}(\Lambda)$. See  Remark~\ref{Rem:entropy and support}.
\end{remark*}

During the proof of Theorem~\ref{Thm:A}, a main novelty is that we prove a $C^r$-connecting lemma for the geometric Lorenz attractors. See Theorem \ref{thm:Lorenz-connecting} for detailed statement.

\begin{theorem}\label{thm:intro-connecting}
	Let $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$ and $X\in\mathscr{X}^r(M^3)$ which exhibits a geometric Lorenz attractor $\Lambda$ with singularity $\sigma\in\sing(\Lambda)$. Assume $z\in W^u(\sigma)$ and $p\in\Lambda$ is a critical element, then there exists $Y\in\mathscr{X}^r(M^3)$ which is arbitrarily $C^r$-close to $X$, such that
	$$
	\orb(z,\phi_t^Y)\subseteq W^u(\sigma,\phi_t^Y)\cap W^s(p,\phi_t^Y),
	$$
	with $\orb(\sigma,\phi_t^X)=\orb(\sigma,\phi_t^Y)$ and $\orb(p,\phi_t^X)=\orb(p,\phi_t^Y)$.
\end{theorem}

Recall that a point $p$ is \emph{a critical element} of $X\in\mathscr{X}^r(M)$ if $p\in\sing(X)$ or $p\in\per(X)$. Here $p$ can be equal to $\sigma$. This theorem shows that a $C^r$-perturbation can generate a homoclinic or heteroclinic orbit for geometric Lorenz attractors.
It has its own interest in the study of homoclinic bifurcations for geometric Lorenz attractors. For instance, we show that $C^r$-densely, the singularity inside a geometric Lorenz attractor admits a homoclinic orbit (Corollary~\ref{Cor:dense-loop}).

For $C^1$-vector fields, we obtain similar conclusions as in Theorem~\ref{Thm:A} and Corollary~\ref{Cor:Cr-entropy-support} for a larger class of  singular hyperbolic attractors.
Given a $d$-dimensional smooth closed Riemannian manifold $M=M^d$ with $d\geq 3$ and given $X\in\mathscr{X}^1(M)$, an invariant compact set $\Lambda$ of $\phi_t^X$ is called a {\it Lorenz-like attractor} of $X$ if
\begin{enumerate}
	\item $\Lambda$ is a singular hyperbolic with $\Lambda\cap\sing(X)\neq\emptyset$;
	\item $\Lambda$ is a robustly transitive attractor: there exist a $C^1$-neighborhood $\mathcal{U}_0$ of $X$ and an attracting region $U_0$ such that the maximal invariant set of $\phi_t^Y$ in $U_0$ is a transitive attractor for any $Y\in\mathcal{U}_0$ and coincides with $\Lambda$ when $Y=X$.
\end{enumerate}
Researches on Lorenz-like attractors about robust transitivity, hyperbolicity and  entropy theory can be found in a series works~\cite{ams,CY-Robust-attractor,mo-pa,sgw,yang,PYY}. Our Theorem~\ref{Thm:B} verifies  the path connectedness of the space of ergodic measures restricted on  Lorenz-like attractors for $C^1$-generic vector fields. Similarly as in Theorem~\ref{Thm:A}, we prove that for $C^1$-dense vector fields admitting a  Lorenz-like  attractor with a co-dimensional two singular hyperbolic splitting, the singular measure is isolated in the space of ergodic measures, which implies in this case that the ergodic measure space is not connected.

\begin{theoremalph}\label{Thm:B}
	There exist a residual subset $\mathscr{R}\subset\mathscr{X}^1(M)$ and a dense subset $\mathscr{D}\subset\mathscr{X}^1(M)$, such that
	\begin{itemize}
		\item if $X\in\mathscr{R}$ and $\Lambda$ is a Lorenz-like attractor of $X$, then $\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{erg}(\Lambda)}=\overline{\mathcal{M}_{per}(\Lambda)}$. Moreover, the space $\mathcal{M}_{erg}(\Lambda)$ is path connected.
		\item if $X\in\mathscr{D}$ and $\Lambda$ is a Lorenz-like attractor of $X$ whose singular hyperbolic splitting $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ satisfies $\dim(E^{cu})=2$, then $\overline{\mathcal{M}_{per}(\Lambda)}\subsetneqq\overline{\mathcal{M}_{erg}(\Lambda)}\subsetneqq\mathcal{M}_{inv}(\Lambda)$. Moreover, the space $\mathcal{M}_{erg}(\Lambda)$ is not connected.
	\end{itemize}
\end{theoremalph}

\begin{remark}
	Concerning  the residual part of Theorem~\ref{Thm:B}, similar properties are obtained in~\cite[Theorem 3.5]{abc} for isolated non-trivial transitive sets of diffeomorphisms. However, for vector fields, we have to conquer the difficulties caused by the existence of singularities, i.e. to show that the atomic measure of the singularity can be accumulated by periodic measures inside the Lorenz-like attractor. Moreover, we obtains the dense part of Theorem~\ref{Thm:B} which is a sharp comparison with the residual part. Very recently, D. Yang and J. Zhang proved $\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{erg}(\Lambda)}=\overline{\mathcal{M}_{per}(\Lambda)}$ for which $\Lambda$ is a non-trivial isolated transitive set for $C^1$-generic vector fields in~\cite[Theorem 3.1]{Yang-Zhang2}.
\end{remark}

The following is a corollary of Theorem~\ref{Thm:B} which concerns the support and entropy of measures on Lorenz-like attractors and whose statement is similar to Corollary~\ref{Cor:Cr-entropy-support}.
\begin{corollary}\label{Cor:entropy-support}
    There exists a residual subset $\mathscr{R}\subset\mathscr{X}^1(M)$, such that for any  $X\in\mathscr{R}$, if $\Lambda$ is a Lorenz-like attractor of $X$, then:
    \begin{enumerate}
        \item\label{item:residual} there exists a residual subset $\mathcal{M}_{res}$ in $\mathcal{M}_{inv}(\Lambda)$ such that every $\mu\in\mathcal{M}_{res}$ is ergodic with $\supp(\mu)=\Lambda$ and $h_{\mu}=0$.

	    \item\label{item:dense} there exits a dense subset $\mathcal{M}_{den}$ in $\mathcal{M}_{inv}(\Lambda)$ such that every $\mu\in\mathcal{M}_{den}$ is ergodic with $h_{\mu}>0$.
\end{enumerate}
\end{corollary}

A related notion with attractors is the {\it Lyapunov stability}. An invariant compact set $\Lambda$ of a vector field $X\in\mathscr{X}^1(M)$ is {\it Lyapunov stable} if for any neighborhood $U$ of $\Lambda$, there exists another neighborhood $V$ of $\Lambda$ such that $\phi_t(V)\subset U$ for any $t\geq 0$.
S. Crovisier and D. Yang~\cite{CY-Robust-attractor} proved recently that there exists an open dense set $\mathcal{U}$ in $\mathscr{X}^1(M)$ such that for any $X\in\mathcal{U}$, any singular hyperbolic Lyapunov stable chain recurrence class $\Lambda$ is a robustly transitive attractor (see Section~\ref{Section:connecting lemmas} for the definition of chain recurrence class). As a consequence, $\Lambda$ either is a Lorenz-like attractor or  a uniformly hyperbolic attractor, depending on whether it contains singularities or not. Thus we have the following conclusion as a corollary of Theorem~\ref{Thm:B} and Corollary~\ref{Cor:entropy-support} combined with Crovisier-Yang's result~\cite{CY-Robust-attractor} .

\begin{theoremb}
	The conclusions of Theorem~\ref{Thm:B} and Corollary~\ref{Cor:entropy-support} are valid if we assume $\Lambda$ is a singular hyperbolic Lyapunov stable chain recurrence class.
\end{theoremb}

\subsection*{Organization of the paper} In Section~\ref{Section:Pre}, we state some preliminaries. Seciton~\ref{Section:measure} is devoted to give some properties of the space of invariant measures supported on certain invariant compact set. We give a detailed characterization of geometric Lorenz attractors and prove Theorem~\ref{Thm:A} together with Corollary~\ref{Cor:Cr-entropy-support} in Section~\ref{Section:Cr-Lorenz} and prove Theorem~\ref{Thm:B} together with Corollary~\ref{Cor:entropy-support} in Section~\ref{Section:C1}. In Appendix, we give a proof of Proposition~\ref{prop:Lorenz} which states that the $C^r(r\geq 2)$-vector fields admitting a geometric Lorenz attractor forms an open subset of $\mathscr{X}^r(M)$.

\subsection*{Acknowledgment} We would like to thank S. Crovisier, S. Gan, G. Liao, P. Varandas, L. Wen, D. Yang, J. Yang and J. Zhang for useful suggestions and comments. This work was prepared  during Y. Shi and X. Wang visiting School of Mathematical Sciences, Fudan University. The two authors would like to thank their hospitality.

\section{Preliminary}\label{Section:Pre}
   Recall that for a $C^r(r\geq 1)$-vector field $X\in\mathscr{X}^r(M)$, we denote by $\phi_t^X\colon M\rightarrow M$ the $C^r$-flow generated by $X$ and by $\Phi_t^X=D\phi_t^X$ the tangent map of $\phi_t^X$. We also use $\phi_t$ and $\Phi_t$ for simplicity if there is no confusion.

\subsection{Singular hyperbolicity}

We give the definition of hyperbolic and singular hyperbolic set.

\begin{definition}\label{Def:hyp}
    Given  a vector field $X\in\mathscr{X}^1(M)$. An invariant compact set $\Lambda$ is {\it hyperbolic} if $\Lambda$ admits a continuous $\Phi_t$-invariant splitting $T_{\Lambda}M=E^s\oplus \langle X\rangle\oplus E^u$, where $\langle X\rangle$ denotes the one-dimensional linear space generated by the flow direction, such that $E^s$ is contracted by $\Phi_t$ and $E^u$ is expanded by $\Phi_t$. To be precise, there exist two constants $C>0$ and $\eta>0$, such that for any $x\in\Lambda$ and any $t\geq 0$, it satisfies:
    \begin{itemize}
    	\item for any $v\in E^s(x)$, $\|\Phi_t(v)\|\leq Ce^{-\eta t}\|v\|$;
    	\item for any $v\in E^u(x)$, $\|\Phi_{-t}(v)\|\leq Ce^{-\eta t}\|v\|$.
    \end{itemize}
The stable dimension $\dim(E^s)$ is called the {\it index} of the hyperbolic splitting.
\end{definition}

Singular hyperbolicity was first introduced by Morales-Pacifico-Pujals~\cite{mpp} to describe the geometric structure of Lorenz attractors and was developed in higher dimensions by~\cite{lgw,Me-Mo}.

\begin{definition}\label{Def:singular-hyp}
	Given  a vector field $X\in\mathscr{X}^1(M)$. An invariant compact set $\Lambda$ is {\it singular hyperbolic} if $\Lambda$ admits a continuous $\Phi_t$-invariant splitting $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ and there exist two constants $C,\eta>0$, such that for any $x\in\Lambda$ and any $t\geq 0$, it satisfies:
	\begin{itemize}
		\item $E^{ss}\oplus E^{cu}$ is a {\it dominated splitting}: $\|\Phi_t|_{E^{ss}(x)}\|\cdot \|\Phi_{-t}|_{E^{cu}(\phi_t(x))}\|< Ce^{-\eta t}$;
		\item $E^{ss}$ is contracted by $\Phi_t$: $\|\Phi_t(v)\|< Ce^{-\eta t}$ for any $v\in E^{ss}(x)$;
		\item $E^{cu}$ is sectionally expanded by $\Phi_t$:  $\|\det\Phi_t(x)|_{V_x}\|> Ce^{\eta t}$ for any 2-dimensional  subspace $V_x\subset E^{cu}_x$.
	\end{itemize}

\end{definition}

\begin{remark}\label{Rem:singular hyperbolicity}
	(1) Given $X\in\mathscr{X}^1(M)$. If an invariant compact set $\Lambda$ is hyperbolic, then $\Lambda$ must contain no singularity. On the other hand, if $\sing(\Lambda)=\emptyset$, then $\Lambda$ is hyperbolic if and only if $\Lambda$ is singular hyperbolic for $X$ or for $-X$.

	(2) By definition, singular hyperbolicity is a robust property. To be precise, if $\Lambda$ is a singular hyperbolic invariant compact set of $X\in\mathscr{X}^1(M)$ associated with splitting $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ and constants $(C,\eta)$, then there exist a neighborhood $U$ of $\Lambda$ and a neighborhood $\mathcal{U}\subset \mathscr{X}^1(M)$ of $X$ such that for any $Y\in\mathcal{U}$, the maximal invariant set of $\phi_t^Y$ in $U$ is singular hyperbolic associated with the same stable dimension and constants $(C,\eta)$.
\end{remark}

\subsection{The geometric Lorenz attractor}

We follow the constructions of Guckenheimer and Williams~\cite{guck,gw,williams} to give the definition of {\it geometric Lorenz attractor} for any $C^r (r\geq 1)$ vector field.
Let $X\in\mathscr{X}^r(M)$ where $r\in\mathbb{N}\cup\{\infty\}$,  an open subset $U\subseteq M$ is called an \emph{attracting region} of $X$, if for every $x\in\partial U$, the vector field $X(x)$ is transverse to $\partial U$ and points to the interior of $U$, and $\phi_t^X(x)\in U$ for every $t>0$.
This implies $\overline{\phi_t^X(U)}=\phi_t^X(\overline{U})\subsetneqq U$ for every $t>0$.
An invariant compact set $\Lambda$ is an {\it attractor} of $X$ if $\Lambda$ is transitive and $\Lambda$ is the maximal invariant set of an attracting region.
Recall that $M^3$ denotes a three dimensional closed smooth Riemannian manifold.

\begin{definition}\label{def:Lorenz}
	Let $r\geq 2$ and $X\in\mathscr{X}^r(M^3)$. We say $X$ admits a \emph{geometric Lorenz attractor} $\Lambda$, if $X$ has an attracting region $U\subset M^3$ such that  $\Lambda=\bigcap_{t>0}\phi^X_t(U)$ is a singular hyperbolic attractor and satisfies the following properties:
	\begin{itemize}
		\item $\Lambda$ contains a unique singularity $\sigma$ with three eigenvalues $\lambda_1<\lambda_2<0<\lambda_3$ with respect to $\Phi_1(\sigma)$ satisfying $\lambda_1+\lambda_3<0$ and $\lambda_2+\lambda_3>0$.
		\item $\Lambda$ admits a $C^r$-smooth cross section $\Sigma$ which is $C^1$-diffeomorphic to $[-1,1]\times[-1,1]$, such that for every $z\in U\setminus W^s_{\it loc}(\sigma)$, there exists $t>0$ such that $\phi_t^X(z)\in\Sigma$, and $l=\{0\}\times[-1,1]=W^s_{\it loc}(\sigma)\cap\Sigma$.
		\item The Poincar\'e map $P:\Sigma\setminus l\rightarrow\Sigma$ is $C^1$-smooth in the coordinate $\Sigma=[-1,1]^2$ and has the form
		$P(x,y)=\big( f(x)~,~H(x,y) \big)$ for every
		$(x,y)\in[-1,1]^2\setminus l$.
		Moreover, it satisfies
		\begin{itemize}
			\item $H(x,y)<0$ for $x>0$, $H(x,y)>0$ for $x<0$, and $\sup_{(x,y)\in\Sigma\setminus l}\big|\partial H(x,y)/\partial y\big|<1$,
			$\sup_{(x,y)\in\Sigma\setminus l}\big|\partial H(x,y)/\partial x\big|<1$;
			\item the one-dimensional quotient map $f:[-1,1]\setminus\{0\}\rightarrow[-1,1]$ is $C^1$-smooth and satisfies
			$\lim_{x\rightarrow0^-}f(x)=1$,
			$\lim_{x\rightarrow0^+}f(x)=-1$, $-1<f(x)<1$ and
			$f'(x)>\sqrt{2}$ for every $x\in[-1,1]\setminus\{0\}$.
		\end{itemize}
	\end{itemize}
\end{definition}

\begin{figure}[htbp]
	\centering
	\includegraphics[width=14cm]{Lorenz-def.jpg}
	\caption{Geometric Lorenz attractor and return map}
\end{figure}

\begin{remark}\label{rk:unstable}
	The unstable manifold of $\sigma$ contains two orbits. We denote $z^+,z^-\in W^u(\sigma)\cap\Sigma$ which satisfy $\phi_t^X(z^{\pm})\notin\Sigma$ for every $t<0$, and
	$$
	z^+=(-1,y^+)\in\Sigma, \qquad {\rm and} \qquad
	z^-=(1,y^-)\in\Sigma.
	$$
	Here we can assume $-1<y^+<0$ and $0<y^-<1$ by changing the $y$-coordinate.
	For every $(x,y)\in\Sigma\setminus l$, if $x\rightarrow0^+$, then $P(x,y)$ approaches $z^+$;  if $x\rightarrow0^-$, then $P(x,y)$ approaches $z^-$.
\end{remark}

\begin{lemma}\label{lem:cone}
	The Poincar\'e map $P(x,y)=\big( f(x),H(x,y) \big):\Sigma\setminus l\rightarrow\Sigma$ satisfies
	$$
	\lim_{x\rightarrow0}f'(x)=+\infty,
	\qquad {\rm and} \qquad
	\lim_{x\rightarrow0}\big|\partial H(x,y)/\partial y\big|=0.
	$$

	Moreover, for every $\alpha>0$, let the cone field $\cC_{\alpha}=\bigcup_{(x,y)\in\Sigma}\cC_{\alpha}(x,y)$ be defined on $\Sigma$ as
	$$
	\cC_{\alpha}(x,y)~=~\big\{~v=a\cdot\partial/\partial x
	+b\cdot\partial/\partial y:~
	|b|\leq\alpha\cdot |a|~\big\}.
	$$
	Then for every $\alpha\geq1/(\sqrt{2}-1)$, the Poincar\'e map $P$ satisfies
	$$
	DP\big(\cC_{\alpha}(x,y)\big)\subseteq\cC_{\alpha}(P(x,y)),
	\qquad \forall (x,y)\in\Sigma\setminus l.
	$$
\end{lemma}

\begin{proof}
	Since $(x,y)\rightarrow W^s_{\it loc}(\sigma)$ as $x\rightarrow0$, the orbit of $(x,y)$ approaches the singularity $\sigma$. As we assumed $\lambda_1<\lambda_2<0$ and $\lambda_2+\lambda_3>0$, thus $\lim\limits_{x\rightarrow0}f'(x)=+\infty$ and $\lim\limits_{x\rightarrow0}\big|\partial H(x,y)/\partial y\big|=0$. Moreover, for every $\alpha\geq1/(\sqrt{2}-1)$ and $v=\partial/\partial x+\alpha\cdot\partial/\partial y\in\cC_{\alpha}$, if we denote  $DP(v)=a\cdot\partial/\partial x+b\cdot\partial/\partial y$, then we have $|a|>\sqrt{2}$ and $|b|\leq 1+\alpha\leq \alpha\cdot\sqrt{2}$. This implies $DP(v)\in\cC_{\alpha}$.
\end{proof}

\begin{remark}\label{rk:H}
	For every point $z\in\Sigma\cap\Lambda$, the cone field $\cC_{\alpha}(z)\oplus\left\langle X(z)\right\rangle$ is a neighborhood of center-unstable bundle $E^{cu}(z)$ of $z$.

	In Definition \ref{def:Lorenz}, the property $\sup_{(x,y)\in\Sigma\setminus l}\big|\partial H(x,y)/\partial x\big|<1$ is not necessary. We only need there exists some constant $K>0$, such that $\sup_{(x,y)\in\Sigma\setminus l}\big|\partial H(x,y)/\partial x\big|<K$, which is automatically holds because $\Lambda$ is singular hyperbolic and $H$ is uniformly $C^1$-smooth. Then we can take $\alpha\geq K/(\sqrt{2}-1)$ and the cone field $\cC_{\alpha}$ is $DP$-invariant.
\end{remark}

It has been showed in~\cite{c1lorenz} that, following Definition~\ref{def:Lorenz}, every geometric Lorenz attractor $\Lambda$ is robustly transitive.
In Section \ref{Section:Cr-Lorenz}, we will introduce a series of properties of geometric Lorenz attractors. In particular, the $C^r$-vector fields exhibiting a geometric Lorenz attractor consists an open subset in $\mathscr{X}^r(M^3)$ for every $r\geq2$ (Proposition \ref{prop:Lorenz}).

\subsection{$C^1$-connecting lemmas}\label{Section:connecting lemmas}
In this subsection, we state some $C^1$-perturbation techniques, mainly the connecting lemma~\cite{hayashi,wen-xia-connecting} by S. Hayashi and L. Wen-Z. Xia, and the chain-connecting lemma~\cite{bc} by C. Bonatti-S. Crovisier.

\begin{theorem}[\cite{hayashi,wen-xia-connecting}]\label{Thm:connecting lemma}
	Assume $X\in\mathscr{X}^1(M)$. For any neighborhood $\mathcal{U}$ of $X$, there exists $L>0$ such that for any neighborhood $U$ of a non-critical point $z$, there exists a neighborhood $V\subset U$ of $z$ satisfying the following property:\\
	For any two points $p,q\notin \bigcup_{t\in[0,L]}\phi^X_{t}(U)$, if there exit $m,n>0$ satisfying $\phi^X_{m}(p)\in V$ and $\phi^X_{-n}(q)\in V$, then there exist $Y\in\mathcal{U}$ and $t>0$ such that $q=\phi_t^Y(p)$ and $Y$ coincides with $X$ on $M\setminus \bigcup_{t\in[0,L]}\phi^X_{t}(U)$.
\end{theorem}

\begin{remark}\label{Rem:uniform connecting}
	The constant $L$ and the neighborhood $V$ of $z$ are valid for vector fields in a neighborhood $\mathcal{U}_1\subset\mathcal{U}$ of $X$, see~\cite{wen-uniform-connecting}.
\end{remark}

In certain cases, we have to perturb a vector field in different open regions. The results in~\cite[Section 2]{PR} guarantee that every vector field $X\in\mathscr{X}^r(M)$ ($r\in\mathbb{N}\cup\{\infty\}$) admits a basis of neighborhoods $\mathcal{U}$ satisfying the following property:
\begin{itemize}
\item[$(F)$]: {\it For any two perturbations $Y_1,Y_2\in \mathcal{U}$ of $X$, if there exists two open sets $W_1,W_2$ such that $Y_i=X|_{M\setminus W_i}$ for $i=1,2$ and $W_1\cap W_2=\emptyset$, then the composed perturbation $Y$ is also in $\mathcal{U}$, where $Y=X|_{M\setminus {W_1\cup W_2}}$ and $Y=Y_i|_{W_i}$ for $i=1,2$.}
\end{itemize}
\medskip

Conley theory decomposes the dynamics into chain recurrence classes~\cite{conley}.
Given a vector field $X\in\mathscr{X}^1(M)$, a point $y$ is {\it chain attainable} from $x$, if for any $\varepsilon>0$, there exist points $x_0=x,x_1,x_2,\cdots,x_n=y$ and constants $t_0,t_2,\cdots,t_{n-1}\geq 1$ where $n\geq 1$ such that $d(\phi_{t_i}(x_i),x_{i+1})<\varepsilon$, for any $0\leq i\leq n-1$.
The collection of pairs $\{x_i,t_i\}_{0\leq i\leq n-1}$ is called an $\varepsilon$-chain from $x$ to $y$.
Given a compact set $K$, we say $y$ is {\it chain attainable from $x$ inside $K$} if the chains from $x$ to $y$ can be chosen inside $K$.
An invariant compact set $\Lambda$ is {\it chain transitive} if $y$ is chain attainable from $x$ inside $\Lambda$  for any $x,y\in\Lambda$. A maximal chain transitive set is called a {\it chain recurrence class}, that is to say it is not a proper subset of any chain transitive set. For any point $x$ that is chain attainable from itself, its chain recurrence class is denoted by $C(x,X)$ or $C(x)$ for simplicity if there is no confusion.

The following is the famous chain-connecting lemma proved by Bonatti-Crovisier~\cite{bc}.
\begin{theorem}[\cite{bc}]\label{Thm:chain-connecting}

	Given $X\in\mathscr{X}^1(M)$ and a compact set $K\subset M$ such that all critical elements in $K$ are hyperbolic. Assume $y$ is chain attainable from $x$ inside $K$. Then for any neighborhood $\mathcal{U}\subset\mathscr{X}^1(M)$ and any neighborhood $U$ of $K$, there exists $Y\in\mathcal{U}$ and $t>0$ such that $y=\phi_t^Y(x)$ and $Y$ coincides with $X$ on $M\setminus U$.

\end{theorem}

\subsection{Linear Poincar\'e flow}

Given $X\in\mathscr{X}^1(M)$ and $x\in M\setminus \sing(X)$. Denote by $\mathcal{N}_x$ the  orthogonal complement space of $\langle X(x)\rangle$, i.e.
$$\mathcal{N}_x=\{v\in T_xM|v\perp X(x)\}.$$
Denote $\mathcal{N}=\bigcup_{x\in M\setminus \sing(X)}\mathcal{N}_x$. We define the {\it linear Poincar\'e flow} $\psi_t\colon \mathcal{N}\rightarrow \mathcal{N}$ and the {\it rescaled linear Poincar\'e flow} $\psi^*_t\colon \mathcal{N}\rightarrow \mathcal{N}$ as follows: for any $x\in M\setminus \sing(X)$ and any $v\in \mathcal{N}_x$,
$$\psi_t(v)=\Phi_t(v)-\frac{\langle\Phi_t(v),X(\phi_t(x))\rangle}{\|X(\phi_t(x))\|^2}X(\phi_t(x)).$$

$$\psi^*_t(v)=\frac{\|X(x)\|}{\|X(\phi_t(x))\|}\psi_t(v)=\frac{1}{\|\Phi_t|_{\langle X(x)\rangle}\|}\psi_t(v).$$

\begin{lemma}\label{Lem:domination for measure}

	Let $X\in\mathscr{X}^1(M)$ and $\Lambda$ be a transitive compact invariant set. Assume that $\Lambda$ is singular hyperbolic. Then for any ergodic measure $\mu\in\mathcal{M}_{erg}(\Lambda)$ which is not any atomic supported on a singularity, the rescaled linear Poincar\'e flow admits a dominated splitting $\mathcal{N}_{\supp(\mu)\setminus\sing(\Lambda)}=\mathcal{E}\oplus \mathcal{F}$ with $\dim(\cal E)=\ind(\mu)$. Moreover, there exist two constants $\eta, T>0$ such that
	$$\int\log\|\psi^*_T|_{\mathcal{E}_x}\|d\mu(x)<-\eta ~~~\text{ and }~~~\int\log m(\psi^*_{T}|_{\mathcal{F}_x})d\mu(x)>\eta.$$
\end{lemma}
\begin{remark*}
	A similar result for star vector fields can be found in~\cite[Lemma 2.6]{lsww} and we only give a sketch here.
\end{remark*}
\begin{proof}[Sketch of proof]
	Let $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ be  the singular hyperbolic splitting.
	Note that for any $\mu\in\mathcal{M}_{erg}(\Lambda)$ which is not the atomic measure supported on any singularity, we have that $\ind(\mu)=\dim(E^{ss})$.

	By~\cite[Lemma 3.4]{bgy}, either $X(x)\in E^{ss}(x)$ for any $x\in\Lambda\setminus\sing(\Lambda)$ or $X(x)\in E^{cu}(x)$ for any $x\in\Lambda\setminus\sing(\Lambda)$. Since $E^{ss}$ is uniformly contracted, we have that $X(x)\in E^{cu}(x)$ for any $x\in\Lambda\setminus\sing(\Lambda)$. By the definition of (scaled) linear Poincar\'e flow, there exists a dominated splitting $\mathcal{N}_{\Lambda\setminus\sing(\Lambda)}=\mathcal{E}\oplus \mathcal{F}$ with $\dim(E)=\dim(E^{ss})$ with respect to $\psi^*_t$~\cite[Lemma 2.3]{bgy}. Applying Mane's Ergodic Closing Lemma~\cite{mane-ergodic closing} and by the definition of singular hyperbolicity, the two inequalities of estimations follow the arguments in~\cite[Theorem 5.6]{sgw}.
\end{proof}

\begin{definition}\label{Def:quasi-hyperbolic segment}
    Given $X\in\mathscr{X}^1(M)$ and two constants $\eta,T_0>0$. For $x\notin\sing(X)$ and $T>0$, the orbit segment $\phi_{[0,T]}(x)$ is called \emph{$(\eta,T_0)^*$-quasi hyperbolic} with respect to a direct sum $\mathcal{N}_{x}=\mathcal{E}_x\oplus\mathcal{F}_x$, if there exists a sequence of constants
    $$0=t_0<t_1<\cdots<t_n=T,~~~\text{ with } t_{k+1}-t_k\in [T_0,2T_0],$$
    such that for any $k=0,1,2,\cdots,n-1$
    $$\prod_{i=0}^{k}\|\psi^*_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(\mathcal{E}_x)}\|\leq e^{-\eta t_{k+1}},$$
    $$\prod_{i=k}^{n-1} m(\psi^*_{t_{i+1}-t_{i}}|_{\psi_{t_{i}}(\mathcal{F}_x)})\geq e^{\eta (t_n-t_{k})},$$
    $$\frac{\|\psi^*_{t_{k+1}-t_k}|_{\psi_{t_k}(\mathcal{E}_x)}\|}{m(\psi^*_{t_{k+1}-t_k}|_{\psi_{t_k}(\mathcal{F}_x)})}\leq e^{-\eta (t_{k+1}-t_k)}.$$
\end{definition}

The following is a shadowing lemma proved by S. Liao~\cite{liao-shadowing1,liao-shadowing2}. A detailed proof can be found in~\cite[Theorem A.6]{yang}.
\begin{theorem}\label{Thm:Liao-shadowing}
	Given $X\in\mathscr{X}^1(M)$, two constants $\eta,T_0>0$ and a compact set $\Lambda\subset M\setminus \sing(X)$. For any $\varepsilon>0$, there exists $\delta>0$ such that for any $(\eta,T_0)^*$-quasi hyperbolic orbit segment $\phi_{[0,T]}(x)$ with respect to a direct sum $\mathcal{N}_{x}=\mathcal{E}_x\oplus\mathcal{F}_x$ contained in $\Lambda$, if $d(\mathcal{E}_{x},\psi_T(\mathcal{E}_x))<\delta$, then there exist a periodic point $p$ of $X$ and a strictly increasing function $\theta\colon [0,T]\rightarrow \mathbb{R}$ satisfying that
	\begin{itemize}
		\item $\theta(0)=0$ and $|\theta'(t)-1|<\varepsilon$ for any $t\in[0,T]$;
		\item $\phi_{\theta(T)}(p)=p$ and $d(\phi_t(x),\phi_{\theta(t)}(p))<\varepsilon |X(\phi_t(x))|$ for any $t\in[0,T]$.
	\end{itemize}
\end{theorem}

\begin{definition}\label{Def:Pliss}
	Let $X\in\mathscr{X}^1(M)$. Assume $\Lambda$ is an invariant compact set such that the rescaled linear Poincar\'e flow $\psi^*_t$ admits a dominated splitting $\mathcal{N}_{\Lambda\setminus\sing(\Lambda)}=\mathcal{E}\oplus \mathcal{F}$. Given $\eta>0,~T>0$.
	A point $x\in\Lambda\setminus\sing(\Lambda)$ is called an {\it $(\eta,T)$-$\mathcal{E}$-Pliss point}, if
	$$\prod_{k=0}^{n-1}\|\psi^*_T|_{\mathcal{E}_{\phi_{kT}(x)}}\|\leq e^{-n\eta T},~~~\text{for any n$\in\mathbb{N}$}.$$
	A point $x\in\Lambda\setminus\sing(\Lambda)$ is called an {\it $(\eta,T)$-$\mathcal{F}$-Pliss point}, if
	$$\prod_{k=0}^{n-1}\|\psi^*_{-T}|_{\mathcal{F}_{\phi_{-kT}(x)}}\|\leq e^{-n\eta T},~~~\text{for any n$\in\mathbb{N}$}.$$
	A point $x\in\Lambda\setminus\sing(\Lambda)$ is called an {\it $(\eta,T)$-bi-Pliss point}, if $x$ is both an $(\eta,T)$-$\mathcal{E}$-Pliss point and an $(\eta,T)$-$\mathcal{F}$-Pliss point.
\end{definition}

Next theorem is proved by S. Liao~\cite{liao-shadowing2}, see also~\cite[Theorem 5.4]{sgw}.
\begin{theorem}\label{Thm:uniform inv-manifold}
	Given $X\in\mathscr{X}^1(M)$ and $\eta>0,~T>0$. Assume $\Lambda$ is an invariant compact set of $X$ such that the rescaled linear Poincar\'e flow $\psi^*_t$ admits a dominated splitting $\mathcal{N}_{\Lambda\setminus\sing(\Lambda)}=\mathcal{E}\oplus \mathcal{F}$.
	Then for any $\varepsilon>0$, there exist $\delta>0$ such that for any $(\eta,T)$-$\mathcal{E}$-Pliss point $x\in\Lambda$, there exists a $C^1$ map $\kappa\colon \mathcal{E}_x(\delta|X(x)|)\rightarrow \mathcal{N}_x$, satisfying that
	\begin{itemize}
		\item $d_{C^1}(\kappa,id)<\varepsilon$,
		\item $\kappa(0)=0$,
		\item $W^{cs}_{\delta|X(x)|}\subset W^s(\orb(x))$ where $W^{cs}_{\delta|X(x)|}=\exp_x(\emph{Image}(\kappa))$.
	\end{itemize}
\end{theorem}

\subsection{Generic properties}
A vector field $X\in\mathscr{X}^1(M)$ is {\it weak Kupka-Smale} if all critical elements of $X$ are hyperbolic.  $X$ is {\it Kupka-Smale} if it is weak Kupka-Smale and the intersection between the stable and unstable manifolds of any two critical elements are transverse. We state the following $C^1$-generic properties for vector fields. Some of them can be found in the book~\cite{bdv}.
\begin{theorem}\label{Thm:generic}
	There exists a residual subset  $\mathcal{R}_0\subset\mathscr{X}^1(M)$, such that every $X\in\mathcal{R}_0$ satisfies the following properties.
	\begin{enumerate}
		\item\label{generic:kupka-smale} $X$ is Kupka-Smale. ~\cite{kupka,Smale}
	    \item\label{generic:chain class} Every chain recurrence class $C$ of $X$ containing a periodic point $p$ coincides with the homoclinic class $H(p)$ and every periodic point $q\in C$ with the same index as $p$ is homoclinically related with $p$. ~\cite{bc}
	    \item\label{generic:lorenz-like is h-class} Every singular hyperbolic Lyapunov stable chain recurrence class is an attractor and contains periodic orbits.~\cite{PYY}
	    \item\label{generic:convex of periodic measrure} For any hyperbolic periodic point $p$,  the closure (in the weak*-topology) of the set of periodic measures supported on $H(p)$ is convex. ~\cite{abc}

	    \item\label{generic:robustly Lyapunov stable} For any hyperbolic critical element $\gamma$ of $X$, if $C(\gamma)$ is Lyapunov stable, then there exist a neighborhood $\mathcal{U}\subset\mathscr{X}^1(M)$ of $X$ such that for any weak Kupka-Smale vector field $Y\in\mathcal{U}$, the chain recurrence class $C(\gamma_Y,Y)$ is Lyapunov stable.~\cite[Lemma 3.15]{GY}
	\end{enumerate}
\end{theorem}

\begin{remark}\label{Rem:robustly Lyapunov stable}
	Note that singular hyperbolicity is a robust property and any periodic orbit of a singular hyperbolic set is hyperbolic. Hence in item~\ref{generic:robustly Lyapunov stable} of Theorem~\ref{Thm:generic}, if $C(\gamma)$ is singular hyperbolic, then $C(\gamma_Y,Y)$ is singular hyperbolic and Lyapunov stable for any $Y\in\mathcal{U}$.
\end{remark}

\section{The space of ergodic measures}\label{Section:measure}
In this section, we obtain several properties of the space of invariant measures.

\subsection{Non-singular ergodic measures}

The following proposition states that any ergodic measure supported on a singular hyperbolic homoclinic class can be accumulated by periodic measures if it is not supported on singularities.
The proof essentially follows the arguments of~\cite[Theorem 5.6]{sgw} and we give an explanation here for completeness.
\begin{proposition}\label{Prop:dense periodic measure}
	Let $X\in\mathscr{X}^1(M)$ and $\Lambda$ be a singular hyperbolic homoclinic class. Assume $\mu\in\mathcal{M}_{erg}(\Lambda)$ is not an atomic measure supported on any singularity. Then $\mu\in\overline{\mathcal{M}_{per}(\Lambda)}$.
\end{proposition}

\begin{proof}
	Let $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ be the singular hyperbolic splitting. Then $\mu$ is a hyperbolic measure with $\ind(\mu)=\dim(E^{ss})$. By Lemma~\ref{Lem:domination for measure}, there exist a $\psi_t^*$-dominated splitting $\mathcal{N}_{\supp(\mu)\setminus\sing(\Lambda)}=\mathcal{E}\oplus \mathcal{F}$ with $\dim(\mathcal{E})=\ind(\mu)=\dim(E^{ss})$ and two constants $\eta, T>0$ such that
	$$\int\log\|\psi^*_T|_{\mathcal{E}_x}\|d\mu(x)\leq-\eta ~~~\text{ and }~~~\int\log\|\psi^*_{-T}|_{\mathcal{F}_x}\|d\mu(x)\leq-\eta.$$
	By Birkhoff Ergodic Theorem, for $\mu$-almost every $x\in M$, we have
	$$\lim_{n\rightarrow}\frac{1}{n}\sum_{i=0}^{n-1}\log \|\psi^*_T|_{\mathcal{E}_{\phi_{iT}(x)}}\|\leq -\eta~~~\text{ and }~~~\lim_{n\rightarrow}\frac{1}{n}\sum_{i=0}^{n-1}\log m(\psi^*_{T}|_{\mathcal{F}_{\phi_{-iT}(x)}})\geq \eta.$$

	Now we follow Katok's arguments~\cite{katok}. As in Pesin theory, denote by $\Lambda_C$ the set of points $x$ in $\supp(\mu)\cap B(\mu,T)$ that satisfies:
	$$\prod_{i=0}^{n-1} \|\psi^*_T|_{\mathcal{E}_{\phi_{iT}(x)}}\|\leq Ce^{-n\eta}~~~\text{ and }~~~\prod_{i=0}^{n-1} m(\psi^*_{T}|_{\mathcal{F}_{\phi_{iT}(x)}})\geq C^{-1}e^{n\eta},~~\text{ for any $n\in\mathbb{N}$,}$$
	where $B(\mu,T)=\{x\in M:\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{\phi_{iT}(x)}\rightarrow \mu\}$. Then $\mu(\Lambda_C)\rightarrow 1$ as $C\rightarrow+\infty$. Take $C$ large enough such that $\mu(\Lambda_C)>0$.   Since $\mu$ is not an atomic measure supported on any singularity, we have that $\mu(\sing(\Lambda))=0$. Thus there exists $\alpha>0$ such that $\mu(\Lambda_{C,\alpha})>0$ where $\Lambda_{C,\alpha}=\Lambda_C\setminus B(\sing(\Lambda),\alpha)$ and $B(\sing(\Lambda),\alpha)$ is the $\alpha$-neighborhood of $\sing(\Lambda)$. Note that $\Lambda_{C,\alpha}$ is a compact set. By Poincar\'e Recurrence Theorem, for $\mu$-almost every $x\in\Lambda_{C,\alpha}$, there exist a sequence of points $\{x_n\}$ and an increasing sequence of integers $\{k_n\}$ satisfying the following properties.
	\begin{itemize}
		\item $x_n,\phi_{k_nT}(x_n)\in\Lambda_{C,\alpha}$ with $k_n\rightarrow+\infty$;
		\item $\lim\limits_{n\rightarrow\infty}x_n=\lim\limits_{n\rightarrow\infty}\phi_{k_nT}(x_n)=x$;
		\item $\lim\limits_{n\rightarrow\infty}\frac{1}{k_n}\sum\limits_{i=0}^{k_n}\delta_{\phi_{iT}(x_n)}\rightarrow\mu$.
	\end{itemize}
    By definition of $\Lambda_{C,\alpha}$, for any $n\in\mathbb{N}$ and any $0\leq m\leq k_n-1$, we have that
    $$\prod_{i=0}^{m} \|\psi^*_T|_{\mathcal{E}_{\phi_{iT}(x_n)}}\|\leq Ce^{-(m+1)\eta}~~~\text{ and }~~~\prod_{i=m}^{k_n-1} m(\psi^*_{T}|_{\mathcal{F}_{\phi_{iT}(x)}})\geq C^{-1}e^{(k_n-m)\eta}.$$

    Note that $k_n\rightarrow+\infty$, hence  replacing $T$ by $T_0=m_0 T$ with $m_0$ large enough if necessary, then the orbit segment $\phi_{[0,k_nT]}(x_n)$ is $(\eta,T_0)$-quasi hyperbolic for every $n$ large. By Theorem~\ref{Thm:Liao-shadowing}, there exists a sequence of periodic points $p_n$ which converges to $\mu$ and the periodic measure $\delta_{\orb(p_n)}$ associated to $\orb(p_n)$ converges to $\mu$ in the weak*-topology. Moreover, the shadowing property guarantees that $p_n$ is a $(\frac{\eta}{2},T_0)$-bi-Pliss point for every $n$ large enough. Then by Theorem~\ref{Thm:uniform inv-manifold}, the  local stable and unstable manifolds of $p_n$ has uniform size and thus are homoclinically related with each other for $n$ large enough. Recall that $z\in\Lambda$ and $\Lambda$ is a homoclinic class, hence $p_n\in\Lambda$ for $n$ large. This implies $\mu\in\overline{\mathcal{M}_{per}(\Lambda)}$.
\end{proof}

\subsection{Path connectedness}
The following proposition gives a general criterion for path connectedness (or arcwise connectedness) of ergodic measures.  A related conclusion for diffeomorphisms can be found in~\cite{Go-Pe}.
\begin{proposition}\label{Prop:connectedness}
	Let $X\in\mathscr{X}^1(M)$ and $\Lambda$ be a  homoclinic class of $X$ such that all periodic points in $\Lambda$  are hyperbolic and any two periodic points $p,q$ in $\Lambda$ are homoclinically related.

	Then the set $\overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$ is path connected.
\end{proposition}

\begin{proof}

	We first prove the following two claims which are the flow version of~\cite[Lemma 3.1, 3.2]{Go-Pe}. For a periodic point $p$, we denote by $\delta_{\orb(p)}$ the periodic measures corresponding to $\orb(p)$.

	\begin{claim}\label{Claim:connect per-measure}
		For any periodic points $p,q\in\Lambda$, there exits a continuous path $\{\nu_t\}_{t\in[0,1]}\subset \overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$ such that $\nu_0=\delta_{\orb(p)}$ and $\nu_1=\delta_{\orb(q)}$.
	\end{claim}
	\begin{proof}
		By assumption, the two orbits $\orb(p)$ and $\orb(q)$ are homoclinically related. Therefore, there exists a hyperbolic horseshoe $K\subset \Lambda$ of $X$ containing $p$ and $q$. Note here that the hypberolic horseshoe $K$ of $X$ is the suspension of a hyperbolic horseshoe $\tilde{K}$ for a diffeomorphism (the time one map of the flow) with a continuous roof function. By~\cite[Theorem B]{Sigmund}, the space  $\mathcal{M}_{erg}(\tilde{K})$ is path connected. On the other hand, there is a one-to-one correspondence between $\mathcal{M}_{erg}(\tilde{K})$ and $\mathcal{M}_{erg}(K)$, see for instance~\cite[Chapter 6]{parry-pollicott}. Hence $\mathcal{M}_{erg}(K)$ is path connected. Moreover, $\mathcal{M}_{per}(K)$ is dense in $\mathcal{M}_{erg}(K)$ since $K$ is a hyperbolic horseshoe. Thus $\delta_{\orb(p)}$ and $\delta_{\orb(q)}$ can be connected by a path contained in $\mathcal{M}_{erg}(K)\subset\overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$.
	\end{proof}

	\begin{claim}\label{Claim:short path}
		For any $\mu\in\mathcal{M}_{erg}(\Lambda)$ and $\varepsilon>0$, there exists $\eta>0$ such that for any periodic points $p,q\in\Lambda$, if $d(\mu,\delta_{\orb(p)})<\eta$ and $d(\mu,\delta_{\orb(q)})<\eta$, then there exists a path $\{\nu_t\}_{t\in[0,1]}\subset \overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$ such that $\nu_0=\delta_{\orb(p)}$, $\nu_1=\delta_{\orb(q)}$ and $d(\mu,\nu_t)<\varepsilon$ for any $t\in[0,1]$.
	\end{claim}
	\begin{proof}
		Since $\orb(p)$ and $\orb(q)$ are homoclinically related, there exist  hyperbolic horseshoes such that every point in which spends arbitrarily large fractions of its orbit shadowing $\orb(p)$ and $\orb(q)$ as closely as we want.

		Thus we can choose a hyperbolic horseshoe $K$ containing $p$ and $q$ to be $2\eta$-close to $\mu$ in the measure sense: any ergodic measure supported on $K$ is contained in the $2\eta$-neighborhood of $\mu$.
		Then Claim~\ref{Claim:short path} concludes if we take $\eta=\varepsilon/2$.
	\end{proof}

	Take two ergodic measures $\mu,\nu\in\overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$. We only need to show there exists a path $\{\mu_t\}_{t\in[0,1]}\subset \overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$ such that $\mu_0=\mu$ and $\mu_1=\nu$. We follow the idea of~\cite[Theorem 1]{Go-Pe} and just give an explanation here.
	Note that there exist two sequences of periodic points $\{p_n\}$ and $\{q_n\}$ in $\Lambda$ such that $\delta_{\orb(p_n)}\rightarrow\mu$ and  $\delta_{\orb(q_n)}\rightarrow\nu$ in the weak*-topology. Similarly with the proof of~\cite[Theorem 1.1]{Go-Pe}, by Claim~\ref{Claim:connect per-measure}, there exits a continuous path $\{\nu_t\}_{t\in[\frac{1}{3},\frac{2}{3}]}\subset \overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$ such that $\nu_{\frac{1}{3}}=\delta_{\orb(p_1)}$ and $\nu_{\frac{2}{3}}=\delta_{\orb(q_1)}$. By Claim~\ref{Claim:short path}, for any $n\geq 1$ there exist two paths
	$$\{\nu_t\}_{t\in[\frac{1}{3^n},\frac{1}{3^{n+1}}]} \text{ and } \{\nu_t\}_{t\in[1-\frac{1}{3^n},1-\frac{1}{3^{n+1}}]} $$
	in $\overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$  connecting $\delta_{\orb(p_n)},\delta_{\orb(p_{n+1})}$ and $\delta_{\orb(q_n)},\delta_{\orb(q_{n+1})}$ respectively and the length of the paths decreases to 0 exponentially when $n$ goes to $\infty$. Therefore  $\{\nu_t\}_{t\in[0,1]}$ is a continuous path in $\overline{\mathcal{M}_{per}(\Lambda)}\cap\mathcal{M}_{erg}(\Lambda)$ satisfying that $\nu_0=\mu$ and $\nu_1=\nu$.
\end{proof}

\subsection{Homoclinic loop and periodic measures}
In this subsection, we prove that a homoclinic loop of a singularity can be accumulated by periodic orbits with the associated periodic measures accumulating to the singular measure by $C^r$-perturbations.

\begin{proposition}\label{Prop:loop}
	Let $r\in\mathbb{N}\cup\{\infty\}$ and $X\in\mathscr{X}^r(M)$ be a $C^r$-vector field on $M$. If $\sigma\in\sing(X)$ is a hyperbolic singularity of $X$, and
	$$
	\Gamma=\orb(z,\phi_t^X)\subset W^s(\sigma,\phi_t^X)\cap W^u(\sigma,\phi_t^X)
	$$
	is a homoclinic orbit of $\sigma$, then there exist a sequence of vector fields $\{X_n\}$ and a sequence of periodic points $p_n\in\per(X_n)$, such that
	\begin{enumerate}
		\item The vector field $X_n$ converges to $X$ in $\mathscr{X}^r(M)$.
		\item $\sigma\in\sing(X_n)$ and the periodic measure
		$\delta_{\orb(p_n,\phi_t^{X_n})}$ converges to $\delta_{\sigma}$.
		\item The orbit $\orb(p_n,\phi_t^{X_n})$ converges to $\Gamma\cup\{\sigma\}$ in the Hausdorff topology.
	\end{enumerate}
\end{proposition}

\begin{proof}
	Let $d=\dim M$ and $s=\dim E^s(\sigma)$ be the index of $\sigma$, then $\dim E^u(\sigma)=d-s$. Since $\Gamma=\orb(z,\phi_t^X)$ is a homoclinic loop of $X$, we have
	$$
	\lim_{t\rightarrow\pm\infty}\phi_t^X(z)=\sigma.
	$$
	By Grobman-Hartman Theorem \cite[4.10 Theorem]{deM-Palis-book}, there exists a small neighborhood $U(\sigma)$ of $\sigma$, such that the flow $\phi_t^X$ is orbit equivalent to the linear system generated by $DX(\sigma)$. Moreover, shrinking $U(\sigma)$ if necessary, we can assume
	there exist $T_1<0<T_2$ satisfying
	$$
	U(\sigma)\cap\Gamma~=~
	\phi_{(-\infty,T_1)}^X(z)~\cup~\phi_{(T_2,+\infty)}^X(z),
	$$
	where $\phi_{T_1}^X(z)\in W^u_{\it loc}(\sigma,\phi_t^X)$ and $\phi_{T_2}^X(z)\in W^s_{\it loc}(\sigma,\phi_t^X)$.

	There exists a $(d-1)$-dimensional $C^\infty$-smooth cross section $\Sigma_1$ which is transverse to $X$ everywhere and satisfies the following properties:
	\begin{itemize}
		\item $\phi_{T_1-1}^X(z)\in\Sigma_1$, and $\phi_{[0,1]}^X(\Sigma_1)\cap\Gamma=\phi_{[T_1-1,T_1]}^X(z)$.
		\item There exists a $s$-dimensional $C^\infty$-smooth disk $\Sigma_1^s\subset\Sigma_1$ such that $\Sigma_1^s$ is transverse to $W^u_{\it loc}(\sigma)$ at $\phi_{T_1-1}^X(z)$.
	\end{itemize}
	Symmetrically, There exists a $(d-1)$-dimensional $C^\infty$-smooth cross section $\Sigma_2$ which is transverse to $X$ everywhere and satisfies the following properties:
	\begin{itemize}
		\item $\phi_{T_2+1}^X(z)\in\Sigma_2$, and  $\phi_{[-1,0]}^X(\Sigma_2)\cap\Gamma=\phi_{[T_2,T_2+1]}^X(z)$;
		\item there exists a $(d-s)$-dimensional $C^\infty$-smooth disk $\Sigma_2^u\subset\Sigma_2$ such that $\Sigma_2^u$ is transverse to $W^s_{\it loc}(\sigma)$ at $\phi_{T_2+1}^X(z)$.
	\end{itemize}

	By the $\lambda$-lemma (Inclination Lemma), see \cite[7.2 Lemma]{deM-Palis-book}, there exist a sequence of points $x_n\in\Sigma_2^u\subseteq\Sigma_2$ and $S_n\rightarrow+\infty$, such that
	\begin{enumerate}
		\item The point $x_n\rightarrow\phi_{T_2+1}^X(z)$ in $\Sigma_2^u$ as $n\rightarrow\infty$.
		\item The point $y_n=\phi_{S_n}^X(z)\in\Sigma_1^s\subseteq\Sigma_1$, and $y_n\rightarrow\phi_{T_1-1}^X(z)$ in $\Sigma_1^s$ as $n\rightarrow\infty$.
		\item The orbit segment $\phi_{[0,S_n]}^X(x_n)\subset U(\sigma)$ for every $n$.
	\end{enumerate}

	Now we consider two flow box $V_1=\phi_{[T_1-1,T_1]}^X(\Sigma_1)$ and $V_2=\phi_{[T_2,T_2+1]}^X(\Sigma_2)$. Since $x_n\rightarrow\phi_{T_2+1}^X(z)$ in $\Sigma_2^u$ and $y_n\rightarrow\phi_{T_1-1}^X(z)$ in $\Sigma_1^s$, there exists a sequence of vector fields $X_n\in\mathscr{X}^r(M)$, such that
	\begin{enumerate}
		\item The vector field $X_n$ converges to $X$ in $C^r$-topology.
		\item $X_n|_{M\setminus(V_1\cup V_2)}\equiv X|_{M\setminus(V_1\cup V_2)}$.
		\item There exist $a_n,b_n\rightarrow 1$ as $n\rightarrow\infty$, such that
		$$
		\phi_{a_n}^{X_n}(y_n)=\phi_{T_1}^X(z),
		\qquad {\rm and} \qquad
		\phi_{b^n}^{X_n}\big( \phi_{T_2}^X(z) \big)=x_n.
		$$
	\end{enumerate}
	This implies the vector field $X_n$ satisfies
	$\phi_{b_n+S_n+a_n}^{X_n}\big(\phi_{T_2}^X(z)\big)=\phi_{T_1}^X(z)$.
	So $\Gamma_n=\orb(z,\phi_t^{X_n})$ is a periodic orbit of $X_n$ with period $\pi_n=a_n+b_n+(T_2-T_1)+S_n$. Moreover, we have
	$$
	\phi_{t+b_n}^{X_n}\big( \phi_{T_2}^X(z) \big)=
	\phi_t^{X_n}(x_n)=\phi_t^X(x_n), ~~\forall t\in[0,S_n],
	\quad {\rm and} \quad
	\phi_t^{X_n}(z)=\phi_t^X(z), ~~\forall t\in[T_1,T_2].
	$$

	Recall that $x_n\rightarrow\phi_{T_2+1}^X(z)\in W^s_{\it loc}(\sigma)$ and $y_n\rightarrow\phi_{T_1-1}^X(z)$. Since $(\pi_n-S_n)$ is uniformly bounded above, the fact that
	$$
	\phi_{[0,S_n]}^{X_n}(x_n)=\phi_{[0,S_n]}^X(x_n)\subseteq U(\sigma)
	$$
	and $\phi_t^X$ is orbit equivalent to the linear system generated by $DX(\sigma)$ in $U(\sigma)$ implies $\Gamma_n$ converges to $\Gamma\cup\{\sigma\}$ in the Hausdorff topology and the periodic measure
	$\delta_{\Gamma_n}$ converges to $\delta_{\sigma}$.
\end{proof}

\subsection{Isolated singular measures}

In this subsection, we show that if the unstable manifold of a codimension one singularity is contained in the stable manifold of a periodic orbit, then the Dirac measure of this singularity is isolated in the space of ergodic measures. The idea of this proposition is inspired by the work of J. Palis \cite{palis}, see also \cite{deM-Palis}.

\begin{proposition}\label{Prop:kick-out singularity}
	Let $X\in\mathscr{X}^1(M)$ and $\sigma\in\sing(X)$ be a hyperbolic singularity satisfying $\dim E^u(\sigma)=1$. Assume there exists a hyperbolic periodic point $p\in\Lambda$ such that $W^u(\sigma)\setminus\{\sigma\}\subset W^s(\orb(p))$. Then the Dirac measure $\delta_{\sigma}$ is isolated in $\mathcal{M}_{erg}(X)$.
\end{proposition}

\begin{proof}
	Assume there exists a sequence of ergodic measures $\mu_n$ approximating the Dirac measure $\delta_{\sigma}$. We only need to find a continuous function $\varphi:M\rightarrow\RR$, such that $\int\varphi{\rm d}\mu_n$ does not converge to $\int\varphi{\rm d}\delta_{\sigma}$, then we get a contradiction.

	Since $\sigma$ is a hyperbolic singularity,
	each $\mu_n$ doesn't support on a singularity.
	Let $x_n\in\supp(\mu_n)$ be a generic point of $\mu_n$, i.e. the empirical measure equidistributed on the orbit segment $\phi_{[0,T]}(x_n)$ converges to $\mu_n$ as $T\rightarrow+\infty$. We must have $\ind(p)<d-1$. Otherwise, $\orb(p)$ is a periodic sink. Hence as the positive orbit of $x_n$ approaches $\sigma$, it also approaches $W^u(\sigma)$ and enters the attracting region of $\orb(p)$. This contradicts to $x_n\in\supp(\mu_n)$ which implies that $x_n$ is a recurrent point.

	As we have assumed $\dim E^u(\sigma)=1$, let $\lambda_1,\cdots,\lambda_d$ be eigenvalues of $DX(\sigma)$ which satisfy
	$|{\rm Re}\lambda_1 |\leq \cdots \leq |{\rm Re}\lambda_{d-1}|
	< 1< |{\rm Re}\lambda_d|=|\lambda_d |$.
	We fix a constant $\epsilon_1$ satisfying $0<\epsilon_1<\min\big\{-\log|{\rm Re}\lambda_{d-1}|,\log|\lambda_d|\big\}$, and let
	\begin{align}\label{mu1}
	u_1=\log|{\rm Re}\lambda_{d-1}|+\epsilon_1<0.
	\end{align}

	\begin{claim}\label{clm:sigma-nbhd}
		There exists a neighborhood $U(\sigma)$ of $\sigma$ admitting a $C^1$-coordinate
		$$
		U(\sigma)~=~
		\left\{(x,y)=(x_1,\cdots,x_{d-1},y)\in\RR^{d-1}\times\RR:
		~\|x\|=\big(\sum_{i=0}^{d-1}x_i^2\big)^{1/2}\leq1,
		~|y|\leq1\right\}
		$$
		such that
		\begin{enumerate}
			\item The singularity $\sigma=(0,0)\in\RR^{d-1}\times\RR$, and the local stable and unstable manifolds of $\sigma$ in $U(\sigma)$ are
			$$
			W^s_{\it loc}(\sigma)=\big\{(x,0):~\|x\|\leq1\big\},
			\qquad {\it and} \qquad
			W^u_{\it loc}(\sigma)=\big\{(0,y):~|y|\leq1\big\}.
			$$
			\item For every $z=(x,y)\in U(\sigma)$, let $\phi_t(z)=\big(x(t),y(t)\big)$ be the orbit of $z$  under $\phi_t$ where $\big(x(0),y(0)\big)=(x,y)$. Assume the orbit segment $\phi_{[T_1,T_2]}(z)\subset U(\sigma)$ where $T_1<0<T_2$, then it satisfies $|y'(t)-\lambda_d\cdot y(t)|\leq\epsilon_1\cdot|y(t)|,~~\forall T_1\leq t \leq T_2$,
			$$
			\|x(t)\|\leq\exp(u_1 t)\|x(0)\|, ~~\forall 0\leq t \leq T_2;
			\quad {\it and} \quad
			\|x(t)\|\geq\exp(u_1 t)\|x(0)\|, ~~\forall T_1\leq t\leq0.
			$$
			\item $U(\sigma)\cap\orb(p,\phi_t)=\emptyset$, and for $z^{\pm}=(0,\pm1)\in\RR^{d-1}\times\RR$, they satisfy
			$\phi_t(z^\pm)\notin U(\sigma)$ for every $t>0$.
		\end{enumerate}
	\end{claim}

	\begin{proof}[Proof of Claim~\ref{clm:sigma-nbhd}]
		The first two items have been proved in Lemma 2.1, Theorem 2.4 and Theorem 2.5 of \cite{Shilnikov}. The third item comes from the fact that $z^\pm\in W^u_{\it loc}(\sigma)$ while their orbits $\orb(z^\pm)\subseteq W^s(\orb(p))$. So we only need to take $U(\sigma)$ small enough.
	\end{proof}

	For every $0<\delta<1$, we denote $U(\sigma,\delta)=\big\{(x,y)\in U(\sigma):\|x\|\leq\delta,|y|\leq\delta\big\}$. Let
	$$
	\Sigma^{\pm}=\big\{(x,\pm1)\in U(\sigma):\|x\|\leq1\big\},
	\qquad {\rm and} \qquad
	\Sigma^{\pm}(\delta)=\big\{(x,\pm1)\in \Sigma^{\pm}:\|x\|\leq\delta\big\}.
	$$
	then $\Sigma^{\pm}\subseteq\partial U(\sigma)$. Moreover, the second item of Claim \ref{clm:sigma-nbhd} shows that the vector field $X$ is transverse to $\Sigma^{\pm}$ and points outside $U(\sigma)$. For every $\delta>0$, $\Sigma^\pm$ are local cross sections of $X$ centered at $z^\pm$ respectively.

	\begin{claim}\label{clm:time-estimate-1}
		For every $0<\delta<1$ and $z=(x_z,y_z)\in U(\sigma,\delta)$ with $x_z\neq 0,y_z>0$, it satisfies
		\begin{enumerate}
			\item  There exist $T_1<0<T_2$ such that $\phi_{T_1}(z)\in\partial U(\sigma)\setminus\Sigma^\pm$,
			$\phi_{T_2}(z)\in\Sigma^+$, and $\phi_t(z)\in {\rm Int}U(\sigma)$ for every $t\in(T_1,T_2)$.
			\item There exist $T_1'\in(T_1,0]$ and $T_2'\in[0,T_2)$, such that $\phi_{[T_1,T_2]}(z)\cap U(\sigma,\delta)=\phi_{[T_1',T_2']}(z)$.
			\item If  $(x_w,1)=\big(x(T_2),y(T_2)\big)=\phi_{T_2}(z)\in\Sigma^+$, then we have
			$\|x_w\|<\delta<1$ and
			$$
			T_2'-T_1'~\leq~T_2-T_1~\leq~
			\frac{1}{u_1}\cdot\big[\log\|x_w\|-\log1\big]
			~=~\frac{1}{u_1}\cdot\log\|x_w\|.
			$$
		\end{enumerate}
		The same conclusion holds for $x_z\neq0,y_z<0$ if we replace $\Sigma^+$ by $\Sigma^-$.
	\end{claim}

	\begin{proof}[Proof of Claim~\ref{clm:time-estimate-1}]
		From the second item of Claim \ref{clm:sigma-nbhd}, for $z=(x,y)$ satisfying $x\neq0$ and $y>0$, the flow $\big(x(t),y(t)\big)$ satisfies
		\begin{itemize}
			\item $y(t)>0$ and monotonous increasing with respect to $t$, which implies $\exists T_2>0$ such that $\phi_{T_2}(z)\in\Sigma^+$;
			\item $\|x(t)\|\geq\exp(u_1 t)\|x(0)\|$ for every $t\leq0$, which implies $\exists T_1<0$, such that $\phi_{T_1}(z)\in\partial U(\sigma)\setminus\Sigma^\pm$.
		\end{itemize}
		Moreover, we have $\phi_t(z)\in {\rm Int}U(\sigma)$ for every $t\in(T_1,T_2)$. This proves the first item. The proof of  second item is the same.

		As $\|x(T_1)\|=1$ and $\|x(T_2)\|=\|x_w\|\leq\exp(u_1 T_2)\|x\|<\delta$,
		we apply the fact that $\|x(t)\|\geq\exp(u_1 t)\|x(0)\|$ for every $t\leq0$, which shows
		$$
		T_2'-T_1'~\leq~T_2-T_1~\leq~
		\frac{1}{u_1}\cdot\big(\log\|x_w\|-\log1\big)
		~=~\frac{1}{u_1}\cdot\log\|x_w\|.
		$$
	\end{proof}

	Recall $p$ is a hyperbolic periodic point of $X$ with index $0<\ind(p)<d-1$. We denote $k=\ind(p)$, then we have the following claim.

	\begin{claim}\label{clm:p-nbhd}
		There exist a cross section $\Sigma_p$ transverse to $X$ everywhere admitting a $C^1$ coordinate
		$$
		\Sigma_p=\left\{ (r,s)\in\RR^k\times\RR^{d-1-k}:
		\|r\|\leq1,\|s\|\leq1 \right\},
		$$
		and two constants $\kappa>1,~T_0>0$,
		such that
		\begin{enumerate}
			\item The orbit $\orb(p,\phi_t)\cap\Sigma_p=\{p\}$ and $p=(0,0)\in\RR^k\times\RR^{d-1-k}$.
			\item The local stable and unstable manifolds of the orbit of $p$ intersect $\Sigma_p$ as
			$$
			W^s_{\it loc}(\orb(p))\cap\Sigma_p=
			\big\{ (r,0):\|r\|\leq1 \big\},
			\qquad
			W^u_{\it loc}(\orb(p))\cap\Sigma_p=
			\big\{ (0,s):\|s\|\leq1 \big\}.
			$$
			\item The Poincar\'e return map
			$$
			R:~\Sigma_p'=
			\big\{(r,s)\in\Sigma_p:\|s\|\leq\kappa^{-1}\big\}
			\longrightarrow \Sigma_p
			$$
			is well defined and $C^1$-smooth on $\Sigma_p'$. For every $(r_0,s_0)\in\Sigma_p'$, if we denote $(r_1,s_1)=R(r,s)$, then
			$$
			\|r_1\|<\|r_0\|, \qquad {\rm and} \qquad
			\|s_1\|\leq\kappa\cdot\|s_0\|.
			$$
			\item For every $z\in\Sigma_p'$, let $T_z>0$ be the first return time of $z$ to $\Sigma$, i.e. $R(z)=\phi_{T_z}(z)\in\Sigma$, then $T_z\geq T_0$ and
			$$
			\left( \bigcup_{z\in\Sigma_p'}\phi_{[0,T_z]}(z)  \right)
			~\cap~ U(\sigma) ~=~\emptyset.
			$$
		\end{enumerate}
	\end{claim}

	\begin{proof}[Proof of Claim~\ref{clm:p-nbhd}]
		Let $\pi(p)$ be the period of $p$ and $u_2>0$ be the largest Lyapunov exponent of $\orb(p)$.

		There exists a $C^1$ cross section $\Sigma_1$ containing $p$ and transversing to $X$ everywhere. By shrinking $\Sigma_1$ if necessary, we can assume $\Sigma_1\cap\orb(p,\phi_t)=\{p\}$.

		Since $p$ is a hyperbolic periodic orbit of $X$, there exists a small neighborhood $\Sigma_2\subseteq\Sigma_1$, such that the Poincar\'e map $R:\Sigma_2\rightarrow\Sigma_1$ is well defined and $C^1$-smooth. Moreover, $p$ is a hyperbolic fixed point of $R$ with largest Lyapunov exponent equal to $\big(\pi(p)u_2\big)$. For every $z\in\Sigma_2$, let $T_z$ be the first return time of $z$ in $\Sigma_1$, i.e. $R(z)=\phi_{T_z}(z)$.
		By shrink $\Sigma_2$ if necessary, there exists $T_0>0$ which is close to $\pi(p)$, such that $T_z\geq T_0$ for every $z\in\Sigma_2$. Moreover, since $U(\sigma)\cap\orb(p)=\emptyset$, we can assume
		$$
		\left( \bigcup_{z\in\Sigma_2}\phi_{[0,T_z]}(z)  \right)
		~\cap~ U(\sigma) ~=~\emptyset.
		$$

		Finally, we fix a small constant $\epsilon_2>0$ and take
		\begin{align}\label{kappa}
		\kappa=\exp\big[\pi(p)\cdot(u_2+\epsilon_2)\big].
		\end{align}
		Similar to Claim \ref{clm:sigma-nbhd}, we can take an adapted $C^1$-coordinate on $\Sigma_2$, such that $p=(0,0)\in\RR^k\times\RR^{d-1-k}$ in this coordinate and the second and third items of the claim are satisfied on
		$$
		\Sigma_p=
		\left\{ (r,s)\in\RR^k\times\RR^{d-1-k}:
		\|r\|\leq1,\|s\|\leq1 \right\}
		~\subseteq~\Sigma_2.
		$$
		See \cite[Theorem 2.4 \& 2.5]{Shilnikov}. This finishes the proof of the claim.
	\end{proof}

	Let $V(p)=\bigcup_{z\in\Sigma_p'}\phi_{[0,T_z]}(z)$ which is a neighborhood of $\orb(p)$.
	Since $W^u(\sigma)\setminus\{\sigma\}\subseteq W^s(\orb(p))$, for $z^\pm=(0,\pm1)\in U(\sigma)\cap W^u_{\it loc}(\sigma)$,  there exist $T^\pm>0$ and $(r^\pm,0)\in{\rm Int}\big(\Sigma_p'\big)$ such that
	$$
	(r^+,0)=\phi_{T^+}(z^+),
	\qquad {\rm and} \qquad
	(r^-,0)=\phi_{T^-}(z^-).
	$$
	Since $\orb^+(z^\pm)\cap U(\sigma)=\emptyset$, there exist two constants $\delta_0>0,K>0$ with $K\cdot\delta_0\ll1$, such that
	\begin{itemize}
		\item The Poincar\'e map
		$P^+:\Sigma^+(\delta_0)\rightarrow\Sigma_p$ is well defined and $C^1$-smooth, satisfying $P^+(z^+)=P^+(0,1)=(r^+,0)$ and for every $w=(x_w,1)\in\Sigma^+(\delta_0)$, if we denote $P^+(w)=(r_w,s_w)$, then it satisfies
		$$
		\|r_w-r^+\|\leq K\cdot\|x_w\|,
		\qquad {\rm and} \qquad
		\|s_w\|\leq K\cdot\|x_w\|.
		$$
		Moreover, if we denote $T^+_w>0$  the time satisfying $P^+(w)=\phi_{T^+_w}(w)$, then the orbit segment $\phi_{(0,T^+_w)}(w)\cap U(\sigma)=\emptyset$.

		\item The Poincar\'e map
		$P^-:\Sigma^-(\delta_0)\rightarrow\Sigma_p$ is well defined and $C^1$-smooth, satisfying $P^-(z^-)=P^-(0,-1)=(r^-,0)$ and for every $w=(x_w,-1)\in\Sigma^-(\delta_0)$, if we denote $P^-(w)=(r_w,s_w)$, then it satisfies
		$$
		\|r_w-r^-\|\leq K\cdot\|x_w\|,
		\qquad {\rm and} \qquad
		\|s_w\|\leq K\cdot\|x_w\|.
		$$
		Moreover, if we denote $T^-_w>0$  the time satisfying $P^-(w)=\phi_{T^-_w}(w)$, then the orbit segment $\phi_{(0,T^-_w)}(w)\cap U(\sigma)=\emptyset$.
	\end{itemize}

	\begin{claim}\label{clm:time-estimate-2}
		For every $z=(x_z,y_z)\in U(\sigma,\delta_0)$ with $x_z\neq0,y_z>0$, let $T_1<0<T_2$ be defined as in Claim \ref{clm:time-estimate-1} and $w=(x_w,1)=\phi_{T_2}(x)\in\Sigma^+(\delta_0)$, then for the point $P^+(w)=(r_w,s_w)\in\Sigma_p$, let
		$$
		T_p=\left(-\frac{\log(K\|x_w\|)}{\log\kappa}-2\right)\cdot T_0.
		$$
		we have $\phi_{[0,T_p]}(P^+(w))\subseteq V(p)$.

		In particular, combined with the third item of Claim \ref{clm:time-estimate-1} that $T_2-T_1\leq \log\|x_w\|/u_1$, we have
		\begin{align}\label{equ:time}
		\frac{T_p}{T_2-T_1}~\geq~
		\frac{-u_1\cdot T_0}{\log\kappa}\cdot
		\left(1+\frac{\log K+2\log\kappa}{\log\|x_w\|}\right).
		\end{align}

		The same conclusion holds for $z=(x_z,y_z)\in U(\sigma,\delta_0)$ with $x_z\neq0,y_z<0$.
	\end{claim}

	\begin{proof}[Proof of Claim~\ref{clm:time-estimate-2}]
		This claim is a direct calculation by applying the third item of Claim \ref{clm:p-nbhd} and $\|s_w\|\leq K\cdot\|x_w\|$.
	\end{proof}

	Now we take $\delta_1\in(0,\delta_0]$ small enough, such that
	$(\log K+2\log\kappa)/\log\delta_1>-1/2$.
	Then for every $z=(x_z,y_z)\in U(\sigma,\delta_1)$, the corresponding estimation of Equation (\ref{equ:time}) is
	\begin{align}\label{equ:time-1}
	T_p~\geq~
	L\cdot(T_2-T_1),
	\qquad {\rm where} \qquad
	L=\frac{-u_1\cdot T_0}{2\log\kappa}.
	\end{align}

	Now we take a $C^1$ smooth function $\varphi:M\rightarrow\RR$ satisfying:
	\begin{enumerate}
		\item $\varphi(z)\geq-1$ for every $z\in M$ and $\varphi(\sigma)=-1$.
		\item $\varphi(z)\geq0$ for every $z\in M\setminus U(\sigma,\delta_1)$.
		\item $\varphi(z)\geq L^{-1}$ for every $z\in V(p)$.
	\end{enumerate}
	Then for   the generic point $x_n$ of $\mu_n$, let
	$$
	0~\leq~ T_1^1<T_2^1<T_3^1<T_4^1~<~
	\cdots\cdots
	~<~T_1^m<T_2^m<T_3^m<T_4^m~<~\cdots\cdots
	$$
	be the time sequence where $\{T_1^m\}_m$ are all the times that $\phi_t(x_n)$ enters $U(\sigma,\delta_1)$ and $T_2^m$ is the time when $\phi_t(x_n)$ enters $U(\sigma,\delta_1)$ after $T_1^m$; $T_3^m$ is the time when $\phi_t(x_n)$ enters $V(p)$ after $T_2^m$ and $T_4^m$ is the time  when $\phi_t(x_n)$ leaves $V(p)$ after $T_3^m$.

	For every $m$, we have
	$$
	(T_4^m-T_3^m)~\geq~L\cdot (T_2^m-T_1^m).
	$$
	This implies
	\begin{align*}
	\int\varphi{\rm d}\mu_n~& =~
	\lim_{T\rightarrow+\infty}\frac{1}{T}
	\int_{0}^{T}\varphi\big(\phi_t(x_n)\big){\rm d}t \\
	& \geq~
	\lim_{T_4^m\rightarrow+\infty}
	\frac{1}{T_4^m}\sum_{i=1}^{m}\left[(-1)\cdot(T_2^m-T_1^m)+ L^{-1}\cdot(T_4^m-T_3^m)\right] \\
	& \geq~0.
	\end{align*}
	However, we have $\int\varphi{\rm d}\delta_{\sigma}=\varphi(\sigma)=-1$. This contradicts to $\mu_n\rightarrow\delta_\sigma$, thus $\delta_\sigma$ is isolated in $\mathcal{M}_{erg}(X)$.
\end{proof}

\section{The geometric Lorenz attractors}\label{Section:Cr-Lorenz}

\subsection{Properties of geometric Lorenz attractors}\label{Section:properties-Lorenz}

In this subsection, we investigate some properties for the geometric Lorenz attractor. We assume $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$ and $X\in\cX^r(M^3)$ which exhibits a geometric Lorenz attractor $\Lambda$ with cross section $\Sigma$. Let $P(x,y)=\big(f(x),H(x,y)\big):\Sigma\setminus l\rightarrow\Sigma$ be the Poincar\'e map of $\phi_t^X$.

Recall that $\Lambda$ is a singular hyperbolic attractor with splitting $T_{\Lambda}M^3=E^{ss}\oplus E^{cu}$ and there exists a H\"older continuous foliation $\cF^{ss}$ tangent to $E^{ss}$ everywhere~\cite{psw}.
Since the strong stable bundle $E^{ss}$ is determined by the positive orbit of a point and $\Lambda$ is an attractor, $E^{ss}$ and $\cF^{ss}$ is well defined in the whole attracting region $U$, see \cite[Proposition 3.2]{am1}. Moreover, we have the following lemma.

\begin{lemma}\label{lem:stable}
	For every $z\in U\setminus\cF^{ss}(\sigma)$, if we define
	$$
	\cF^s(z)=\big(\bigcup_{t\in\RR}\phi_t^X(\cF^{ss}(z))\big)\cap U,
	$$
	then $\cF^s$ is a 2-dimensional $\phi_t^X$-invariant foliation in $U\setminus\cF^{ss}(\sigma)$. Moreover, each leaf of $\cF^s$ is $C^r$-smooth and the holonomy maps of $\cF^s$ is $C^{1}$-smooth.
\end{lemma}

\begin{proof}
	The definition of $\cF^s$ implies it is a 2-dimensional $\phi_t^X$-invariant foliation in $U\setminus\cF^{ss}(\sigma)$. The classical invariant manifold theorem \cite{hps} shows each leaf of $\cF^{ss}$ is $C^r$-smooth, so does each leaf of $\cF^s$. The holonomy maps of $\cF^s$ is $C^{1}$-smooth\footnote{In fact, \cite[Lemma 7.1]{am2} proves more that the holonomy maps of $\cF^s$ is $C^{1+}$-smooth.} has been proved in \cite[Lemma 7.1]{am2}.
\end{proof}

\begin{lemma}\label{lem:big-section}
	There exists a $C^r$-smooth cross section $\Sigma_1$ of $\phi_t^X$ which is an extension of $\Sigma$ and $C^1$-diffeomorphic to $[-1-\epsilon,1+\epsilon]^2$ for some $\epsilon>0$ such that
		$$
		l_1=\{0\}\times[-1-\epsilon,1+\epsilon]
		 =\Sigma_1\cap W^s_{\it loc}(\sigma,\phi_t^X),
		\quad {\rm and} \quad
		\Sigma=[-1,1]^2\subset
		[-1-\epsilon,1+\epsilon]^2=\Sigma_1.
		$$
	The Poincar\'e map $P:\Sigma\setminus l\rightarrow\Sigma$ smoothly extends to a Poincar\'e map
	$P_1:\Sigma_1\setminus l_1\rightarrow\Sigma_1$, which has the form $P_1(x,y)=\big( f_1(x), H_1(x,y) \big)$ for every $(x,y)\in\Sigma\setminus l_1$ and satisfies
	$P_1|_{\Sigma\setminus l}\equiv P$.
	Moreover, they satisfy:
	\begin{itemize}
		\item $H_1(x,y)>0$ for $x>0$, $H_1(x,y)<0$ for $x<0$, and $\sup_{(x,y)\in\Sigma_1\setminus l_1}\big|\partial H_1(x,y)/\partial y\big|<1$,
		$\sup_{(x,y)\in\Sigma\setminus l}\big|\partial H(x,y)/\partial x\big|<1$;
		\item the one-dimensional quotient map $f_1:[-1-\epsilon,1+\epsilon]\setminus\{0\}\rightarrow[-1,1]$ is $C^1$-smooth and satisfies $f_1|_{[-1,1]\setminus\{0\}}\equiv f$, $-1<f(x)<1$ and
		$f'(x)>\sqrt{2}$ for every $x\in[-1-\epsilon,1+\epsilon]\setminus\{0\}$.
	\end{itemize}
    In particular, the image of the Poincar\'e map $P_1$ satisfies $\overline{P_1(\Sigma_1\setminus l_1)} \subseteq [-1,1]\times(-1,1)$.
\end{lemma}

\begin{proof}
	Since $\Sigma$ is a $C^r$-smooth cross section of $\phi_t^X$, we can extend $\Sigma$ to a larger surface $\Sigma_1$ transverse to $\phi_t^X$.
	Moreover, Lemma \ref{lem:stable} shows that the stable foliation $\cF^s$ intersects $\Sigma_1$ induced a $C^{1}$-smooth foliation in $\Sigma_1$, where we already have $\cF^s\cap\Sigma=\big\{x\times[-1,1]:x\in[-1,1]\big\}$. So we can  $C^1$-smoothly extend the coordinate of $\Sigma$ to $\Sigma_1=[-1-\epsilon,1+\epsilon]^2$ for some $\epsilon>0$, such that
	$$
	\cF^s\cap\Sigma_1~=~
	\big\{x\times[-1-\epsilon,1+\epsilon]:~
	x\in[-1-\epsilon,1+\epsilon]\big\},
	$$
	and $l_1=0\times[-1-\epsilon,1+\epsilon]=W^s_{\it loc}(\sigma,\phi_t^X)\cap\Sigma_1$.

	Let $P_1:\Sigma_1\setminus l_1\rightarrow \Sigma_1$ be the Poincar\'e map associated to $\Sigma_1$, then it has the form $P_1(x,y)=(f_1(x), H_1(x,y))$ and $P_1|_{\Sigma\setminus l}\equiv P$. In particular, $f_1$ is $C^1$-smooth and $f_1|_{[-1,1]\setminus\{0\}}\equiv f$.
	From the definition of geometric Lorenz map, the one-dimensional quotient map $f:[-1,1]\setminus\{0\}\rightarrow[-1,1]$ satisfies $-1<f(x)<1$ and $f'(x)>\sqrt{2}$ everywhere. This implies $-1<f(-1)<0<f(1)<1$. Thus if we choose $\epsilon$ small enough, we also have $-1<f_1(1-\epsilon)<0<f(1+\epsilon)<1$. Shrinking $\epsilon$ if necessary, $P_1$ is well defined on $\Sigma_1\setminus l_1$, and $H_1,f_1$ satisfies all estimations of $H,f$. In particular, we have
	$$
	\overline{P_1(\Sigma_1\setminus l_1)}=
	P_1(\Sigma_1\setminus l_1)\cup\{z^+,z^-\}
	\subseteq[-1,1]\times(-1,1).
	$$
\end{proof}

\begin{notation}\label{notat:section}
	From now on, we use $\Sigma,P,f,H$ to denote $\Sigma_1,P_1,f_1,H_1$ respectively for the simplicity of notations. We denote $\pi_x:\Sigma\rightarrow[-1-\epsilon,1+\epsilon]$ the projection to $x$-coordinate on $\Sigma$.
	Since $\lim_{x\rightarrow0}f'(x)=+\infty$, there exists $\lambda_0>\sqrt{2}$, such that $f'(x)>\lambda_0$ for every $x\in[-1-\epsilon,1+\epsilon]\setminus\{0\}$.
	On the other hand, since $f(-1)>-1$, we must have $\lambda_0<2$.

	We fix $\alpha=1/(\sqrt{2}-1)$. The proof of Lemma \ref{lem:cone} shows that the cone field $\cC_{\alpha}$ satisfies
	$$
	DP(\cC_{\alpha}(x,y))~\subseteq~
	 \cC_{\sqrt{2}\alpha/\lambda_0}(P(x,y)),
	 \qquad \forall (x,y)\in\Sigma\setminus l.
	$$
\end{notation}

\begin{definition}\label{def:map}
	Let $\mathscr{L}^r~(r\geq 1)$ be the set consisting of $C^r$-maps $f:[-1,1]\setminus\{0\}\rightarrow(-1,1)$ which satisfies
	\begin{itemize}
		\item $\lim_{x\rightarrow0^-}f(x)=1$, and $\lim_{x\rightarrow0^-}f'(x)=+\infty$;
		\item $\lim_{x\rightarrow0^+}f(x)=-1$, and $\lim_{x\rightarrow0^+}f'(x)=+\infty$;
		\item $-1<f(x)<1$ and $f'(x)>\sqrt{2}$
		for every $x\in[-1,1]\setminus\{0\}$.
	\end{itemize}
	We call $f\in\mathscr{L}^r$ a \emph{Lorenz expanding map}.
\end{definition}

\begin{figure}[htbp]
	\centering
	\includegraphics[width=7cm]{lorenz_map.jpg}
	\caption{The Lorenz expanding map}
\end{figure}

The following lemma is standard in the study of geometric Lorenz attractors. We include its proof which illustrates the main idea of proving Claim \ref{clm:cover} in the proof of Theorem \ref{thm:Lorenz-connecting}.

\begin{lemma}\label{lem:evenually-onto}
	Let $f\in\mathscr{L}^r$, then $f$ is eventually onto, i.e. for every open interval $J\subset[-1,1]\setminus\{0\}$, there exists an integer $N>0$, such that $\bigcup_{i=0}^Nf^i(J)=(-1,1)$.
\end{lemma}

\begin{proof}[New proof of Lemma \ref{lem:evenually-onto}]
	For every $n\geq 1$, we consider the map $f^n:J\rightarrow[-1,1]$.

	Let $$D_n=\{t\in J: f^m(t)=0, \text{ for some $1\leq m\leq n-1$}\}~~\text{for $n\geq 2$ and } D_1=\emptyset.$$
	We call $t\in D_n$ a discontinuity point of $f^n$. Then $D_n$ is a finite subset of $J$ with $D_n\subseteq D_{n+1}$, and $f^n$ is well-defined and $C^1$-smooth on each connected component of $J\setminus D_n$. Moreover, if $(a,b)\subseteq J$ is a connected component of $J\setminus D_n$, then $f^n|_{(a,b)}$ is a diffeomorphism from $(a,b)$ to its image
	$$
	f^n(a,b)~=~
	\big(\lim_{x\rightarrow a^+}f^n(x)~,~
	    \lim_{x\rightarrow b^-}f^n(x)\big)
	\quad {\rm  and} \quad
	(f^n)'(x)\geq \lambda_0^n, \quad \forall x\in(a,b).
	$$

	\begin{claim}\label{clm:counting}
		For every $n\geq1$, assume $I=(a,b)$ is a connected component of
		$J\setminus D_n$, then either $\#\big(I\cap D_{n+2}\big)\leq1$, or
		 $(-1,1)\subseteq f^{n+1}(I)\cup f^{n+2}(I)$.
	\end{claim}
	\begin{proof}[Proof of the claim]

		Recall that $f^n$ is well-defined and $C^1$-smooth on $I$. Let $(c,d)=f^n(I)$. If $0\notin(c,d)$, then $I\cap D_{n+1}=\emptyset$ and $f^{n+1}(I)=f(c,d)$ is an open interval in $(-1,1)$. Thus there exists at most one point $t\in I$, such that $f^{n+1}(t)=0$. This implies $\#\big(I\cap D_{n+2}\big)\leq1$.

		If $0\in(c,d)=f^n(I)$, then we denote $t_0\in I$ satisfying $f^n(t_0)=0$. Then the two open intervals $(a,t_0)$ and $(t_0,b)$ are two connected components of $J\setminus D_{n+1}$. Moreover, their images by $f^{n+1}$ satisfies
		$$
		f^{n+1}(a,t_0)=\big(\lim_{x\rightarrow a^+}f(x),~1\big),
		\qquad {\rm and } \qquad
		f^{n+1}(t_0,b)=\big(-1,~\lim_{x\rightarrow a^+}f(x)\big).
		$$
		If neither of them contains $0$, then we have $\#\big(I\cap D_{n+2}\big)=\#\big(I\cap D_{n+1}\big)=1$.

		Otherwise, we have either  $0\in f^{n+1}(a,t_0)$ or $0\in f^{n+1}(t_0,b)$. Thus either
		$$
		[0,1)\subset f^{n+1}(a,t_0)\subset f^{n+1}(I),
		\qquad {\rm or} \qquad
		(-1,0]\subset f^{n+1}(t_0,b)\subset f^{n+1}(I).
		$$
		In both cases, iterating by $f$ again, we have
		$(-1,1)\subseteq f^{n+1}(I)\cup f^{n+2}(I)$.
	\end{proof}

	Now we show that there exists some $n>0$ and a connected component $I$ of $J\setminus D_n$, such that $(-1,1)\subseteq f^{n+1}(I)\cup f^{n+2}(I)$. Otherwise, Claim \ref{clm:counting} implies
	$$
	\# D_2\leq1, \qquad
	\# D_4\leq\# D_2+\# D_2+1\leq 3,
	\qquad \cdots \qquad
	\# D_{2n}\leq \# D_{2n-2}+\# D_{2n-2}+1\leq 2^n-1.
	$$
	This implies $J\setminus D_{2n}$ has at most $2^n$ connected components. So there exists a connected component $I_{2n}\subset J\setminus D_{2n}$, such that its length satisfies $|I_{2n}|\geq 2^{-n}|J|$.

	Since $f$ is a diffeomorphism on each connected component, and satisfies $(f^{2n})'>\lambda_0^{2n}$, thus the length of the interval $f^{2n}(I_{2n})$ satisfies
	$$
	\left|f^{2n}(I_{2n})\right|
	~\geq~ \lambda_0^{2n}\cdot|I_{2n}|
	~\geq~ \left(\frac{\lambda_0^2}{2}\right)^n\cdot|J|
	~\rightarrow~+\infty,
	\qquad {\rm as}\quad n\rightarrow+\infty.
	$$
	This is absurd since $f^{2n}(I_{2n})\subseteq(-1,1)$. We take $N=n+2$ finishing the proof of this lemma.
\end{proof}

The following proposition shows that every Lorenz attractor is a homoclinic class and every pair of periodic orbits are homoclinic related. Moreover, for every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, the set of $C^r$ vector fields exhibiting a geometric Lorenz attractor is an open subset of $\mathscr{X}^r(M^3)$.

\begin{proposition}\label{prop:Lorenz}
	Let $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$  and $X\in\mathscr{X}^r(M^3)$. If $X$ exhibits a geometric Lorenz attractor $\Lambda$ with attracting region $U$, then $\Lambda$ is a singular hyperbolic homoclinic class of $X$, and the orbits of every two periodic points $p,q\in\Lambda$ are homoclinic related.

	Moreover, there exists a $C^r$-neighborhood $\cU$ of $X$ in $\mathscr{X}^r(M^3)$, such that for every $Y\in\cU$, $U$ is an attracting region of $Y$, and the maximal invariant set $\Lambda_Y=\bigcap_{t>0}\phi_t^Y(U)$ is a geometric Lorenz attractor.

\end{proposition}

\begin{remark}
	The proof of Proposition~\ref{prop:Lorenz} is in the Appendix.
	The first part  is classical and the ``moreover'' part

	 relies on a beautiful observation in \cite{am2}, which shows the stable foliation of the Poincar\'e map in the cross section is $C^{1+}$-smooth.
\end{remark}

\subsection{A $C^r$-connecting theorem for geometric Lorenz attractors}\label{subsec:connecting}

This subsection is devoted to prove the following $C^r$-connecting theorem for geometric Lorenz attractors which is a key technique for proving Theorem~\ref{Thm:A}.

\begin{theorem}\label{thm:Lorenz-connecting}
	Assume $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$ and $X\in\mathcal{X}^r(M^3)$ admits a geometric Lorenz attractor $\Lambda$ with singularity $\sigma$ and attracting region $U$. Let $\Sigma$ be the cross section of $\Lambda$ from Notation~\ref{notat:section}, and $z\in W^u(\sigma)\cap\Sigma$ be a point satisfying $\phi_t^X(z)\notin\Sigma$ for every $t<0$.

	For every $C^r$-neighborhood $\cV$ of $X$, every critical element $p\in\Lambda$ and every $\delta_0>0$, there exist $0<\delta<\delta_0$ and $Y\in\cV$, such that the maximal invariant set $\Lambda_Y=\bigcap_{t>0}\phi_t^Y(U)$ is a geometric Lorenz attractor satisfying
	\begin{enumerate}
		\item The vector field $Y$ and $\delta$-neighborhood $B(z,\delta)$ of $z$ satisfy
		$$
		Y|_{M^3\setminus B(z,\delta)}~\equiv~ X|_{M^3\setminus B(z,\delta)},
		\qquad {\it and} \qquad
		B(z,\delta)~\cap~\left(\orb(p,\phi_t^X)\cup\{\sigma\}\right)
		~=~\emptyset.
		$$
		This implies
		$\sigma_Y=\sigma\in\sing(\Lambda_Y)$, and
		$p\in\per(\Lambda_Y)$ with
		$\orb(p,\phi_t^Y)=\orb(p,\phi_t^X)$.

		\item The point $z\in W^u(\sigma,\phi_t^Y)$ and there exists $T>0$, such that
		$\phi^Y_T(z)\in W^s_{\it loc}(p,\phi_t^Y)$. This implies
		$$
		\orb(z,\phi_t^Y)~\subset~
		W^u(\sigma,\phi_t^Y) \cap W^s(\orb(p),\phi_t^Y).
		$$
	\end{enumerate}
\end{theorem}

Before proving Theorem~\ref{thm:Lorenz-connecting}, we would like to state the following direct corollary by considering the critical element $p$ as the singularity $\sigma$ itself, which shows that $C^r$-densely, the singularity in the geometric Lorenz attractor admits a homoclinic orbit.

\begin{corollary}\label{Cor:dense-loop}

	Assume $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$.  There exists a dense subset $\mathcal{D}^r_h\subset \mathscr{X}^r(M^3)$, such that if $X\in \mathcal{D}^r_h$ admits a geometric Lorenz attractor with singularity $\sigma$, then $\sigma$ has a homoclinic orbit.
\end{corollary}

Now we give the proof of Theorem~\ref{thm:Lorenz-connecting}.
\begin{proof}[Proof of Theorem~\ref{thm:Lorenz-connecting}]

	By Proposition \ref{prop:Lorenz}, we can shrink the $C^r$-neighborhood $\cV$ of $X$ if necessary, such that for every $Y\in\cV$, the maximal invariant set $\Lambda_Y=\bigcap_{t>0}\phi_t^Y(U)$ is a geometric Lorenz attractor.

	There are two points $z^\pm\in W^u(\sigma)\cap\Sigma$ satisfying $\phi_t^X(z^\pm)\notin\Sigma$ for every $t<0$.
	We assume $z=z^+=(-1,y^+)\in\Sigma$.
	The proof for $z=z^-$ is the same.

	For the critical element $p\in\Lambda$, there are two possibilities: either $p=\sigma$, or $p$ is a periodic point.
	\begin{itemize}
		\item  If $p=\sigma$, there exists $\delta\in(0,\delta_0)$, such that  $B(z,\delta)\cap W^s_{\it loc}(\sigma)=\emptyset$. We only need to show there exists a vector field $Y\in\cV$ which satisfies $Y|_{M^3\setminus B(z,\delta)}~\equiv~ X|_{M^3\setminus B(z,\delta)}$, such that for some $T>0$,
		$$
	    z\in W^u(\sigma,\phi_t^Y)\cap\Sigma,
	    \qquad {\rm and} \qquad
	    \phi_T^Y(z) \in l=
	    \Sigma\cap W^s_{\it loc}(\sigma,\phi_t^Y).
    	$$
	    \item	If $p$ is a periodic point, then we can assume $p\in\Sigma$ with
	    $$
    	p=(x_p,y_p)\in\big((-1,1)\setminus\{0\}\big)\times[-1,1]
	    \quad {\rm and} \quad
	    l_p=\{x_p\}\times[-1-\epsilon,1+\epsilon]\subset W^s_{\it loc}(\orb(p),\phi_t^X).
	    $$
	    There exists $\delta\in(0,\delta_0)$, such that
	    $B(z,\delta)\cap
	    \left(\bigcup_{t\geq0}\phi_t^X(l_p)\cup\{\sigma\}\right)
	    =\emptyset$.
	    We need to show there exists a vector field $Y\in\cV$ which  satisfies
	    $Y|_{M^3\setminus B(z,\delta)}~\equiv~ X|_{M^3\setminus B(z,\delta)}$, such that for some $T>0$,
	    $$
    	z\in W^u(\sigma,\phi_t^Y)\cap\Sigma,
    	\qquad {\rm and} \qquad
    	\phi_T^Y(z)\in l_p
    	\subset\Sigma\cap W^s_{\it loc}(\orb(p),\phi_t^Y).
    	$$
	\end{itemize}
	From now on, we fix $\delta\in(0,\delta_0)$ which satisfies the above property. Recall that the cross section $\Sigma=[-1-\epsilon,1+\epsilon]^2$ for some $\epsilon>0$ and $P:\Sigma\setminus l\rightarrow\Sigma$ is the Poincar\'e map.

	\begin{claim}\label{clm:nbhd}
		There exist $\eta\in(0,\epsilon)$ and $t_0>0$, such that the cross section
		$\Sigma_z=[-1-\eta,-1+\eta]\times[y^+-\eta,y^++\eta] \subset\Sigma$ at $z$ satisfies
		 \begin{enumerate}
		 	\item\label{item:1} $P(\Sigma_z)\cap\Sigma_z=\emptyset$ and there exists $\eta'>0$ such that $P^{-1}(\Sigma_z)\subset(0,\eta')\times[-1-\epsilon,1+\epsilon]$ and
		 	$f'(x)>10$ for every $x\in(0,\eta')$.
		 	\item\label{item:2} The set $\phi_{[0,3t_0]}^X(\Sigma_z)=\bigcup_{t\in[0,3t_0]}\phi_t^X(\Sigma_z)~\subset~ B(z,\delta)$ and $\phi_t^X(x)\notin\phi_{[0,3t_0]}^X(\Sigma_z)$ for each $t<0$.
		 	\item\label{item:3} We have $(1-10^{-3})f'(x_1)<f'(x_2)<(1+10^{-3})f'(x_1)$ for every $x_1,x_2\in[-1-\eta,-1+\eta]$.
		 \end{enumerate}
	\end{claim}

	\begin{proof}
		The fact that $f(-1)>-1$ implies if $\eta$ is small enough, then $f([-1-\eta,-1+\eta])\cap[-1-\eta,-1+\eta]=\emptyset$. Thus $P(\Sigma_z)\cap\Sigma_z=\emptyset$.

		Since $P(\Sigma)\subset[-1,1]\times(-1,1)$, if a point $w\in\Sigma$ satisfies $P(w)\in\Sigma_z$, then $\pi_x\circ P(w)\in(-1,-1+\eta]$.
		The one-dimensional map $f$ satisfies $f(-1-\epsilon)>-1$, so if $\eta<f(-1-\epsilon)+1$, then there exists some $\eta'>0$, such that $P^{-1}(\Sigma_z)\subset   (0,\eta']\times[-1-\epsilon,1+\epsilon]$.
		If $\eta\rightarrow 0^+$, then $\eta'\rightarrow0^+$, and $f'(x)\rightarrow+\infty$ as $x\in(0,\eta')$. So we only need to shrink $\eta$ to take $\eta'$ small enough, such that $f'(x)>10$ for every $x\in(0,\eta')$. This shows item~\ref{item:1}.

		Finally, since the quotient map $f$ is $C^1$-smooth and there exists $t_0>0$ such that $\phi_{[-t_0,4t_0]}^X(z)\subset B(z,\delta)$. Hence shrinking $\eta$ again if necessary, item~\ref{item:2} and~\ref{item:3} are satisfied.
	\end{proof}

    The subset $\phi_{[0,3t_0]}^X(\Sigma_z)$ admits a natural coordinate: for every $w\in\phi_{[0,3t_0]}^X(\Sigma_z)$,
    there exist a unique point $w_0=(x_0,y_0)\in\Sigma_z=[-1-\eta,-1+\eta]\times[y^+-\eta,y^++\eta]$ and $t_w\in[0,3t_0]$, such that $w=\phi_{t_w}^X(w_0)$. Then we denote
    $$
    w=(x_0,y_0,t_w)~\in~ \phi_{[0,3t_0]}^X(\Sigma_z)
    =[-1-\eta,-1+\eta]\times[y^+-\eta,y^++\eta]\times[0,3t_0].
    $$
    We point out here that this coordinate is only $C^1$-smooth.

    Let $Z\in\mathscr{X}^\infty(M^3)$ be a smooth vector field satisfying $\supp(Z)\subset\phi_{[t_0,2t_0]}^X(\Sigma_z)\subset B(z,\delta)$ such that if we represent $Z$ in the coordinate $\{(x,y,t)\}$ as
    $$
    Z(x,y,t)=
      a(x,y,t)\cdot\frac{\partial}{\partial x}+
      b(x,y,t)\cdot\frac{\partial}{\partial y}+
      c(x,y,t)\cdot\frac{\partial}{\partial t}
    $$
    then for every $(x,y,t)\in\phi_{[t_0,2t_0]}^X(\Sigma_z)$, it satisfies the following properties
    \begin{enumerate}
    	\item $\max\big\{ |b(x,y,t)|,|c(x,y,t)| \big\}\leq  \beta|a(x,y,t)|$ and $|a(x,y,t)|<\beta$

    	where $0<\beta<10^{-3}$ is a small constant to be determined in Claim~\ref{clm:vector-field-family} below.
    	\item $a(x,y,t)\geq0$ and $a(-1,y^+,3t_0/2)>0$.
    \end{enumerate}
    Here since the coordinate $\{(x,y,t)\}$ is $C^1$-smooth, the three functions $a,b,c$ are only uniformly continuous.

    \begin{claim}\label{clm:vector-field-family}
    	Let $X_s=X+s\cdot Z\in\mathscr{X}^r(M^3)$ for $s\in[0,1]$. Then there exists $\beta>0$ such that the $C^r$-vector field family $\{X_s\}_{s\in[0,1]}$ satisfies the following properties:
    	\begin{enumerate}
    		\item\label{item:convergence} $X_0=X$ and $X_s$ converges to $X$ in $C^r$-topology as $s\rightarrow 0$.
    		\item\label{item:unstable} For every $s\in[0,1]$, $\sigma$ is the singularity of $X_s$ and $z\in W^u(\sigma,\phi_t^{X_s})$.
    		\item\label{item:support} For every $s\in[0,1]$, $X_s|_{\phi_{[t_0,2t_0]}^X(\Sigma_z)}\equiv X|_{\phi_{[t_0,2t_0]}^X(\Sigma_z)}$.
    		\item\label{item:return map} For every $s\in[0,1]$, the first return map $P_s:\Sigma\setminus l\rightarrow\Sigma$ is well-defined for the flow $\phi_t^{X_s}$, and satisfies
    		    $$
    		    P_s|_{\Sigma\setminus\Sigma_z}~\equiv~
    		    P|_{\Sigma\setminus\Sigma_z}.
    		    $$
    		\item\label{item:curve} For every $w\in\Sigma_z$, the curve $P_s(w):[0,1]\rightarrow\Sigma$ is $C^r$-smooth with respect to $s$, and  satisfies
    		$$
    		\frac{{\rm d}\pi_x\circ P_s(w)}{{\rm d}s}\geq 0,
    		\qquad \text { and } \qquad
    		\frac{{\rm d}P_s(w)}{{\rm d}s}|_{s=s'}\in
    		\cC_{\alpha}(P_{s'}(w)), ~\forall s'\in[0,1].
    		$$
    		Moreover, for the point $z$, the curve $P_s(z):[0,1]\rightarrow\Sigma$ satisfies
    		${\rm d}\pi_x\circ P_s(z)/{\rm d}s>0$ at the point $s=0$.
    	\end{enumerate}
    \end{claim}

    \begin{proof}[Proof of the claim]
    	The definitions of $Z$ and $X_s$ imply the first three items directly. The vector field $X$ on $\phi_{[0,3t_0]}^X(\Sigma_z)$ is equal to $\partial/\partial t$ everywhere. For every $w$ in the boundary of $\Sigma_z$, $Z|_{\phi_{[0,3t_0]}^X(w)}\equiv 0$, this implies $\phi_t^{X_s}(w)=\phi_t^X(w)$ for every $t\in[0,3t_0]$. Moreover, since
    	$$
    	\max_{(x,y,t)\in\phi_{[0,3t_0]}^X(\Sigma_z)}
    	  \big\{~|a(x,y,t)|~,~|b(x,y,t)|~,~|c(x,y,t)|~\big\}
    	   ~<~\beta,
    	$$
    	hence taking $0<\beta<10^{-3}$ small, for every $s\in[0,1]$ and every $w\in\Sigma_z$, there exists some $t=t(w,s)$ which is close to $3t_0$, such that $\phi_t^{X_s}(w)\in\phi_{3t_0}^X(\Sigma_z)$. Thus the Poincar\'e map from $\Sigma_z$ to $\Sigma$ associated to $\phi_t^{X_s}$ is well defined for every $s\in[0,1]$.

    	For every $w\in\Sigma\setminus\Sigma_z$, its positive orbit does not intersect $\phi_{[0,3t_0]}^X(\Sigma_z)$ before it enters $\Sigma$ again. This implies for every $s\in[0,1]$, the first return map $P_s:\Sigma\setminus l\rightarrow\Sigma$ is well-defined for the flow $\phi_t^{X_s}$, and satisfies
    	$P_s|_{\Sigma\setminus\Sigma_z}~\equiv~
    	P|_{\Sigma\setminus\Sigma_z}$. This proves item~\ref{item:return map}.

    	\vskip 2mm

    	Let
    	$$
    	P_{s,1}:\Sigma_z\rightarrow\phi_{3t_0}^X(\Sigma_z),
    	\qquad {\rm and} \qquad
    	P_{s,2}:\phi_{3t_0}^X(\Sigma_z)\rightarrow\Sigma
    	$$
    	be Poincar\'e maps of $\phi_t^{X_s}$ from $\Sigma_z$ to $\phi_{3t_0}^X(\Sigma_z)$ and from $\phi_{3t_0}^X(\Sigma_z)$ to $\Sigma$ respectively. Then we have $P_s=P_{s,2}\circ P_{s,1}$.
    	Let the coordinate on $\phi_{3t_0}^X(\Sigma_z)$ be induced  by pushing the coordinate of $\Sigma_z\subset\Sigma$ by $\phi_{3t_0}^X$, then $P_{s,2}\equiv P|_{\Sigma_z}$ in this coordinate.

    	Since $X_s\in \mathscr{X}^r(M^3)$ is depends continuously with respect to $s$ in $\mathscr{X}^r(M^3)$ and the cross section $\Sigma$ is a $C^r$ smooth surface, the curve $P_s(w):[0,1]\rightarrow\Sigma$ is $C^r$-smooth with respect to $s$ for every $w\in\Sigma_z$.

    	The property $a(x,y,t)\geq 0$ everywhere implies for every $w\in\Sigma_z$, ${\rm d}\pi_x\circ P_{s,1}(w)/{\rm d}s\geq0$ everywhere. Here $\pi_x$ is the projection to $x$-coordinate on $\phi_{3t_0}^X(\Sigma_z)$. Since $f'(x)\geq\lambda_0$ everywhere, we have
    	$$
    	\frac{{\rm d}\pi_x\circ P_s(w)}{{\rm d}s}~=~
    	\frac{{\rm d}f}{{\rm d}x}\cdot
    	   \frac{{\rm d}\pi_x\circ P_{s,1}(w)}{{\rm d}s}
    	 ~\geq~0.
    	$$
    	Moreover, since $|b(x,y,t)|\leq\beta\cdot|a(x,y,t)|\leq10^{-3}\cdot|a(x,y,t)|$ for every $(x,y,t)$, hence for every $w\in\Sigma_z$ and every $0\leq s_1<s_2\leq 1$, we have
    	$$
    	\left|\pi_y\circ P_{s_2,1}(w)-\pi_y\circ P_{s_1,1}(w)\right|
    	~\leq~ 10^{-3}\cdot
    	\left|\pi_x\circ P_{s_2,1}(w)-\pi_x\circ P_{s_1,1}(w)\right|.
    	$$
    	This implies ${\rm d}P_{s,1}/{\rm d}s$ is tangent to the cone field $D\phi_{3t_0}^X\left(\cC_{\alpha}|_{\Sigma_z}\right)$ everywhere on $\phi_{3t_0}^X(\Sigma_z)$. Since $P_{s,2}\equiv P|_{\Sigma_z}$, and $P$ preserves the cone field $\cC_{\alpha}$(Lemma \ref{lem:cone} and Notation \ref{notat:section}), the curve $P_s(w):[0,1]\rightarrow\Sigma$ is tangent to $\cC_{\alpha}$ everywhere.

    	For the point $z=(-1,y^+)\in\Sigma_z$, since $a(-1,y^+,3t_0/2)>0$, there exists a neighborhood of $(-1,y^+,3t_0/2)$, such that for every $(x,y,t)$ in this neighborhood, we have $a(x,y,t)>0$. Thus there exists some $s'>0$ and $\kappa>0$, such that for every $0<s<s'$, we have $\pi_x\circ P_{s,1}(z)\geq-1+\kappa\cdot s$. Thus we have
    	$$
    	\frac{{\rm d}\pi_x\circ P_s(z)}{{\rm d}s}|_{s=0}~=~
    	\frac{{\rm d}f}{{\rm d}x}|_{x=-1}\cdot
    	\frac{{\rm d}\pi_x\circ P_{s,1}(w)}{{\rm d}s}|_{s=0}
    	~\geq~\lambda_0\cdot\kappa>0.
    	$$
    	Then item~\ref{item:curve} is proved and the proof of the claim is completed.
    \end{proof}

    From Notation \ref{notat:section}, there exists $\lambda_0\in(\sqrt{2},2)$, such that $f'(x)>\lambda_0$ for every $x\in[-1-\epsilon,1+\epsilon]$. We fix the $C^\infty$-vector field $Z$ to satisfy Claim~\ref{clm:vector-field-family} and a constant $\lambda\in(\sqrt{2},\lambda_0)$.

    \begin{claim}\label{clm:small-perturb}
    	There exists a constant $\tau\in(0,1)$ such that $X_s\in\cV$ for every $s\in[0,\tau]$, and satisfies the following two properties:
    	\begin{enumerate}
    		\item\label{item:esti-return-map} For every $s\in[0,\tau]$, the Poincar\'e return map $P_s:\Sigma\setminus l\rightarrow\Sigma$ satisfies for every $(x,y)\in\Sigma\setminus l$
    		\begin{align*}
    		\frac{\partial\pi_x\circ P_s(x,y)}{\partial x}>\lambda,
    		\qquad &{\it and} \qquad
    		\frac{\partial\pi_x\circ P_s(x,y)}{\partial y}
    		  <10^{-3}(\lambda-\sqrt{2}); \\
    	    \left|\frac{\partial\pi_y\circ P_s(x,y)}{\partial x}\right|<1,
    	    \qquad &{\it and} \qquad
    	    \left|\frac{\partial\pi_y\circ P_s(x,y)}{\partial y}\right|
    	    <1.
    		\end{align*}
    		\item\label{item:esti-curve} Let $\gamma_0:[0,\tau]\rightarrow\Sigma$ be defined as $\gamma_0(s)=P_s(z)$, then we have
    		$$
    		\frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=s'}
    		~\geq~
    		\frac{1}{2}\cdot
    		\frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=0},
    		\qquad \forall s'\in[0,\tau]
    		$$
    	\end{enumerate}
    \end{claim}

    \begin{proof}[Proof of the claim]
    	Since $X_s\rightarrow X$ in $\mathscr{X}^r(M^3)$ and $P_s|_{\Sigma\setminus\Sigma_z}\equiv P|_{\Sigma\setminus\Sigma_z}$, so we have $P_s\rightarrow P$ in $C^1$-topology under the coordinate $\Sigma=[-1-\epsilon,1+\epsilon]^2$. On the other hand,
    	$$
    	\frac{\partial\pi_x\circ P(x,y)}{\partial x}
    	   \equiv f'(x)\geq\lambda_0,
    	 \qquad {\rm and} \qquad
    	\frac{\partial\pi_x\circ P(x,y)}{\partial y}
    	   \equiv \frac{f(x)}{y} \equiv 0
    	 \qquad \forall (x,y)\in\Sigma\setminus l.
    	$$
    	So there exists $\tau_1>0$, such that for every $s\in[0,\tau]$, $X_s\in\cV$ whose Poincar\'e map $P_s$ satisfies
    	$\partial\pi_x\circ P_s(x,y)/\partial x>\lambda$
    	and
    	$\partial\pi_x\circ P_s(x,y)/\partial y<10^{-3}(\lambda-\sqrt{2})$
    	 for every $(x,y)\in\Sigma\setminus l$. The proof for
    	 $|\partial\pi_y\circ P_s(x,y)/\partial x|<1$ and $|\partial\pi_y\circ P_s(x,y)/\partial y|<1$ is the same.

    	Since $P_s(z)$ is $C^r$-smooth with respect to $s\in[0,1]$, so the function $\pi_x\circ P_s(z)$ is $C^1$-smooth with respect to $s$. There exists $\tau_2>0$, such that for every $s'\in[0,\tau_2]$, it satisfies
    	$$
    	\frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=s'}
    	~\geq~\frac{1}{2}\cdot
    	\frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=0}
    	$$
    	We take $\tau=\min\{\tau_1,\tau_2\}$ finishing the proof of the claim.
    \end{proof}

    Fix $\tau\in(0,1)$ from Claim~\ref{clm:small-perturb}. For every $n\geq1$, let $D_n\subset[0,\tau]$ be the set consisting of points $s$ satisfying there exists $m<n$, such that $P_s^m\circ\gamma_0(s)\in l$. Then $D_n$ is a finite subset of $[0,\tau]$ with $D_n\subseteq D_{n+1}$.

    \begin{claim}\label{clm:1-dim-expanding}
    	For every $n\geq0$, the map
    	$$
    	\gamma_n(s)=\left(x_n(s),~y_n(x)\right)=P_s^n\circ\gamma_0(s):
    	~[0,\tau]\setminus D_n\longrightarrow\Sigma
    	$$
    	is well defined. For each connected component $J$ of $[0,\tau]\setminus D_n$, $\gamma_n$ is a diffeomorphism from $J$ to its image, and satisfies
    	$$
    	\gamma_n'(s)~\in~\cC_{\alpha}(\gamma_n(s)),
    	\qquad {\it and } \qquad
    	\frac{{\rm d}\pi_x\circ\gamma_n(s)}{{\rm d}s}~=~x_n'(s)~\geq~\lambda^n\cdot x_0'(s),
    	\qquad \forall s\in J.
    	$$
    \end{claim}

    \begin{proof}[Proof of the claim]
    	We proof this claim by induction. When $n=0$, the item~\ref{item:curve} of Claim \ref{clm:vector-field-family} shows that $\gamma_0'(s)\in\cC_{\alpha}(\gamma_0(s))$ for every $s\in[0,\tau]$. And $x_0'(s)\geq\lambda^0 x_0'(s)$ automatically holds for every $s\in[0,\tau]$.
    	Now we assume for every $0\leq i\leq n-1$ and every $s\in[0,\tau]\setminus D_i$,
    	$$
    	\gamma_i'(s)\in\cC_{\alpha}(\gamma_i(s)).
    	\qquad {\rm and } \qquad
    	x_i'(s)~\geq~\lambda^i\cdot x_0'(s).
    	$$
    	For every $s\in[0,\tau]\setminus D_n$, we have
    	$$
    	x_n(s)=\pi_x\circ P_s\left(x_{n-1}(s),y_{n-1}(s)\right),
    	\qquad {\rm and} \qquad
    	y_n(s)=\pi_y\circ P_s\left(x_{n-1}(s),y_{n-1}(s)\right).
    	$$

    	Now we fix $s_0\in[0,\tau]\setminus D_n$.
    	There are two possibilities: either $(x_{n-1}(s_0),y_{n-1}(s_0))\notin\Sigma_z$, or $(x_{n-1}(s_0),y_{n-1}(s_0))\in\Sigma_z$. If $(x_{n-1}(s_0),y_{n-1}(s_0))\notin\Sigma_z$, then there exists a neighborhood of $(x_{n-1}(s_0),y_{n-1}(s_0))$, such that $P_s\equiv P$ in this neighborhood. Lemma \ref{lem:cone} shows that
    	$$
    	\gamma_n'(s_0)~=~DP(\gamma_{n-1}'(s_0))
    	~\in~\cC_{\alpha}(\gamma_n(s_0)).
    	$$

    	If $(x_{n-1}(s_0),y_{n-1}(s_0))\in\Sigma_z$, item~\ref{item:1} of Claim \ref{clm:nbhd} implies $(x_{n-2}(s_0),y_{n-2}(s_0))\notin\Sigma_z$, $0<x_{n-2}(s_0)<\eta'$ and $f'(x_{n-2}(s_0))>10$. If we denote
    	\begin{align*}
    	    \gamma_{n-2}'(s_0)~&=~
    	    x_{n-2}'(s_0)\cdot\frac{\partial}{\partial x}+
    	    y_{n-2}'(s_0)\cdot\frac{\partial}{\partial y}~\in~
    	    \cC_{\alpha}(\gamma_{n-2}(s_0)). \\
    	    \gamma_{n-1}'(s_0)~&=~
    	    x_{n-1}'(s_0)\cdot\frac{\partial}{\partial x}+
    	    y_{n-1}'(s_0)\cdot\frac{\partial}{\partial y}~\in~
    	    \cC_{\alpha}(\gamma_{n-1}(s_0)).
    	\end{align*}
    	Then $\gamma_{n-1}'(s_0)=
    	\left(DP|_{(x_{n-2}(s_0),y_{n-2}(s_0))}\right) \big(\gamma_{n-2}'(s_0)\big)$. Thus we have
    	\begin{align*}
    	    \left|y_{n-2}'(s_0)\right|\leq
    	    \alpha\cdot\left|x_{n-2}'(s_0)\right|,
    	    \qquad
    	    |x_{n-1}'(s_0)|=f'(x_{n-2}(s_0))\cdot|x_{n-2}'(s_0)|
    	    \geq 10|x_{n-2}'(s_0)|,
    	\end{align*}
    	and
    	\begin{align*}
    	    |y_{n-1}'(s_0)|~&\leq~
    	    \frac{\partial H}{\partial x}|_{\gamma_{n-2}(s_0)}
    	    \cdot|x_{n-2}'(s_0)|~+~
    	    \frac{\partial H}{\partial y}|_{\gamma_{n-2}(s_0)}
    	    \cdot|y_{n-2}'(s_0)| \\
    	    &\leq~ |x_{n-2}'(s_0)|~+~|y_{n-2}'(s_0)| \\
    	    &\leq~(1+\alpha)|x_{n-2}'(s_0)|.
    	\end{align*}

    	Now we calculate $\gamma_n'(s_0)$. Since
    	$x_n(s)=\pi_x\circ P_s\left(x_{n-1}(s),y_{n-1}(s)\right)$
    	and
    	$y_n(s)=\pi_y\circ P_s\left(x_{n-1}(s),y_{n-1}(s)\right)$,
    	if we denote $w_{n-1}=\left(x_{n-1}(s_0),y_{n-1}(s_0)\right)$, then
    	\begin{align*}
    	    x_n'(s_0)=
    	     \frac{\partial \pi_x\circ P_{s_0}}{\partial x}
    	     \cdot x_{n-1}'(s_0) +
    	     \frac{\partial \pi_x\circ P_{s_0}}{\partial y}
    	     \cdot y_{n-1}'(s_0) +
    	     \frac{\partial \pi_x\circ P_s(w_{n-1})}{\partial s}
    	     |_{s=s_0}.
    	\end{align*}
    	From item~\ref{item:esti-return-map} of Claim \ref{clm:small-perturb}, we have
    	\begin{align*}
    	     \frac{\partial \pi_x\circ P_{s_0}}{\partial x}
    	       \cdot x_{n-1}'(s_0)
    	       \geq \lambda\cdot x_{n-1}'(s_0)
    	     \quad {\rm and} \quad
    	     \left|\frac{\partial \pi_x\circ P_{s_0}}{\partial y}
    	       \cdot y_{n-1}'(s_0)\right|
    	       \leq 10^{-3}(\lambda-\sqrt{2})\cdot|y_{n-1}'(s_0)|.
    	\end{align*}
    	Since $|y_{n-1}'(s_0)|\leq\alpha\cdot x_{n-1}'(s_0)$ and $10^{-3}\alpha<1$, we have
    	\begin{align}\label{equ:x-expanding}
    	       x_n'(s_0)\geq
    	       \sqrt{2}\cdot x_{n-1}'(s_0) + \frac{\partial \pi_x\circ P_s(w_{n-1})}{\partial s}
    	       |_{s=s_0}
    	\end{align}

    	On the other hand, we have
    	$$
    	y_n'(s_0)=\frac{\partial \pi_y\circ P_{s_0}}{\partial x}
    	\cdot x_{n-1}'(s_0) +
    	\frac{\partial \pi_y\circ P_{s_0}}{\partial y}
    	\cdot y_{n-1}'(s_0) +
    	\frac{\partial \pi_y\circ P_s(w_{n-1})}{\partial s}
    	|_{s=s_0}.
    	$$
    	Again, item~\ref{item:esti-return-map} of Claim \ref{clm:small-perturb} shows that $|\partial \pi_y\circ P_{s_0}/\partial x|<1$ and $|\partial \pi_y\circ P_{s_0}/\partial y|<1$. Thus we have
    	\begin{align}\label{equ:y-small}
    	|y_n'(s)|<(1+\alpha)\cdot x_{n-1}'(s_0)+
    	 \left|\frac{\partial \pi_y\circ P_s(w_{n-1})}{\partial s}
    	 |_{s=s_0}\right|.
    	\end{align}

    	Item~\ref{item:curve} of Claim \ref{clm:vector-field-family} shows
    	that
    	$$
    	\frac{{\rm d}P_s(w_{n-1})}{{\rm d}s}|_{s=s_0}\in
    	\cC_{\alpha}(P_{s_0}(w))
    	\qquad {\rm and} \qquad
    	\frac{\partial \pi_x\circ P_s(w_{n-1})}{\partial s}|_{s=s_0}\geq 0,
    	$$
    	Which implies
    	$$
    	\left|\frac{\partial \pi_y\circ P_s(w_{n-1})}{\partial s}|_{s=s_0}\right|
    	~\leq~\alpha\cdot
    	\frac{\partial \pi_x\circ P_s(w_{n-1})}{\partial s}|_{s=s_0}.
    	$$
    	Since $\alpha=1/(\sqrt{2}-1)$ which is equivalent to $(1+\alpha)/\sqrt{2}=\alpha$, Equation \ref{equ:x-expanding} and Equation \ref{equ:y-small} imply
    	$$
    	|y_n'(s_0)|~\leq~ \alpha\cdot x_n'(s_0),
    	\qquad {\it i.e.} \qquad
    	\gamma_n'(s_0)\in\cC_{\alpha}(\gamma_n(s_0)).
    	$$

    	Finally, Equation \ref{equ:x-expanding} shows that
    	$$
    	x_n'(s_0)~\geq~\sqrt{2}\cdot x_{n-1}'(s_0)
    	  ~\geq~ 10\sqrt{2}\cdot x_{n-2}'(s_0)
    	  ~>~ \lambda^2\cdot\lambda^{n-2}
    	  ~=~ \lambda^n.
    	$$
    	This finishes the proof of the claim.
    \end{proof}

    We can finish the proof of this theorem by using the idea of Lemma \ref{lem:evenually-onto}.

    \begin{claim}\label{clm:cover}
    	There exists some integer $N>0$, such that
    	$$
    	(-1,1)~\subseteq~
    	 \bigcup_{i=0}^N\pi_x\circ\gamma_n\big([0,\tau]\big).
    	$$
    \end{claim}

    \begin{proof}[Proof of the claim]
    	If $(a,b)\subseteq[0,\tau]$ is a connected component of $[0,\tau]\setminus D_n$, then
    	$$
    	\pi_x\circ\gamma_n(s)=\pi_x\circ P_s^n\circ\gamma_0(s):
    	~[0,\tau]\rightarrow[-1,1]
    	$$
    	is a diffeomorphism from $(a,b)$ to its image.

    	Similar to Claim \ref{clm:counting}, we show that
    	for every $n\geq1$, let $I=(a,b)$ be a connected component of
    	$[0,\tau]\setminus D_n$, then either $\#\big(I\cap D_{n+2}\big)\leq1$ or
    	$(-1,1)\subseteq \pi_x\circ\gamma_{n+1}(I)\cup \pi_x\circ\gamma_{n+2}(I)$.

    	Actually, let $(c,d)=\pi_x\circ\gamma_n(I)$. If $0\notin(c,d)$, then $I\cap D_{n+1}=\emptyset$ and
    	$$
    	\pi_x\circ\gamma_{n+1}(I)=
    	 \pi_x\circ P_s^{n+1}\circ\gamma_0(I)
    	$$
    	is an open interval in $(-1,1)$. Thus there exists at most one point $s_0\in I$ satisfying $\pi_x\circ\gamma_{n+1}(s_0)=0$. This implies $\#\big(I\cap D_{n+2}\big)\leq1$.

    	If $0\in(c,d)$, then we denote $s_0\in I$ satisfying $\pi_x\circ\gamma_n(s_0)=\pi_x\circ P_{s_0}^n\circ\gamma_0(s_0)=0$. Then two open intervals $(a,s_0)$ and $(s_0,b)$ are two connected components of $I\setminus D_{n+1}$. Moreover, item~\ref{item:return map} of Claim \ref{clm:vector-field-family} shows that
    	$P_s|_{\Sigma\setminus\Sigma_z}\equiv
    	P|_{\Sigma\setminus\Sigma_z}$. This implies
    	$$
    	\lim_{s\rightarrow s_0^-}\pi_x\circ\gamma_{n+1}(s)=1,
    	\qquad {\rm and} \qquad
    	\lim_{s\rightarrow s_0^+}\pi_x\circ\gamma_{n+1}(s)=-1.
    	$$
    	Thus there exists $e,f\in(-1,1)$, such that
    	$$
    	\pi_x\circ\gamma_{n+1}\big( (a,s_0) \big)=(e,1),
    	\qquad {\rm and} \qquad
    	\pi_x\circ\gamma_{n+1}\big( (s_0,b) \big)=(-1,f).
    	$$
    	If both of them don't contain $0$, then we have $\#\big(I\cap D_{n+2}\big)=\#\big(I\cap D_{n+1}\big)=1$.

    	Otherwise, we have either  $0\in(e,1)$ or $0\in(-1,f)$. Thus either
    	$$
    	[0,1)\subset \pi_x\circ\gamma_{n+1}\big( (a,s_0) \big)
    	  \subset \pi_x\circ\gamma_{n+1}(I),
    	\qquad {\rm or} \qquad
    	(-1,0]\subset \pi_x\circ\gamma_{n+1}\big( (s_0,b) \big)
    	   \subset \pi_x\circ\gamma_{n+1}(I).
    	$$
    	In both cases, we can apply $P_s|_{\Sigma\setminus\Sigma_z}\equiv
    	P|_{\Sigma\setminus\Sigma_z}$ and show that
    	$$
    	(-1,1)\subseteq \pi_x\circ\gamma_{n+1}(I)\cup \pi_x\circ\gamma_{n+2}(I).
    	$$

    	Now we claim there exists some $n>0$ and a connected component $I$ of $[0,\tau]\setminus D_n$, such that $(-1,1)\subseteq \pi_x\circ\gamma_{n+1}(I)\cup \pi_x\circ\gamma_{n+2}(I)$. Otherwise, we have
    	$$
    	\# D_2\leq1, \qquad
    	\# D_4\leq\# D_2+\# D_2+1\leq 3,
    	\qquad \cdots \qquad
    	\# D_{2n}\leq \# D_{2n-2}+\# D_{2n-2}+1\leq 2^n-1
    	$$
    	This implies $[0,\tau]\setminus D_{2n}$ has at most $2^n$ connected components. So there exists a connected component $I_{2n}\subset [0,\tau]\setminus D_{2n}$, such that its length satisfies $|I_{2n}|\geq 2^{-n}\tau$.

    	Since $\pi_x\circ\gamma_{2n}$ is a diffeomorphism on each connected component of $[0,\tau]\setminus D_{2n}$, and Claim \ref{clm:1-dim-expanding} shows that
    	$$
    	\frac{{\rm d}\pi_x\circ\gamma_n(s)}{{\rm d}s}~=~x_n'(s)~\geq~\lambda^n\cdot x_0'(s),
    	\qquad \forall s\in I_{2n},
    	$$
    	the length of the interval $\pi_x\circ\gamma_n(I_{2n})$ satisfies
    	$|\pi_x\circ\gamma_n(I_{2n})|
    	  \geq \lambda^{2n}\cdot|\gamma_0(I_{2n})|$.

    	Finally,  item~\ref{item:esti-curve} of Claim \ref{clm:small-perturb} shows that
    	$$
    	|\gamma_0(I_{2n})|~\geq~\frac{1}{2}\cdot
    	 \frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=0}
    	 \cdot|I_{2n}|
    	 ~\geq\frac{1}{2}\cdot
    	 \frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=0}\cdot 2^{-n}\tau.
    	$$
    	Thus we have
    	$$
    	|\pi_x\circ\gamma_n(I_{2n})|~\geq~
    	\frac{\tau}{2}\cdot
    	\frac{{\rm d}\pi_x\circ \gamma_0(s)}{{\rm d}s}|_{s=0}
    	\cdot\left(\frac{\lambda^2}{2}\right)^n
    	~\longrightarrow~+\infty,
    	\qquad {\rm as} \quad n\rightarrow+\infty.
    	$$
    	This is absurd because $\pi_x\circ\gamma_n(I_{2n})\subseteq(-1,1)$. So there exists $n>0$ and a connected component $I$ of $[0,\tau]\setminus D_n$, such that $(-1,1)\subseteq \pi_x\circ\gamma_{n+1}(I)\cup \pi_x\circ\gamma_{n+2}(I)$. We take $N=n+2$ to complete the proof of this claim.
    \end{proof}

    For the critical point $p\in\Lambda$, recall that $l_p=\{x_p\}\times[-1-\epsilon,1+\epsilon]\subset\Sigma$  with $x_p\in(-1,1)$ (if $p=\sigma$, then $x_p=0$). Claim \ref{clm:cover} shows there exist $s\in[0,\tau]$ and $n>0$, such that
    $$
    x_p~=~\pi_x\circ P_s^n\circ\gamma_0(s)
     ~=~\pi_x\circ P_s^n\circ P_s(z).
    $$
    This implies for the vector field $Y=X_s\in\cV$, there exists $T>0$, such that $\phi_T^Y(z)\in l_p$.

    Since $X_s|_{M^3\setminus B(z,\delta)}\equiv X|_{M^3\setminus B(z,\delta)}$ and
    $B(z,\delta)\cap
    \left(\bigcup_{t\geq0}\phi_t^X(l_p)\cup\{\sigma\}\right)
    =\emptyset$, we have
    $l_p\subset W^s_{\it loc}(\orb(p),\phi_t^Y)$.
    The fact $z\in W^u(\sigma,\phi_t^Y)=W^u(\sigma,\phi_t^{X_s})$ implies
    $$
    \orb(z,\phi_t^Y)~\subset~
    W^u(\sigma,\phi_t^Y) \cap W^s(\orb(p),\phi_t^Y).
    $$
    This completes the proof of this theorem.
\end{proof}

\subsection{Proof of Theorem~\ref{Thm:A} and Corollary~\ref{Cor:Cr-entropy-support}}

We split the proof of Theorem~\ref{Thm:A} into two parts: the residual part and the dense part. We first prove the following lemma which is key for the residual part.

\begin{lemma}\label{Lem:density in erg}
	For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, there exists a $C^r$-residual subset $\mathcal{R}^r\subset \mathscr{X}^r(M^3)$ such that if $X\in\mathcal{R}^r$ admits a Lorenz attractor $\Lambda$, then $\mathcal{M}_{erg}(\Lambda)\subset\overline{\mathcal{M}_{per}(\Lambda)}$.
\end{lemma}

\begin{proof}
	Assume $X\in\mathscr{X}^r(M^3)$ admits a Lorenz attractor $\Lambda$ with attracting region $U$ and let $\mathcal{U}$ be the $C^r$ neighborhood of $X$ satisfying Proposition~\ref{prop:Lorenz} for $\Lambda$ and $U$.
    Recall that  $\Lambda$ is a homoclinic class and any two periodic orbits are homoclinically related by Proposition~\ref{prop:Lorenz}.
    Applying Corollary~\ref{Cor:dense-loop} and Proposition~\ref{Prop:loop}, we have the following claim.
	\begin{claim}\label{Claim:approach singularity}
		For any $C^r$-neighborhood $\mathcal{V}$ of $X$ and any $\varepsilon>0$, there exists $Y\in\mathcal{V}$ which admits a periodic point $p$ satisfying that  $d(\delta_{\orb(p,\phi^Y_t)},\delta_{\sigma_Y})<\varepsilon$.
	\end{claim}

    Lemma~\ref{Lem:density in erg} concludes by a standard Baire argument. Take a countable basis $\{O_n\}_{n\in\mathbb{N}^+}$ of $M^3$ and let $\{U_n\}_{n\in\mathbb{N}^+}$ be the countable family which consists of all the possible unions of finitely many elements in $\{O_n\}_{n\in\mathbb{N}^+}$. For each pair of integers $n,m\in\mathbb{N}^+$, we define the following two subsets of $\mathscr{X}^r(M^3)$:
    \begin{itemize}
    	\item $\mathcal{F}_{m,n}$: $Y\in \mathcal{F}_{m,n}$ if and only if $Y$ admits a hyperbolic singularity $\sigma_Y$ in $U_m$ and there exists a periodic point $q_Y$ of $Y$ such that $d(\delta_{\orb(q_Y,\phi^Y_t)},\delta_{\sigma_Y})<\frac{1}{n}$;

    	\item $\mathcal{N}_{m,n}$: $Y\in \mathcal{F}_{m,n}$ if and only if there exists a neighborhood $\mathcal{V}_Y\subset\mathscr{X}^r(M^3)$ of $Y$ such that for any $Z\in\mathcal{V}_Y$, one of the following cases satisfies
    	\begin{itemize}
    		\item[--] either $Z$ admits no hyperbolic singularity in $U_m$;

    		\item[--] or for any singularity $\sigma_Z$ of $Z$ in $U_m$ and for any periodic point $q_Z$ of $Z$, it satisfies $d(\delta_{\orb(q_Z,\phi^Z_t)},\delta_{\sigma_Z})\geq\frac{1}{n}$.
    	\end{itemize}
    \end{itemize}

    \begin{claim}\label{Claim:Cr-open-dense}
    	$\mathcal{F}_{m,n}\cup \mathcal{N}_{m,n}$ is open and dense in $\mathscr{X}^r(M^r)$.
    \end{claim}
    \begin{proof}
    	By the robustness of hyperbolic critical elements, we have that $\mathcal{F}_{m,n}$ is open in $\mathscr{X}^r(M^3)$. The openness of $\mathcal{N}_{m,n}$ is by definition.

    	For any vector field $Y\in\mathscr{X}^r(M^3)\setminus\mathcal{N}_{m,n}$, there exists a sequence of vector fields $Y_i\in\mathscr{X}^r(M^3)$ such that $Y_i$ admits a hyperbolic singularity $\sigma_{Y_i}$ in $U_m$ and there exists a periodic point $q_{Y_i}$ of $Y_i$ such that $d(\delta_{\orb(q_{Y_i})},\delta_{\sigma_{Y_i}})<\frac{1}{n}$. Again by the robustness of hyperbolic critical elements, we have that $Y_i\in \mathcal{F}_{m,n}$ which implies that $Y\in \overline{\mathcal{F}_{m,n}}$. This shows that $\mathcal{F}_{m,n}\cup \mathcal{N}_{m,n}$ is dense in $\mathscr{X}^r(M^3)$.
    \end{proof}

    By Claim~\ref{Claim:Cr-open-dense}, the set
    $$\mathcal{R}^r=\bigcap_{m,n\in\mathbb{N}^+}(\mathcal{F}_{m,n}\cup \mathcal{N}_{m,n})$$
    is a residual subset in $\mathscr{X}^r(M^3)$.
    Let $X\in\mathcal{R}^r$ and $\Lambda$ be a Lorenz-like attractor of $X$ with singularity $\sigma$. We show that $\mathcal{M}_{erg}(\Lambda)\subset \overline{\mathcal{M}_{per}(\Lambda)}$.

    Take $\mu\in \mathcal{M}_{erg}(\Lambda)$. If $\mu\neq\delta_{\sigma}$, then $\mu\in\overline{\mathcal{M}_{per}(\Lambda)}$ by Proposition~\ref{Prop:dense periodic measure}. If otherwise $\mu=\delta_\sigma$. By the hyperbolicity of $\sigma$, there exist a neighborhood $U_\sigma$ of $\sigma$ and a neighborhood $\mathcal{U}_X\subset\mathscr{X}^r(M^3)$ of $X$ such that any $Y\in\mathcal{U}_X$ admits a unique singularity $\sigma_Z$ in $U_\sigma$ where $\sigma_Z$ is the continuation of $\sigma$. Take $m\in\mathbb{N}^+$ such that $U_m\subset U_{\sigma}$ is a neighborhood of $\sigma_Z$ for all $Z\in \mathcal{U}_X$. By Claim~\ref{Claim:approach singularity}, we have that $X\notin \mathcal{N}_{m,n}$ for any $n\in\mathbb{N}^+$. Hence $X\in \mathcal{F}_{m,n}$ for any $n\in\mathbb{N}^+$. This implies that there exists a sequence of periodic points $p_n$ of $X$ such that $\delta_{\orb(p_n)}$ converges to $\delta_{\sigma}$ in the weak*-topology. Moreover, since $\Lambda$ is a Lorenz attractor, we have that $p_n\in\Lambda$ for each $n$ large enough. This shows that $\delta_{\sigma}\in \overline{\mathcal{M}_{per}(\Lambda)}$ and completes the proof of Lemma~\ref{Lem:density in erg}.
\end{proof}

Recall by Proposition~\ref{prop:Lorenz}, any geometric Lorenz attractor of a vector field $X\in\mathscr{X}^r(M^3)$ is a homoclinic class and any two periodic orbits are homoclinically related. Hence we have the following lemma which is a combination of~\cite[Proposition 4.7 $\&$ Remark 4.6]{abc}.

\begin{lemma}\label{Lem:convex of periodic measure}
	Assume $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$ and $X\in\mathscr{X}^r(M^3)$ admits a geometric Lorenz attractor $\Lambda$, then $\overline{\mathcal{M}_{per}(\Lambda)}$ is convex.
\end{lemma}

Now we manage to prove  Theorem~\ref{Thm:A} and Corollary~\ref{Cor:Cr-entropy-support}.

\begin{proof}[Proof of Theorem~\ref{Thm:A}]
	Let $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$. We splits the proof of Theorem~\ref{Thm:A} into residual part and dense part.
	\paragraph{Residual part:}
	Let $\mathcal{R}^r\subset\mathscr{X}^r(M^3)$ be the residual subset from Lemma~\ref{Lem:density in erg}. Assume $X\in\mathcal{R}^r$ admits a geometric Lorenz attractor $\Lambda$.
	We first show that $$\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{erg}(\Lambda)}=\overline{\mathcal{M}_{per}(\Lambda)}.$$

	Take $\mu\in \mathcal{M}_{inv}(\Lambda)$.
	By the Ergodic Decomposition Theorem, there exists a sequence of measures $\mu_n$ converging to $\mu$ in the weak*-topology where $\mu_n\in\mathcal{M}_{inv}(\Lambda)$ is a convex sum of finitely many ergodic measures supported on $\Lambda$ for each $n\in\mathbb{N}$. By Lemma~\ref{Lem:density in erg}, there exists a sequence of measures $\nu_n$ converging to $\mu$ in the weak*-topology where $\nu_n\in\mathcal{M}_{inv}(\Lambda)$ is a convex sum of finitely many periodic measures supported on $\Lambda$ for each $n\in\mathbb{N}$.
	By Lemma~\ref{Lem:convex of periodic measure},  we have that $\overline{\mathcal{M}_{per}(\Lambda)}$ is convex. Hence $\mu\in\overline{\mathcal{M}_{per}(\Lambda)}$. This implies $\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{erg}(\Lambda)}=\overline{\mathcal{M}_{per}(\Lambda)}$.

	Note that $\mathcal{M}_{erg}(\Lambda)$ coincides with $\overline{\mathcal{M}_{per}(\Lambda)}\cap \mathcal{M}_{erg}(\Lambda)$. Hence the path connectedness of $\mathcal{M}_{erg}(\Lambda)$ follows directly from Proposition~\ref{prop:Lorenz} and Proposition~\ref{Prop:connectedness}.

    \paragraph{Dense part:}

	By Proposition~\ref{prop:Lorenz} and robustness of hyperbolic singularities, one can easily prove that there exists an open and dense subset $\mathcal{O}$ in $\mathscr{X}^r(M^3)$ such that the number of geometric Lorenz attractors of every $X\in\mathcal{O}$ is robustly constant. That is to say, for any $X\in\mathcal{O}$, there exists a $C^r$ neighborhood $\mathcal{V}_X$ of $X$ such that the number of geometric Lorenz attractors of any $Z\in \mathcal{V}_X$ is a constant which equals that of $X$.

	To prove the dense part, we only need to show that for any $X\in\mathcal{O}$ and for any $C^r$ neighborhood $\mathcal{V}$ of $X$, there exists a vector field $Y\in\mathcal{V}$ such that every geometric Lorenz attractor $\Lambda_Y$ of $Y$ satisfies
	$$\overline{\mathcal{M}_{per}(\Lambda_Y)}\subsetneqq \overline{\mathcal{M}_{erg}(\Lambda_Y)}\subsetneqq \mathcal{M}_{inv}(\Lambda_Y). $$

	By hyperbolicity of singularities, we have that $X$ admits  only finitely many geometric Lorenz attractors $\Lambda_1,\Lambda_2,\cdots,\Lambda_k$ with pairwise disjoint attracting regions $U_1,U_2,\cdots,U_k$. Shrinking $\mathcal{V}$ if necessary, we assume that every $Z\in\mathcal{V}$ admits exactly $k$ geometric Lorenz attractors $\Lambda_{1,Z},\Lambda_{2,Z},\cdots,\Lambda_{k,Z}$ with the same attracting regions $U_1,U_2,\cdots,U_k$. Moreover, we assume that $\mathcal{V}$ satisfies the property $(F)$ in Section~\ref{Section:connecting lemmas}.
	For each $Z\in\mathcal{V}$ and for each $i=1,2,\cdots,k$, we denote by $\sigma_i^{Z}$ the unique singularity in $\Lambda_{i,Z}$ and denote by $\sigma_i=\sigma_i^{X}$ for simplicity.
	We first show the following claim.

	\begin{claim}\label{Claim:Y_i}
		For each $i=1,2,\cdots,k$, there exists $Y_i\in\mathcal{V}$ satisfying that $Y_i|_{M\setminus U_i}=X|_{M\setminus U_i}$ and $W^u(\sigma_i^{Y_i},\phi_t^{Y_i})\setminus \{\sigma_i^{Y_i}\}\subset W^s(\orb(p_i^{Y_i},\phi_t^{Y_i}))$ where $p_i^{Y_i}$ is a hyperbolic periodic point of $\phi_t^{Y_i}$.
	\end{claim}
    \begin{proof}[Proof of Claim~\ref{Claim:Y_i}]
    	Fix $i\in\{1,2,\cdots,k\}$. We denote by $\Lambda=\Lambda_i,U=U_i$ and $\sigma=\sigma_i$ to simplify notations.

    	Let $\Sigma\subset U$ be the cross section of $\Lambda$ and $z^+,z^-\in W^u(\sigma)\cap\Sigma$ be the two different points satisfying $\phi_t^X(z^+),\phi_t^X(z^-)\notin\Sigma$ for any $t<0$.
    	Take a hyperbolic periodic point $p\in\Lambda$.
    	Take a constant $\delta^+>0$ such that
    	\begin{itemize}
    	    \item $B(z^{+},\delta^+)\subset U\setminus (\orb(p,\phi^X_t)\cup\{\sigma\})$,

    	    \item $\phi^X_t(z^-)\notin B(z^{+},\delta^+)$ for any $t\leq 0$.
    	\end{itemize}

    	By Theorem~\ref{thm:Lorenz-connecting}, there exists $Y^+\in\mathcal{V}$ satisfying that
    	$$Y^+|_{M\setminus B(z^{+},\delta^+)}=X|_{M\setminus B(z^{+},\delta^+)} \text{~~and~~} \orb(z^+,\phi_t^{Y^+})\subset W^u(\sigma,\phi_t^{Y^+})\cap W^s(\orb(p),\phi_t^{Y^+}).$$

        Note that $\mathcal{V}$ is also a $C^r$-neighborhood of $Y^+$ with $z^-\in W^u(\sigma,\phi^{Y^+}_t)$ and $p\in\per(\phi^{Y^+}_t)$. Moreover, by the choice of $\mathcal{V}$, the maximal invariant compact set of $\phi^{Y^+}_t$ is still a geometric Lorenz attractor with singularity $\sigma$ and containing $\orb(p)$ in it.
        We take another constant $\delta^->0$ satisfying that
        \begin{itemize}
        	\item $B(z^{-},\delta^-)\subset U\setminus (\orb(p,\phi^X_t)\cup\{\sigma\})$,

        	\item $\phi^{Y^+}_t(z^+)\notin B(z^{-},\delta^-)$ for any $t\in\mathbb{R}$.
        \end{itemize}
    	Applying Theorem~\ref{thm:Lorenz-connecting} again for the vector field $Y^+$ and the neighborhood $\mathcal{V}$, there exists $Y_i\in\mathcal{V}$ such that
    	$$Y_i|_{M\setminus B(z^{-},\delta^-)}=Y^+|_{M\setminus B(z^{-},\delta^-)} \text{~~and~~} \orb(z^-,\phi_t^{Y_i})\subset W^u(\sigma,\phi_t^{Y_i})\cap W^s(\orb(p),\phi_t^{Y_i}).$$
    	By the choice of $\delta^-$, we know that it satisfies also
    	$$\orb(z^+,\phi_t^{Y_i})\subset W^u(\sigma,\phi_t^{Y_i})\cap W^s(\orb(p),\phi_t^{Y_i}).$$
    	The vector field $Y_i$ satisfies Claim~\ref{Claim:Y_i}.
    \end{proof}

	Finally, let $Y$ be the composed perturbation of $X$ satisfying that
	$$Y|_{U_i}=Y_i|_{U_i}, ~\text{ for each~} i=1,2,\cdots,k,~~~\text{ and~~~}  Y|_{M\setminus (\bigcup_{1\leq i\leq k}U_i)}=X|_{M\setminus (\bigcup_{1\leq i\leq k}U_i)}.$$
	By the property $(F)$, we have that $Y\in\mathcal{V}$ since the attracting regions $\{U_i\}_{1\leq i\leq k}$ are pairwise disjoint. By Proposition~\ref{Prop:kick-out singularity}, for each $1\leq i\leq k$, the Dirac measure $\delta_{\sigma_i}=\delta_{\sigma_i^Y}$ is isolated in $\mathcal{M}_{erg}(Y)$.  In particular, we have
	$$\overline{\mathcal{M}_{per}(\Lambda_{i,Y})}\subsetneqq \overline{\mathcal{M}_{erg}(\Lambda_{i,Y})}\subsetneqq \mathcal{M}_{inv}(\Lambda_{i,Y}). $$
	This completes the dense part and hence the proof of Theorem~\ref{Thm:A}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{Cor:Cr-entropy-support}]
	Let $\mathcal{R}^r$ be the residual subset of $\mathscr{X}^r(M^3)$ from Theorem~\ref{Thm:A}.
	Assume  $X\in \mathcal{R}^r$ admits a geometric Lorenz attractor $\Lambda$. By~\cite[Proposition 5.1]{abc}, $\mathcal{M}_{erg}(\Lambda)$ is a $G_{\delta}$ subset in $\mathcal{M}_{inv}(\Lambda)$. Since $\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{erg}(\Lambda)}$, we have that $\mathcal{M}_{erg}(\Lambda)$ is a residual subset in $\mathcal{M}_{inv}(\Lambda)$. On the other hand, periodic orbits are dense in $\Lambda$ since $\Lambda$ is a homoclinic class. Then by~\cite[Proposition 5.3]{abc}, the set of invariant measures with support equal to $\Lambda$ is a residual subset $\mathcal{M}_1$ in $\mathcal{M}_{inv}(\Lambda)$. By~\cite{PYY}, the entropy map
	$$h_{(\cdot)}\colon \mathcal{M}_{inv}(\Lambda)\rightarrow \mathbb{R}$$
	$$~~~~~~~~~~\mu\mapsto h_{\mu}(X)$$
	is upper semi-continuous. Hence the continuity points of the entropy map $h_{(\cdot)}$ form a residual subset $\mathcal{M}_2$ in $\mathcal{M}_{inv}(\Lambda)$. Moreover, since $\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{per}(\Lambda)}$ and every periodic measure has $0$-entropy, we have that every $\mu\in\mathcal{M}_2$ satisfies $h_{\mu}(X)=0$.
	Let $$\mathcal{M}_{res}=\mathcal{M}_{erg}(\Lambda)\cap \mathcal{M}_1\cap \mathcal{M}_2.$$
	Then $\mathcal{M}_{res}$ is a residual subset in $\mathcal{M}_{inv}(\Lambda)$ and the residual part  concludes.

	To prove the dense part, since $\mathcal{M}_{inv}(\Lambda)=\overline{\mathcal{M}_{per}(\Lambda)}$, we only need to show that any periodic measure supported on $\Lambda$ is approximated by ergodic measures in $\mathcal{M}_{erg}(\Lambda)$ with positive entropy.
	The idea follows~\cite[Theorem 3.5 part (c)]{abc} and we give an explanation here.
	For any periodic orbit $\orb(p)$ in $\Lambda$, we have that $\Lambda=H(p)$ by Proposition~\ref{prop:Lorenz}. Hence $\orb(p)$ exhibits some transverse homoclinic orbit which implies that there are hyperbolic horseshoes contained in $\Lambda$ arbitrarily close to this homoclinic orbit. That means points in the hyperbolic horseshoes spends arbitrarily large fractions of their orbits shadowing $\orb(p)$ as closely as required. Hence measures supported on such hyperbolic horseshoes are close to $\delta_{\orb(p)}$ in the weak*-topology. On the other hand, every hyperbolic horseshoe must support an ergodic measure with positive entropy. Thus $\delta_{\orb(p)}$ is approximated by ergodic measures in $\mathcal{M}_{erg}(\Lambda)$ with positive entropy.
\end{proof}

\begin{remark}\label{Rem:entropy and support}
	In the proof of Corollary~\ref{Cor:Cr-entropy-support}, the residual subsets $\mathcal{M}_1$ and $\mathcal{M}_2$ of $\mathcal{M}_{inv}(\Lambda)$ exist for every geometric Lorenz attractor $\Lambda$ of every $X\in\mathscr{X}^r(M^3)$. Therefore the measures with zero entropy and full support form a residual subset $\mathcal{M}_0$ in $\mathcal{M}_{inv}(\Lambda)$ for every geometric Lorenz attractor $\Lambda$. Here is an explanation:

	\noindent First, periodic points are dense in $\Lambda$ since it is a homoclinic class by Proposition~\ref{prop:Lorenz}. By ~\cite[Proposition 5.3]{abc}, the set of invariant measures with full support  is a residual subset $\mathcal{M}_1$ in $\mathcal{M}_{inv}(\Lambda)$. Second, by Proposition~\ref{Prop:dense periodic measure},  the set $\mathcal{M}_{per}(\Lambda)\cup \{\delta_{\sigma}\}$ is dense in $\mathcal{M}_{erg}(\Lambda)$. This implies that every $\mu\in\mathcal{M}_{inv}(\Lambda)$ is the limit of invariant measures $\{\mu_n\}$ where each $\mu_n$ is the convex sum of finitely many periodic measures  supported on $\Lambda$ together with $\delta_{\sigma}$. Thus by the upper semi-continuity of the entropy map~\cite{PYY}, the invariant measures with zero entropy form a residual subset $\mathcal{M}_2$ in $\mathcal{M}_{inv}(\Lambda)$.
\end{remark}

\section{Lorenz-like attractors for $C^1$-vector fields}\label{Section:C1}
In this section, we prove  Theorem~\ref{Thm:B} and Corollary~\ref{Cor:entropy-support}.
The idea follows similar arguments as in Theorem~\ref{Thm:A} and Corollary~\ref{Cor:Cr-entropy-support}. For completeness, we give short explanations in this section. We state the following two lemmas as a preparation.
The first one is similar to Lemma~\ref{Lem:density in erg}.

\begin{lemma}\label{Lem:density of ergodic measures}
	There exists a residual subset  $\mathcal{R}\subset\mathscr{X}^1(M)$, such that for any $X\in\mathcal{R}$,  if $\Lambda$ is a  Lorenz-like attractor of $X$, then $\mathcal{M}_{erg}(\Lambda)\subset\overline{\mathcal{M}_{per}(\Lambda)}$.
\end{lemma}

\begin{proof}
	Let $\mathcal{R}_0\subset\mathscr{X}^1(M)$ be the residual subset from Theorem~\ref{Thm:generic}.
	Assume first that $X\in\mathcal{R}_0$ and $\Lambda$ is a Lorenz-like attractor of $X$.
	Let $T_{\Lambda}M=E^{ss}\oplus E^{cu}$ be the singular hyperbolic splitting over $\Lambda$.
	By item~\ref{generic:chain class} and~\ref{generic:lorenz-like is h-class} of Theorem~\ref{Thm:generic}, $\Lambda$ is a homoclinic class.

	We prove the following claim first which compares to Claim~\ref{Claim:approach singularity}.
	\begin{claim}\label{Claim:C1-singular measure}
		Assume $\sigma$ is a hyperbolic singularity of $X$ in $\Lambda$. For any neighborhood $\mathcal{U}\subset\mathscr{X}^1(M)$ of $X$ and for any $\eta>0$, there exists $Y\in \mathcal{U}$ which admits a hyperbolic periodic point $p$ such that $Y$ coincides with $X$ on a small neighborhood of $\sigma$ and $d(\mu_{p},\delta_\sigma)<\eta$, where $\mu_{p}$ is the periodic measure associated to $\orb(p,Y)$.
	\end{claim}
	\begin{proof}
		Replacing $\mathcal{U}$ with a subset if necessary, we assume that $\mathcal{U}$ satisfies the property $(F)$ in Section~\ref{Section:connecting lemmas}.

		Let $L>0$ be the  constant from Theorem~\ref{Thm:connecting lemma} associated to $(X,\mathcal{U})$ and let $\mathcal{U}_1\subset\mathcal{U}$ be the neighborhood from Remark~\ref{Rem:uniform connecting}.
		Since $\Lambda=C(\sigma)$ is a nontrivial chain recurrence class, there exist $x\in \Lambda\cap( W^u(\sigma)\setminus\{\sigma\})$ and $y\in \Lambda\cap(W^s(\sigma)\setminus\{\sigma\})$ (see~\cite[Lemma 2.7]{bgy}). By Theorem~\ref{Thm:chain-connecting}, there exists $Y_1\in\mathcal{U}_1$ such that $y$ is on the positive orbit of $x$ under $\phi_t^{Y_1}$ and $Y_1=X|_{U}$ where $U$ is a neighborhood of $\sigma$. That is to say $\orb(x,\phi_t^{Y_1})\subset W^u(\sigma,\phi_t^{Y_1})\cap W^s(\sigma,\phi_t^{Y_1})$. A detailed explanation can be found in~\cite[Lemma 4.2]{sgw}. Then Claim~\ref{Claim:C1-singular measure} follows from Proposition~\ref{Prop:loop}.
	\end{proof}

	Lemma~\ref{Lem:density of ergodic measures} follows by a standard Baire argument as we have done in Lemma~\ref{Lem:density in erg}. Take a countable basis $\{O_n\}_{n\in\mathbb{N}^+}$ of $M$ and let $\{U_n\}_{n\in\mathbb{N}^+}$ be the countable family which consists of all the possible unions of finitely many elements in $\{O_n\}_{n\in\mathbb{N}^+}$. For each pair of integers $n,m\in\mathbb{N}^+$, we define the following two subsets of $\mathscr{X}^1(M)$:
	\begin{itemize}
		\item $\mathcal{F}_{m,n}$: $Y\in \mathcal{F}_{m,n}$ if and only if $Y$ admits a hyperbolic singularity $\sigma_Y$ in $U_m$ and there exists a periodic point $q_Y$ of $Y$ such that $d(\delta_{\orb(q_Y)},\delta_{\sigma_Y})<\frac{1}{n}$;

		\item $\mathcal{N}_{m,n}$: $Y\in \mathcal{F}_{m,n}$ if and only if there exists a neighborhood $\mathcal{V}_Y\subset\mathscr{X}^1(M)$ of $Y$ such that for any $Z\in\mathcal{V}_Y$, one of the following cases satisfies
		\begin{itemize}
			\item[--] either $Z$ admits no hyperbolic singularity in $U_m$;

			\item[--] or for any singularity $\sigma_Z$ of $Z$ in $U_m$ and for any periodic point $q_Z$ of $Z$, it satisfies $d(\delta_{\orb(q_Z)},\delta_{\sigma_Z})\geq\frac{1}{n}$.
		\end{itemize}
	\end{itemize}

	Similar to Claim~\ref{Claim:Cr-open-dense}, we have the following claim and we omit the proof.

	\begin{claim}\label{Claim:C1-open-dense}
		$\mathcal{F}_{m,n}\cup \mathcal{N}_{m,n}$ is open and dense in $\mathscr{X}^1(M)$.
	\end{claim}

	By Claim~\ref{Claim:C1-open-dense}, the set
	$$\mathcal{R}=\mathcal{R}_0\cap\left(\bigcap_{m,n\in\mathbb{N}^+}(\mathcal{F}_{m,n}\cup \mathcal{N}_{m,n})\right)$$
	is a residual subset in $\mathscr{X}^1(M)$.
	Let $X\in\mathcal{R}$ and $\Lambda$ be a Lorenz-like attractor of $X$. With similar arguments as done in the last paragraph of the proof of Lemma~\ref{Lem:density in erg}, we have that $\mathcal{M}_{erg}(\Lambda)\subset \overline{\mathcal{M}_{per}(\Lambda)}$.

\end{proof}

Next is a perturbation lemma in the $C^1$-topology to make the two branches of the  unstable manifolds of a singularity with co-index one lie in the stable manifold of a periodic orbit.
\begin{lemma}\label{Lem:C1-connection-S-P}
	Let $\mathcal{R}_0$  be the residual subset of $\mathscr{X}^1(M)$ from Theorem~\ref{Thm:generic}.
	Assume $X\in\mathcal{R}_0$ and $\sigma\in\sing(X)$ such that $C(\sigma)$ is a Lorenz-like attractor with attracting region $U_0$ whose singular hyperbolic  splitting $T_{C(\sigma)}M=E^{ss}\oplus E^{cu}$ satisfies $\dim(E^{cu})=2$. Then for any neighborhood $\mathcal{U}$ of $X$ in $\mathscr{X}^1(M)$, there exists $Y\in\mathcal{U}$ such that $Y$ coincides with $X$ on $M\setminus U_0$ and $W^u(\sigma_Y)\setminus\{\sigma_Y\}\subset W^s(\orb(p))$ where $\sigma_Y$ is the continuation of $\sigma$ and $p\in C(\sigma_Y)\cap \per(Y)$. Moreover, $C(\sigma_Y)$ is a Lorenz-like attractor of $Y$.
\end{lemma}

\begin{proof}
	Take $\mathcal{U}_0$ to be the neighborhood of $X$ from the definition of Lorenz-like attractor for $\Lambda$. That is to say for any $Y\in \mathcal{U}_0$, the maximal $\phi_t^Y$-invariant set in $U_0$ is transitive.
	Without loss of generality, we assume that $\mathcal{U}\subset \mathcal{U}_0$.
	Then $C(\sigma)$ contains a hyperbolic periodic point $p$ and $C(\sigma)=H(p)$ by item~\ref{generic:lorenz-like is h-class} of Theorem~\ref{Thm:generic}.
	Let $T_{\sigma}M=E^s(\sigma)\oplus E^u(\sigma)$ be the hyperbolic splitting. Notice that $T_{\sigma}M=E^{ss}(\sigma)\oplus E^{cu}(\sigma)$ which is associated to the singular hyperbolic splitting over $C(\sigma)$. Since $\dim(E^{cu})=2$, we have that $\dim(E^u(\sigma))=1$. Thus $W^u(\sigma)\setminus\{\sigma\}$ decomposes into two components that are contained in $\Lambda$ and let $\orb(x)$ and $\orb(y)$ be the two orbits.

	Shrinking $\mathcal{U}$ if necessary, we assume that $\mathcal{U}$ satisfies the property {\it (F)} in Section~\ref{Section:connecting lemmas}.
	Let $L$ be the constant associated to $(X,\mathcal{U})$ by Theorem~\ref{Thm:connecting lemma} and let $\mathcal{U}_1$ be the neighborhood of $X$ from Remark~\ref{Rem:uniform connecting} such that $L$ is valid for every vector field in $\mathcal{U}_1$ with respect to $(X,\mathcal{U})$. Take $L_1$ to be the constant associated to $(X,\mathcal{U}_1)$ by Theorem~\ref{Thm:connecting lemma}. Let $z\in W^s(\orb(p))\cap W^u(\orb(p))$ be a transverse homoclinic point of $\orb(p)$. Then $z\in H(p)=C(\sigma)$.

	\begin{claim}
		There exits $Y_1\in\mathcal{U}_1$ satisfying that
		\begin{itemize}
			\item $Y_1|_{M\setminus U_0}=X|_{M\setminus U_0}$;
			\item $\phi_t^{Y_1}$ coincides with $\phi_t^X$ on $\{\sigma\}\cup\orb(p)\cup\orb(z)$;
			\item $\phi_t^{Y_1}$ coincides with $\phi_t^X$ on $\orb^-(x,\phi_t^X)$ and $\orb^-(y,\phi_t^X)$, which implies $\orb(x,\phi_t^{Y_1})$ and $\orb(y,\phi_t^{Y_1})$ are still the two components of $W^u(\sigma,\phi_t^{Y_1})\setminus\{\sigma\}$;
			\item $x\in W^s(\orb(p),\phi_t^{Y_1})$.
		\end{itemize}
	\end{claim}
	\begin{proof}
		Denote by $x_1=\phi_1^X(x)$. Take a neighborhood $U_{x_1}$ of $x_1$, a neighborhood $U_\sigma$ of $\sigma$ and a neighborhood $U_{\orb(p)}$ of $\orb(p)$ such that denote by  $\Delta_1=\bigcup_{0\leq t\leq L_1}\phi_t^X(U_{x_1})$, then
		\begin{itemize}
			\item[--] $\Delta_1\subset U_0$;
			\item[--] $U_\sigma\cap U_{\orb(p)}=\emptyset$ and $\Delta_1\cap (U_\sigma\cup U_{\orb(p)})=\emptyset$;
			\item[--] $\Delta_1$ is disjoint with $\orb^-(x,\phi_t^X)$, $\orb^-(\phi_{L+1}^X(y),\phi_t^X)$ and $\orb(z)$.
		\end{itemize}
		Take a smaller neighborhood $V_{x_1}\subset U_{x_1}$ of $x_1$ that satisfies the properties in Theorem~\ref{Thm:connecting lemma} associated to $(x_1,X,\mathcal{U}_1)$.

		Recall that $C(\sigma)=H(p)$. Hence there exist a point $z_1\in W^s(\orb(p),\phi_t^X)\cap U_{\orb(p)}$ and $T_1>0$ such that $\phi_{-T_1}^X(z_1)\in V_{x_1}$ and $\phi_{t}^X(z_1)\in U_{\orb(p)}$ for all $t\geq 0$. By Theorem~\ref{Thm:connecting lemma}, there exits $Y_1\in\mathcal{U}_1$ such that $Y_1$ coincides with $X$ on $M\setminus\Delta_1$ and $z_1$ is on the positive orbit of $x$ under $\phi_t^{Y_1}$. The properties in the claim are satisfied by the choice of the neighborhoods $U_{x_1}$, $U_\sigma$ and $U_{\orb(p)}$.
	\end{proof}

	By the choice of $\mathcal{U}_0$ and $U_0$, we have that the maximal invariant set in $U_0$ with respect to $\phi_t^{Y_1}$ is still a singular hyperbolic  attractor and coincides with $C(\sigma,\phi_t^{Y_1})$. Thus $\orb(y,\phi_t^{Y_1})\subset W^u(\sigma,\phi_t^{Y_1})\subset C(\sigma,\phi_t^{Y_1})$ and $p\in C(\sigma,\phi_t^{Y_1})$. Moreover, the point $z$ is still a homoclinic point of $\orb(p)$, and hence $z\in C(\sigma,\phi_t^{Y_1})$.

	Recall that $\Delta_1$ is disjoint with $\orb^-(\phi_{L+1}^{Y_1}(y),\phi_t^X)$ and $\orb(z,\phi_t^{Y_1})$. Denote by $y_1=\phi_{1}^{Y_1}(y)$ and $z_1=\phi_{-L-1}^{Y_1}(z)$. We can take neighborhoods $U_{y_1}$ of $y_1$ and $U_{z_1}$ of $z_1$ such that denote by $\Delta_2=\bigcup_{0\leq t\leq L}\phi_t^{Y_1}(U_{y_1})$ and $\Delta_3=\bigcup_{0\leq t\leq L}\phi_t^{Y_1}(U_{z_1})$, then it satisfies:
	\begin{itemize}
		\item[--] $\Delta_2\cup\Delta_3\subset U_0$ and $\Delta_1,\Delta_2,\Delta_3$ are pairwise disjoint;
		\item[--] $\Delta_2\cup\Delta_3$ is disjoint with $\orb^-(y,\phi_t^{Y_1})$ and $\orb^+(z,\phi_t^{Y_1})$;
		\item[--] $(\Delta_2\cup\Delta_3)\cap \overline{\orb(x,\phi_t^{Y_1})}=\emptyset$.
	\end{itemize}
	Take $V_{y_1}\subset U_{y_1}$ and $V_{z_1}\subset U_{z_1}$ be smaller neighborhoods of $y_1$ and $z_1$ respectively such that the two triples $(y_1,U_{y_1},V_{y_1})$ and $(z_1,U_{z_1},V_{z_1})$ satisfy Theorem~\ref{Thm:connecting lemma} with respect to $(Y_1,\mathcal{U})$. Since $C(\sigma,Y_1)$ is transitive, there exists a point $w\in V_{y_1}$ and a constant $T>0$ such that $\phi_{T}^{Y_1}(w)\in V_{z_1}$. By Theorem~\ref{Thm:connecting lemma}, there exists $Y_2\in\mathcal{U}$ such that the point $\phi_{T}^{Y_1}(w)$ is on the positive orbit of $y$ under $\phi_{t}^{Y_2}$ and $Y_2$ coincides with $Y_1$ outside $\Delta_2$. Since $\mathcal{U}$ satisfies the property {\it (F)} and $\Delta_1,\Delta_2,\Delta_3$ are pairwise disjoint, by Theorem~\ref{Thm:connecting lemma} again, there exists $Y\in\mathcal{U}$ such that $Y$ coincides with $Y_2$ outside $\Delta_3$ and $z$ is on the positive orbit of $y$ under $\phi_t^Y$.

	To conclude, both the two components $\orb(x,\phi_t^Y)$ and $\orb(y,\phi_t^Y)$ of $W^u(\sigma,\phi_t^Y)\setminus\{\sigma\}$ lie in $W^s(\orb(p),\phi_t^Y)$ and $Y_{M\setminus U_0}=X_{M\setminus U_0}$. Moreover, $C(\sigma,Y)$ is a Lorenz-like attractor since $Y\in\mathcal{U}_0$.
\end{proof}

Now we manage to prove Theorem~\ref{Thm:B} and Corollary~\ref{Cor:entropy-support}.
\begin{proof}[Proof of Theorem~\ref{Thm:B} and Corollary~\ref{Cor:entropy-support}]
	Let $\mathcal{R}$ be the residual subset of $\mathscr{X}^1(M)$ obtained from Lemma~\ref{Lem:density of ergodic measures}. Note that every $X\in\mathcal{R}$ also satisfies Theorem~\ref{Thm:generic}. Thus any Lorenz like attractor of any $X\in\mathcal{R}$  is a homoclinic class.
	Then the residual part of Theorem~\ref{Thm:B} follows the same arguments as that of Theorem~\ref{Thm:A} except in one place where  we apply Lemma~\ref{Lem:convex of periodic measure}  in Theorem~\ref{Thm:A} while here we should apply  item~\ref{generic:chain class},~\ref{generic:lorenz-like is h-class} and~\ref{generic:convex of periodic measrure}  of Theorem~\ref{Thm:generic}.

	The dense part of of Theorem~\ref{Thm:B} follows the same arguments as that of Theorem~\ref{Thm:A} by applying Lemma~\ref{Lem:C1-connection-S-P} here instead of Theorem~\ref{thm:Lorenz-connecting}.

	The proof of Corollary~\ref{Cor:entropy-support} follows the same arguments as Corollary~\ref{Cor:Cr-entropy-support}.  For the residual part, we take the residual subset $\mathcal{R}$ of $\mathscr{X}^1(M)$ from Theorem~\ref{Thm:B}. The dense part is the same as in Corollary~\ref{Cor:Cr-entropy-support}.
\end{proof}

\appendix
\section{Appendix: Robustness of geometric Lorenz attractors}

In the appendix, we prove Proposition \ref{prop:Lorenz}, which shows the geometric Lorenz attractor is a homoclinic class and the vector fields exhibiting a geometric Lorenz attractor forms a $C^2$-open subset in $\mathscr{X}^r(M^3)$ for every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$.

\begin{proof}[Proof of Proposition \ref{prop:Lorenz}]
	The proof of $\Lambda$ is a homoclinic class and every two periodic orbits of $\Lambda$ are homoclinic relies on the fact that the one-dimensional quotient map $f$ is eventually onto (Lemma \ref{lem:evenually-onto}). We refer to \cite[Theorem 1]{lorenz-homoclinic} for showing that $\Lambda$ is a homoclinic class. For every two periodic points $p,q\in\Lambda$, we can assume $p,q\in\Sigma$. Let $\gamma^u\subset W^u_{\it loc}(\orb(p),\phi_t^X)\cap\Sigma$ be the segment containing $p$ in $\Sigma$. Let $q=(x_q,y_q)\in\Sigma$, then $\{x_q\}\times[-1,1]\subset W^s_{\it loc}(\orb(q),\phi_t^X)$. We only need to show there exists some $n>0$ such that
	$$
	P^n(\gamma^u)~\cap~\{x_q\}\times[-1,1]~\neq~\emptyset.
	$$
	Here $P:\Sigma\setminus l\rightarrow\Sigma$ is the Poincar\'e map. Let $J=\pi_x(\gamma^u)$. This is equivalent to there exists $n>0$ such that $x_q\in f^n(J)$, which has been proved in Lemma \ref{lem:evenually-onto}.

	\vskip1mm

	Now we prove the second part of this proposition. We only need to show for every $X\in\mathscr{X}^2(M^3)$ which exhibits a geometric Lorenz attractor with attracting region $U$, there exists a $C^2$-neighborhood $\cU$ of $X$, such that for every $Y\in\cU$, the maximal invariant set $\Lambda_Y=\bigcap_{t>0}\phi_t^Y(U)$ is a geometric Lorenz attractor.

	Let $\lambda_0>\sqrt{2}$ be the constant fixed in Notation \ref{notat:section} satisfying $f'(x)>\lambda_0$ for every $x\in[-1-\epsilon,1+\epsilon]\setminus\{0\}$. Let $\eta>0$ be a fixed constant satisfying
	\begin{align}\label{equ:eta}
	(1+\eta)^{-6}\cdot\lambda_0>\sqrt{2}
	\qquad {\rm and} \qquad
	\sup_{(x,y)\in\Sigma\setminus l}\big|\partial H(x,y)/\partial x\big|<(1+\eta)^{-6}.
	\end{align}

	Since $U\subseteq M^3$ is an attracting region of $X$ and$\Lambda=\bigcap_{t>0}\phi_t^X(U)$ is singular hyperbolic $T_{\Lambda}M^3=E^{ss}\oplus E^{cu}$, there exists a $C^2$-neighborhood $\cU_1$ of $X$, such that for every $Y\in\cU_1$, $U$ is an attracting region of $Y$ and $\Lambda_Y=\bigcap_{t>0}\phi_t^Y(U)$ is singular hyperbolic with splitting $T_{\Lambda_Y}M^3=E^{ss}_Y\oplus E^{cu}_Y$. Moreover, The singularity $\sigma_Y\in\Lambda_Y$ also has three eigenvalues $\lambda_{1,Y}<\lambda_{2,Y}<0<\lambda_{3,Y}$ satisfying $\lambda_{1,Y}+\lambda_{3,Y}<0$ and $\lambda_{2,Y}+\lambda_{3,Y}>0$ for every $Y\in\cU_1$.

	Let $\Sigma=[-1-\epsilon,1+\epsilon]^2$ be the cross section of $X$ and $\Lambda$ defined in Lemma \ref{lem:big-section}.
	By the continuity of the singularity $\sigma_Y$ and the continuity of the local stable and unstable manifolds of $\sigma_Y$, shrinking $\cU_1$ if necessary, we can further assume that for every $Y\in\cU_1$, it satisfies
	\begin{enumerate}
		\item For every $z\in U\setminus W^s_{\it loc}(\sigma_Y,\phi_t^Y)$, there exists $T>0$, such that $\phi_T^Y(z)\in\Sigma$.
		\item The local stable manifold of $\sigma_Y$ intersects $\Sigma$ with $l_Y=W^s_{\it loc}(\sigma_Y,\phi_t^Y)\cap\Sigma$ being $C^2$-close to $l=\{0\}\times[-1-\epsilon,1+\epsilon]$.
		\item The unstable manifold of $\sigma_Y$ intersects $\Sigma$ with $z_Y^+,z_Y^-\in\Sigma$ which are close to $z^+,z^-$ respectively.
		\item The Poincar\'e map $P_Y:\Sigma\setminus l_Y\rightarrow\Sigma$ is well defined and satisfies
		$$
		\overline{P_Y(\Sigma\setminus l_Y)}~=~
		P_Y(\Sigma\setminus l_Y)\cup\{z_Y^+,z_Y^-\}~\subseteq~
		{\rm Int}(\Sigma).
		$$
		\item For every $(x,y)\in\Sigma\setminus l_Y$, we have
		$$
		\frac{\partial\pi_x\circ P_Y}{\partial x}(x,y)>(1+\eta)^{-1}\lambda_0, \quad
		\frac{\partial\pi_y\circ P_Y}{\partial x}(x,y)<(1+\eta)^{-3},
		\quad {\rm and} \quad
		\frac{\partial\pi_y\circ P_Y}{\partial y}(x,y)<1.
		$$
	\end{enumerate}

	For every $Y\in\cU_1$, we denote $\cF^{ss}_Y$ the strong stable foliation of $Y$ tangent to $E^{ss}_Y$ in $U$, and $\cF^s_Y=\bigcup_{t\in\RR}\phi_t^Y(\cF^{ss}_Y)\cap U$ the 2-dimensional stable foliation in $U\setminus\cF^{ss}_Y(\sigma_Y)$ defined as Lemma \ref{lem:stable}. Let
	$$
	\cF^s_{Y,\Sigma}~=~\cF^s_Y~\cap~\Sigma
	$$
	be the stable foliation on $\Sigma$ induce by $\cF^s_Y$.
	By the classical stable manifold theorem \cite{hps},
	for every $z=(x_z,y_z)\in\Sigma$, the leaf $\cF^s_{Y,\Sigma}(z)$ converges to $\{x_z\}\times[-1-\epsilon,1+\epsilon]$ in $C^2$-topology as $Y\rightarrow X$ in $\mathscr{X}^2(M^3)$. To be precise, shrinking $\cU$ if necessary, for every $x_z\in[-1-\epsilon/2,1+\epsilon/2]$, there exists a $C^2$-smooth function $\varphi^s_z:[-1-\epsilon,1+\epsilon]\rightarrow(-\epsilon/2,\epsilon/2)$, such that
	$$
	\cF^s_{Y,\Sigma}(z)=
	{\rm Graph}\big(\varphi^s_z\big)=
	\big\{x_z+\varphi^s_z(y):~y\in[-1-\epsilon,1+\epsilon]\big\}.
	$$
	The function $\varphi^s_z$ satisfies $\varphi^s_z(y_z)=0$ and $\varphi^s_z\rightarrow 0$ in $C^2$-topology as $Y\rightarrow X$ in $\mathscr{X}^2(M^3)$.

	Let $h^s_{Y,\Sigma}$ be the holonomy map induced by $\cF^s_{Y,\Sigma}$ on $\Sigma$. Now we show $h^s_{Y,\Sigma}$ converges to the holonomy map of the foliation
	$\big\{\{x\}\times[-1-\epsilon,1+\epsilon]:~x\in[-1-\epsilon,1+\epsilon]\big\}$ in $C^1$-topology.

	\begin{claim}\label{clm:holonomy}
		The holonomy map $h^s_{Y,\Sigma}$ is $C^{1+}$-smooth on $\Sigma$. There exists a $C^2$-neighborhood $\cU_2$ of $X$, such that for every $Y\in\cU_2$ and every two points $z_i=(x_i,y_i)\in\cF^s_{Y,\Sigma}(z)$, $i=1,2$, let
		$$
		J_i=(a_i,b_i)\times\{y_i\}~\subset~ [-1-\epsilon/2,1+\epsilon/2]\times[-1-\epsilon,1+\epsilon]
		$$
		be two intervals containing $x_i\in(a_i,b_i)$,
		such that $h^s_{Y,\Sigma}(z_1)=z_2$ and $h^s_{Y,\Sigma}(J_1)=J_2$, then
		$$
		(1+\eta)^{-1}~<~
		Dh^s_{Y,\Sigma}(z_1)
		~<~(1+\eta).
		$$
		This implies $h^s_{Y,\Sigma}$ converges to $h^s_{X,\Sigma}\equiv{\rm Id}_x$ in $C^1$-topology as $Y$ converges to $X$ in $\mathscr{X}^2(M^3)$.
	\end{claim}

	\begin{proof}[Proof of the claim]
		Due to a beautiful observation in \cite[Lemma 7.1]{am2},
		the holonomy map $h^s_Y$ of the stable foliation $\cF^s_{Y,\Sigma}$ is $C^{1+}$-smooth in the cross section $\Sigma$. The key fact is that $h^s_Y$ is absolutely continuous with respect to the transversal Lebesgue measure. Thus the fact that $\cF^s_{Y,\Sigma}$ is codimension one implies $h^s_Y$ is $C^{1+}$-smooth.
		Following the estimation in \cite[Lemma 7.1]{am2}, we can show that the holonomy map $h^s_Y$ converges to $h^s_X$ in $C^1$-topology as $Y\rightarrow X$ in $\mathscr{X}^2(M^3)$.

		Recall that the stable foliation $\cF^{ss}_Y$ is H\"older continuous. There exists a H\"older continuous function $T:J_2\rightarrow\RR$, such that for every $w_1\in J_1$ and $w_2=h^s_Y(w_1)\in J_2$, they satisfy
		$$
		\phi_{T(w_2)}^X(w_2)~\in~\cF^{ss}_{Y,{\it loc}}(w_1).
		$$
		Here we can assume the function $|T|<1$ on $J_2$ by flowing the cross section $\Sigma$.

		Since the strong stable foliation
		$\cF^{ss}_Y$ is preserved by $\phi_t^Y$, for $t$ small enough, we have
		$$
		\phi_{T(w_2)+t}^X(w_2)~\in~
		\cF^{ss}_{Y,{\it loc}}(\phi_t^Y(w_1)).
		$$
		This implies there exists $\delta>0$, such that if
		$$
		I_{\delta}~=~[x_1-\delta,x_1+\delta]\times\{y_1\}
		~\subset~(a_1,b_1)\times\{y_1\}~=~J_1,
		$$
		and $D^Y_{\delta}=\bigcup_{t\in[-\delta,\delta]}\phi_t^Y(I_\delta)$, then the holonomy map of the local strong stable foliation $\cF^{ss}_Y$ is well defined:
		$$
		h^{ss}_Y:~D^Y_{\delta}\longrightarrow
		D^Y_2=\bigcup_{t\in[-1,1]}\phi_t^Y(J_2),
		$$
		and satisfies $h^{ss}_Y(w_1)=\phi_{T(w_2)}^X(w_2)$ where $w_2=h^s_Y(w_1)\in J_2$.

		The holonomy map $h^{ss}_Y:D^Y_{\delta}\rightarrow D^Y_2$ is absolutely continuous, see \cite{BP,pugh-shub}. Moreover, let $m_1$ and $m_2$ be Lebesgue measures induced by the Riemannian metric restricted on $D^Y_{\delta}$ and $D^Y_2$ respectively. Then for every $w_1\in D^Y_{\delta}$ and $w_2=h^{ss}_Y(w_1)\in D^Y_2$,
		the Jacobian of $h^{ss}_Y(w_1)$ is
		\begin{align}\label{equ:abso-continuity}
		\jac\left(h^{ss}_Y\right)(w_1)~=~
		\prod_{i=0}^{+\infty}
		\frac{\jac\left(D\phi_{-1}^Y|_{T_{\phi_i^Y(w_2)}\phi_i^Y(D^Y_2)}\right)}{\jac\left(D\phi_{-1}^Y|_{T_{\phi_i^Y(w_1)}\phi_i^Y(D^Y_{\delta})}\right)},
		\end{align}
		see \cite[Theorem 7.1 \& Remark 7.2]{Pe}. Notice here the infinite product must convergence since $w_2=h^{ss}_Y(w_1)\in\cF^{ss}_{Y,{\it loc}}(w_1)$.

		For every $\kappa>0$, there exists a $C^2$-neighborhood $\cV_{\kappa,1}\subset\mathscr{X}^2(M^3)$ of $X$, and an integer $k>0$, such that for every $Y\in\cV_{\kappa,1}$ and $w_2=h^{ss}_Y(w_1)$ with $w_1\in D^Y_{\delta}$, they satisfies
		\begin{align}\label{equ:tail}
		\sum_{i=k+1}^{+\infty}
		\left| \log\jac\left(D\phi_{-1}^Y|_{T_{\phi_i^Y(w_2)}\phi_i^Y(D^Y_2)}\right)-\log\jac\left(D\phi_{-1}^Y|_{T_{\phi_i^Y(w_1)}\phi_i^Y(D^Y_{\delta})}\right) \right| <\frac{\kappa}{4}.
		\end{align}
		Here we use the fact that the distance between $\phi_i^Y(w_1)$ and $\phi_i^Y(w_2)$, and the distance between the tangent bundles $T_{\phi_i^Y(w_1)}\phi_i^Y(D^Y_{\delta})$ and $T_{\phi_i^Y(w_2)}\phi_i^Y(D^Y_2)$ are both exponential convergent to zero.

		For every $w\in I_{\delta}$ and every $t\in[-\delta,\delta]$, we denote $w_t^X=\phi_t^X(w)\in D_{\delta}^X$ and $w_t^Y=\phi_t^Y(w)\in D_{\delta}^Y$ for $Y\in\cV_{\kappa,1}$.
		Let $w_2^X=h^{ss}_X(w_t^X)$ and $w_2^Y=h^{ss}_Y(w_t^Y)$.
		Equation \ref{equ:abso-continuity} and Equation \ref{equ:tail} show that
		\begin{align*}
		&\left| \log\jac\left(h^{ss}_Y\right)(w_t^Y)~-~
		\log\jac\left(h^{ss}_X\right)(w_t^X) \right| \\
		~\leq~&\sum_{i=0}^{k} \left|\log\jac\left(D\phi_{-1}^Y|_{T_{\phi_i^Y(w_2^Y)}\phi_i^Y(D^Y_2)}\right)- \log\jac\left(D\phi_{-1}^X|_{T_{\phi_i^X(w_2^X)}\phi_i^X(D^X_2)}\right)\right| \\
		&+\sum_{i=0}^{k} \left|\log\jac\left(D\phi_{-1}^Y|_{T_{\phi_i^Y(w_t^Y)}\phi_i^Y(D^Y_{\delta})}\right)- \log\jac\left(D\phi_{-1}^X|_{T_{\phi_i^X(w_t^X)}\phi_i^X(D^X_{\delta})}\right)\right|+\frac{\kappa}{2}.
		\end{align*}
		When $Y\rightarrow X$ in $C^2$-topology, we have $D\phi_{-1}^Y\rightarrow D\phi_{-1}^X$, and for every $i=0,\cdots,k$,
		\begin{itemize}
			\item $\phi_i^Y\big(D^Y_{\delta}\big)\rightarrow \phi_i^X\big(D^X_{\delta}\big)$ and $\phi_i^Y\big(D^Y_2\big)\rightarrow \phi_i^X\big(D^X_2\big)$ in $C^1$-topology;
			\item $\phi_i^Y\big(w_t^Y\big)\rightarrow \phi_i^X\big(w_t^X\big)$ and
			$\phi_i^Y\big(w_2^Y\big)\rightarrow \phi_i^X\big(w_2^X\big)$.
		\end{itemize}
		So there exists a $C^2$-neighborhood $\cV_{\kappa,2}\subset\cV_{\kappa,1}$ of $X$, such that for every $Y\in\cV_{\kappa,2}$,
		$$
		\left| \log\jac\left(h^{ss}_Y\right)(w_t^Y)~-~
		\log\jac\left(h^{ss}_X\right)(w_t^X) \right|<\kappa.
		$$
		Thus we have $\jac\left(h^{ss}_Y\right)\rightarrow\jac\left(h^{ss}_X\right)$ as $Y\rightarrow X$ in $\mathscr{X}^2(M^3)$.

		\vskip1mm

		Let $\big\{(x,t): x\in[x_1-\delta,x_1+\delta],t\in[-\delta,\delta]\big\}$ be the coordinate on $D_{\delta}^Y$, which is defined as $w=(x,t)$ if $w_0=(x,y_1)\in I_{\delta}$ and $w=\phi_t^Y(w_0)$. Then there exists a smooth function $v^Y_1$ on $D_{\delta}^Y$, such that the Lebesgue measure $m_1$ on $D_{\delta}^Y$ induced by the Riemannian metric is
		$$
		m_1(x,t)~=~v^Y_1(x,t){\rm d}x{\rm d}t.
		$$
		Similarly, we have the coordinate $\big\{(x,t): x\in(a_2,b_2),t\in[-1,1]\big\}$ on $D_2$ and the smooth function $v^Y_2$ satisfying the Lebesgue measure $m_2(x,t)=v^Y_2(x,t){\rm d}x{\rm d}t$ on $D_2^Y$.

		According to the proof of \cite[Lemma 7.1]{am2}, for every $w_1=(x_1,y_1)\in J_1$ and $w_2^Y=(x_2^Y,y_2)=h^s_{Y,\Sigma}(w_1)\in J_2$, let $\phi_{T(w_2^Y)}^Y(w_2)\in\cF^{ss}_{Y,{\it loc}}(w_1)\cap D^Y_2$, then $w_1=(x_1,0)$ in the $(x,t)$-coordinate in $D^Y_{\delta}$, and $h^{ss}_Y(w_1)=\phi_{T(w_2^Y)}^Y(w_2)=(x_2,T(w_2^Y))$ in the $(x,t)$-coordinate in $D^Y_2$. The derivate of the holonomy map $h^s_{Y,\Sigma}$ is equal to
		\begin{align}\label{equ:holo-deriv}
		Dh^s_{Y,\Sigma}(w_1)~=~
		\lim_{\delta\rightarrow 0}
		\frac{\int_{D^Y_{\delta}}\jac\left(h^{ss}_Y\right)v_1(x,t) ~{\rm d}x{\rm d}t}{\int_{h^{ss}_Y\left(D^Y_{\delta}\right)}v_2(x,t) ~{\rm d}x{\rm d}t}
		~=~\jac\left(h^{ss}_Y\right)(w_1)\cdot
		\frac{v_1(x_1,0)}{v_2(x_2,T(w_2^Y))}.
		\end{align}
		When $Y\rightarrow X$ in $\mathscr{X}^2(M^3)$, $h^{ss}_Y(w_1)\rightarrow h^{ss}_X(w_1)$, so $x_2^Y\rightarrow x_2^X$ and $T(w_2^Y)\rightarrow T(w_2^X)$. Moreover, $D^Y_{\delta}\rightarrow D^X_{\delta}$ and $D^Y_2\rightarrow D^X_2$ imply $v^Y_i\rightarrow v^X_i$ for $i=1,2$. Thus Equation \ref{equ:holo-deriv} implies
		$$
		Dh^s_{Y,\Sigma}\longrightarrow Dh^s_{X,\Sigma}\equiv{\rm Id}_x
		\qquad {\rm as} \qquad
		Y\longrightarrow X~{\rm in}~\mathscr{X}^2(M^3).
		$$
		Thus for $\eta>0$ decided in (\ref{equ:eta}), there exists a $C^2$-neighborhood $\cU_2$ of $X$, such that for every $Y\in\cU_2$, it satisfies
		$$
		(1+\eta)^{-1}~<~
		Dh^s_{Y,\Sigma}
		~<~(1+\eta).
		$$
		This finishes the proof of Claim~\ref{clm:holonomy}.
	\end{proof}

	For every $Y\in\cU_1\cap\cU_2$, we give a new coordinate $\{(x',y')\}$ on the cross section $\Sigma$, such that the Poincar\'e map of $\phi_t^Y$ associated to $\Sigma$ satisfies the definition of Lorenz attractors in this coordinate. For every $z=(x_z,y_z)\in\Sigma$, we fix the $y'$-coordinate of $z$ to $y'_z=y_z$.

	For $\eta>0$ defined in (\ref{equ:eta}), there exists $\epsilon_{\eta}>0$, such that for every $x^-,x^+,x_0\in\RR$ satisfying
	$$
	|x^--(-1)|<\epsilon_{\eta}, \qquad
	|x^+-1|<\epsilon_{\eta}, \qquad {\rm and} \qquad
	|x_0|<\epsilon_{\eta},
	$$
	there exists a diffeomorphism $h:\RR\rightarrow\RR$ satisfying
	$$
	h(x^-)=-1, \quad
	h(x^+)=1, \quad
	h(x_0)=0, \quad {\rm and} \quad
	(1+\eta)^{-1}<h'(x)<(1+\eta),
	\quad\forall x\in\RR.
	$$

	The unstable manifolds of $\sigma_Y$ intersects $\Sigma$ with $z^+_Y\rightarrow z^+=(-1,y^+)$ and $z^-_Y\rightarrow z^-=(1,y^+)$ as $Y\rightarrow X$. Moreover, the intersection between the local stable manifolds of $\sigma_Y$ and $\Sigma$ satisfies $W^s_{\it loc}(\sigma_Y,\phi_t^Y)\cap\Sigma=l_Y\rightarrow l=\{0\}\times[-1-\epsilon,1+\epsilon]$ as $Y\rightarrow X$. There exists a $C^2$-neighborhood $\cU_3$ of $X$, such that for every $Y\in\cU_3$, let $\cF^s_{Y,\Sigma}$ be the stable foliation on $\Sigma$ and $J_0=[-1-\epsilon,1+\epsilon]\times\{0\}$, then the coordinates
	$$
	\cF^s_{Y,\Sigma}(z^+_Y)\cap J_0=(x^-_Y,0), \quad
	\cF^s_{Y,\Sigma}(z^-_Y)\cap J_0=(x^+_Y,0), \quad {\rm and} \quad
	l_Y\cap J_0=(x_{0,Y},0)
	$$
	satisfy
	$$
	|x^-_Y-(-1)|<\epsilon_{\eta}, \qquad
	|x^+_Y-1|<\epsilon_{\eta}, \qquad {\rm and} \qquad
	|x_{0,Y}|<\epsilon_{\eta}.
	$$
	Let $h_Y:\RR\rightarrow\RR$ be the diffeomorphism satisfying
	$$
	h_Y(x^-_Y)=-1, \quad
	h_Y(x^+_Y)=1, \quad
	h_Y(x_{0,Y})=0, \quad {\rm and} \quad
	(1+\eta)^{-1}<h_Y'(x)<(1+\eta),
	\quad\forall x\in\RR.
	$$
	Then we define $x'$-coordinate on $J_0$ as following: for every $z=(x_z,0)\in J_0$, the $(x',y')$-coordinate of $z$ is
	$$
	(~x'_z,~y'_z~)~=~\big(~h_Y(x_z),~0~\big).
	$$

	Let $J_Y=\{(x,0):~-1\leq h_Y(x)\leq 1\}\subseteq J_0$, and
	$$
	\Sigma_Y~=~\left(\bigcup_{z\in J_Y}\cF^s_{Y,\Sigma}\right)
	~\cap~
	\big\{(x,y)\in\Sigma:|y|\leq1\big\}.
	$$
	For every $z=(x_z,y_z)\in\Sigma_Y$, let $w=(x_w,0)=\cF^s_{Y,\Sigma}(z)\cap J_0$, then we define the $x'$-coordinate of $z$ as
	$$
	(~x'_z,~y'_z~)~=~(~x'_w,~y_z~)~=~\big(~h_Y(x_w),~y_z~\big).
	$$
	Thus in the coordinate $\big\{(x',y')\big\}$, $\Sigma_Y$ is $C^1$-diffeomorphic to $[-1,1]^2$.

	For every $Y\in\cU_1\cap\cU_2\cap\cU_3$,
    the stable foliation $\cF^s_{Y,\Sigma}$ is invariant under the action of Poincar\'e map, hence the Poincar\'e map $P_Y:\Sigma_Y\setminus l_Y\rightarrow\Sigma_Y$ is well defined and has the form
	$$
	P_Y(x',y')=\big(~f_Y(x'),~H_Y(x',y')~\big),
	\qquad\forall (x',y')\in\Sigma_Y\setminus l_Y.
	$$

	From the construction of $(x',y')$-coordinate on $\Sigma_Y$, the one dimensional quotient map $f_Y$ is $C^1$-smooth and satisfies
	$\lim_{x'\rightarrow0^-}f_Y(x')=1$, $\lim_{x'\rightarrow0^+}f_Y(x')=-1$, and $-1<f_Y(x')<1$ for every $x'\in[-1,1]\setminus\{0\}$. Since $Y\in\cU_1$, the Poincar\'e map $P_Y$ satisfies $\partial\pi_x\circ P_Y/\partial x>(1+\eta)^{-1}\lambda_0$ everywhere in the $(x,y)$-coordinate. So for every $z=(x_z,y_z)=(x'_z,y_z)\in\Sigma_Y\setminus l_Y$, it satisfies
	$$
	x'_z=h_Y\circ h^s_{Y,\Sigma}(x_z),
	\qquad i.e. \qquad
	x_z=\big( h^s_{Y,\Sigma} \big)^{-1}\circ h_Y^{-1}(x'_z).
	$$
	Let $w=P_Y(z)=(x_w,y_w)=(f_Y(x'_z),y_w)\in\Sigma_Y$, then
	$$
	x_w=\pi_x\circ P_Y(z),
	\qquad {\rm and } \qquad
	f_Y(x'_z)=h_Y\circ h^s_{Y,\Sigma}(x_w)
	=h_Y\circ h^s_{Y,\Sigma}\circ\pi_x\circ P_Y(x_z,y_z).
	$$
	Since $y_z=y_z'$, we have
	\begin{align*}
	f_Y'(x'_z) ~&=~h'_Y \cdot Dh^s_{Y,\Sigma} \cdot
	\frac{\partial \pi_x\circ P_Y}{\partial x} \cdot
	D\big( h^s_{Y,\Sigma} \big)^{-1} \cdot
	\big(h_Y^{-1}\big)'(x'_z) \\
	~&\geq~ (1+\eta)^{-2}\cdot (1+\eta)^{-1}\lambda_0 \cdot (1+\eta)^{-2} \\
	~&\geq~(1+\eta)\sqrt{2}.
	\end{align*}
	This shows that one-dimensional quotient map $f_Y$ satisfies the property of geometric Lorenz attractors.

	Since $P_Y$ is uniformly contracting in each leaf of $\cF^s_{Y,\Sigma}$, the function $H_Y$ is contracting along the $y'$-coordinate. The proof of rest properties for $H_Y$ is the same with $f_Y$. Thus we take $\cU=\cU_1\cap\cU_2\cap\cU_3$ which is a $C^2$-neighborhood of $X$, and $\Lambda_Y$ is a geometric Lorenz attractor for every $Y\in\cU$.    This completes the proof of Proposition~\ref{prop:Lorenz}.
\end{proof}

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