text:
\section{Mathematical model}
\label{sec:theory}

A basic question in the theory of neural networks is as follows:
\begin{quote}
    How far can we perturb the synapses before we damage the network as a whole?
\end{quote}
In this paper, this question is important on two fronts: our learning rule must not destroy the information contained in the synapses, and our numerical representation must be precise enough to encode non-destructive perturbations. Once it has been established that multiplicative updates are a good learning rule (addressing the first point) it becomes natural to represent them using a logarithmic number system (addressing the second). Therefore, this section shall focus on establishing---first as a sketch, then rigorously---the benefits of multiplicative updates for learning compositional functions.

\subsection{Sketch of the main idea}

The \textit{raison d'\^{e}tre} of a synaptic weight is to support learning in the network as a whole. In machine learning, this is formalised by constructing a loss function $\mathcal{L}(W)$ that measures the error of the network in weight configuration $W$. Learning proceeds by perturbing the synapses $W\rightarrow W+\Delta W$ in order to reduce the loss function. A good perturbation direction is the negative gradient of the loss: $\Delta W \propto -\nabla_W \mathcal{L}(W)$. But how far can this direction be trusted?

Intuitively, the negative gradient should only be trusted until its approximation quality breaks down. This breakdown could be measured by the Hessian of the loss function, but this is intractable for large networks since it involves \emph{all pairs} of weights. Instead, \citet{fromage2020} suggest how to operate without the Hessian. To get a handle on exactly how this is done, consider an $L$ layer multilayer perceptron $f(x)$ and the gradient $g_k(W)$ of its loss with respect to the weights at the $k$th layer $W_k$:
\begin{align}\label{eq:backprop}
    g_k(W) := \nabla_{W_k} \mathcal{L}(W) = \pypx{\mathcal{L}}{f}\cdot\pypx{f}{h_k}\cdot\pypx{h_k}{W_k},
\end{align}
where $h_k$ denotes the activations at the $k$th hidden layer of the network. By the backpropagation algorithm \citep{backprop}---and ignoring the nonlinearity for the sake of this sketch---the second term in Equation~\ref{eq:backprop} depends on the product of weight matrices over layers $k+1$ to $L$, and the third term depends on the product of weight matrices over layers 1 to $k-1$. It is therefore natural to model the relative change in the whole expression via the formula for the relative change of a product:
$$\frac{\lnorm{g_k(W+\Delta W) - g_k(W)}_F}{\lnorm{g_k(W)}_F}
\sim \frac{\lnorm{\prod_{l=1}^L (W_l+\Delta W_l)-\prod_{l=1}^L W_l}_F}{\lnorm{\prod_{l=1}^L W_l}_F}
\sim \prod_{l=1}^L \left(1 + \frac{\norm{\Delta W_l}_F}{\norm{W_l}_F}\right) -1.$$
This neglects the specific choice of loss function which enters via the first term $\partial \mathcal{L} / \partial f$ in Equation \ref{eq:backprop}.

To recap, we have proposed that neural networks ought to be trained by following the gradient of their loss until that gradient breaks down. We then sketched that the relative breakdown in gradient depends on a product over relative perturbations to each network layer. The simplest perturbation that follows the gradient direction \textit{and} keeps the layerwise relative perturbation small was first introduced by \citet{You:EECS-2017-156}. Letting $\eta$ be a positive scalar known as the \textit{learning rate}, the update is given by:
$$W_k \rightarrow W_k - \eta \,\frac{\norm{W_k}_F}{\norm{g_k}_F}g_k \quad \text{ for each layer } k=1,...,L.$$
The downside of this rule is that it requires knowing precisely which weights act together as a layer and normalising those updates jointly. It is difficult to imagine this happening in the brain, and for exotic artificial networks it is sometimes unclear what constitutes a layer \citep{densenet}. A convenient way to sidestep this issue is to update each individual weight $w$ multiplicatively, via:
$$w \rightarrow w (1\pm\eta) \approx w \econst^{\pm\eta} \quad \text{ for each weight } w.$$
This update ensures that $\norm{\Delta W_*}_F/ \norm{W_*}_F$ is small \textit{for every subset} $W_*$ of the weights $W$ whilst only using information local to a synapse. Therefore, by appropriate choice of the signs of the multiplicative factors, it can be arranged that this perturbation is roughly aligned with the negative gradient whilst keeping the relative breakdown in gradient small.

On the next page, we shall develop this sketch into a rigorous optimisation theory.

\subsection{First-order optimisation of continuously differentiable functions}

To elucidate the connection between gradient breakdown and optimisation, consider the following inequality. Though it applies to all continuously differentiable functions, we will think of $\mathcal{L}$ as a loss function measuring the performance of a neural network of depth $L$ at some task.

\begin{restatable}{lemma}{ftc}\label{lem:ftc}
    Consider a continuously differentiable function $\mathcal{L}:\R^n\rightarrow \R$ that maps $W \mapsto \mathcal{L}(W)$. Suppose that parameter vector $W$ decomposes into $L$ parameter groups: $W = (W_1,W_2,...,W_L)$, and consider making a perturbation $\Delta W = (\Delta W_1,\Delta W_2,...,\Delta W_L)$. Let $\theta_k$ measure the angle between $\Delta W_k$ and negative gradient $-g_k(W):=-\nabla_{W_k}\mathcal{L}(W)$. Then:
    \begin{gather*}\mathcal{L}(W+\Delta W) - \mathcal{L}(W)\leq - \sum_{k=1}^L \norm{g_k(W)}_F \norm{\Delta W_k}_F\left[\cos \theta_k - \max_{t\in[0,1]}\frac{\norm{g_k(W+t\Delta W)-g_k(W)}_F}{\norm{g_k(W)}_F}\right].\end{gather*}
\end{restatable}

The proof is in Appendix \ref{app:proof}. The result says: \textit{to reduce a function, follow its negative gradient until it breaks down}. Descent is formally guaranteed when the bracketed terms are positive. That is, when:
\begin{equation}\label{eq:descent}
    \max_{t\in[0,1]}\frac{\norm{g_k(W+t\Delta W)-g_k(W)}_F}{\norm{g(W)}_F}<\cos\theta_k \qquad (\text{for }k=1,...,L).
\end{equation}

According to Equation~\ref{eq:descent}, to guarantee descent for neural networks, we must bound their relative breakdown in gradient. To this end, \citet{fromage2020} propose the notion of \emph{deep relative trust}.

\begin{restatable}{assumption}{drt}[Deep relative trust]\label{ass:drt}
Consider a neural network with $L$ layers and parameters $W = (W_1,W_2,...,W_L)$. Consider parameter perturbation $\Delta W = (\Delta W_1,\Delta W_2,...,\Delta W_L)$. Let $g_k(W):=\nabla_{W_k}\mathcal{L}(W)$ denote the gradient of the loss. Then the gradient breakdown is bounded by:
$$\frac{\norm{g_k(W+\Delta W)-g_k(W)}_F}{\norm{g_k(W)}_F} \leq \prod_{l=1}^L \left(1+\frac{\norm{\Delta W_l}_F}{\norm{W_l}_F}\right) - 1\qquad (\text{for }k=1,...,L).$$
\end{restatable}

While deep relative trust is based on a perturbation analysis of multilayer perceptrons with leaky relu nonlinearity, it may be seen as a model of more general neural networks since the product reflects their compositional structure. Crucially, compared to a Hessian that may contain as many as $10^{18}$ entries for modern networks (since the Hessian's size is quadratic in the number of parameters), deep relative trust gives a tractable analytic model for the breakdown in gradient.

\subsection{Descent via multiplicative weight updates}

Since deep relative trust penalises the relative size of the perturbation to each layer, it is natural that our learning algorithm would bound these perturbations on a relative scale. A simple way to achieve this is via the following \textit{multiplicative} update rule:
\begin{equation}\label{eq:mult}
    W \rightarrow W+\Delta W = W \odot \big[1-\eta\,\sign W \odot\,\sign g(W)\big],
\end{equation}
where the sign is taken elementwise and $\odot$ denotes elementwise multiplication. Synapses shrink where the signs of $W$ and $g(W)$ agree and grow where the signs differ. In the following theorem, we establish descent under this update for compositional functions described by deep relative trust.

\begin{restatable}{theorem}{descentlemma}\label{thm:relative}
    Let $\mathcal{L}$ be the continuously differentiable loss function of a neural network of depth $L$ that obeys deep relative trust. For $k=1,...,L$, let $0\leq\gamma_k\leq\tfrac{\pi}{2}$ denote the angle between $\abs{g_k(W)}$ and $\abs{W_k}$ (where $\abs{\cdot}$ denotes the elementwise absolute value). Then the multiplicative update in Equation \ref{eq:mult} will decrease the loss function provided that:
    $$\eta < \left(1 + \cos \gamma_k \right)^\frac{1}{L} -1 \qquad (\text{for all }k=1,...,L).$$
\end{restatable}

Theorem \ref{thm:relative} tells us that for small enough $\eta$, multiplicative updates achieve descent. It also tells us \textit{on what scale} $\eta$ must be small. The proof is given in Appendix \ref{app:proof}.

We can bring Theorem \ref{thm:relative} to life by plugging in numbers. First, we must consider what values the angles $\gamma_k$ are likely to take. Since $\abs{g_k(W)}$ and $\abs{W_k}$ are nonnegative vectors, the angle between them can be no larger than $\pi/2$, which happens when the support of the two vectors is disjoint. A simple model to consider is two high-dimensional Gaussian vectors with iid components, for which the angle between their absolute values satisfies $\cos \gamma_k\approx 0.64$. Taking this value, Theorem \ref{thm:relative} implies that for a 40 layer network, setting $\eta=0.01$ guarantees descent. We find that $\eta=0.01$ works well in all our experiments with the \textit{Madam} optimiser in later sections.


TITLE: Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)
QUESTION [0 upvotes]: A matrix is symmetric if $A^T$=A
And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site).
My question: How do i describe the orthogonal complement of W?
Thanks in advance.

REPLY [4 votes]: A basis for the $2\times 2$ symmetric matrices is given by
$$ \def\p#1#2#3#4{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}}
  \p1000, \quad \p0001, \quad \p0110 $$
The inner product on ${\rm Mat}_2(\mathbf R)$ is given by $\def\<#1>{\left<#1\right>}\<A,B> := \def\tr{\mathrm{tr}}\tr(A^tB)$, now let $A = \p abcd$ be given, we have
\begin{align*}
  \<A, \p 1000> &= \tr\p a0b0\\
                &= a\\
  \<A, \p0001> &= \tr\p 0c0d\\
                &= d\\
  \<A, \p0110> &= \tr\p cadb\\
                &= b+c
\end{align*}
Hence $A$ is orthogonal to all symmetric matrices iff $a = d = b+c = 0$, so we have
$$ {\rm Sym}_2(\mathbf R)^\bot = \left\{\p0a{-a}0\biggm| a \in \mathbf R\right\}, $$
the antisymmetric matrices.


text:
\begin{document}

\maketitle\

\bigskip

\centerline{\scshape Peng Sun}
\medskip
{\footnotesize

 \centerline{China Economics and Management Academy}
   \centerline{Central University of Finance and Economics}
   \centerline{Beijing 100081, China}
}

\bigskip

\begin{abstract}

We explore an approach to the conjecture of Katok on intermediate
entropies that based on uniqueness of equilibrium states,
provided the entropy function is upper semi-continuous.
As an application, we prove Katok's conjecture for Ma\~n\'e diffeomorphisms.
\end{abstract}

\bigskip

\section{Introduction}

Let $(X,d)$ be a compact metric space and
$f:X\to X$ be a continuous map.
Denote by $\cm(f)$
the subspace of all invariant measures for the dynamical system
$(X,f)$ and by $\cm_e(f)$ the subset of all ergodic measures.
Denote by $C(X)$ the space of all continuous potentials on $X$,
equipped with the $C^0$ (supremum) norm $\|\cdot\|$.
For
$\phi\in C(X)$,

denote
by $P(\phi)=P(f,\phi)$ the topological pressure of $(X,f,\phi)$

and by
$P_\mu(\phi)=P_\mu(f,\phi):=h_\mu(f)+\int\phi d\mu$
the pressure
of $\mu\in\cm(f)$.
We usually omit $f$ when it is clearly fixed.
The Variational Principle states that
\begin{equation*}
P(\phi)=\sup\{P_\mu(\phi):\mu\in\cm(f)\}.
\end{equation*}
For convenience, we say that $(X,f)$ is a \emph{USC} system if the entropy function
$\mu\mapsto h_\mu(f)$ is upper semi-continuous, where $h_\mu(f)$
denotes the metric entropy of $(X,f)$ with respect to $\mu\in\cm(f)$.
Denote the set of equilibrium states for $(X,f,\phi)$ by
$$\cE(\phi):=\{\mu\in\cm(f):P_\mu(\phi)=P(\phi)\}.$$

It is well-known that when $(X,f)$ is a USC system,
$\cE(\phi)$ is nonempty for every $\phi\in C(X)$.

Study on equilibrium states has a long history.
Existence, uniqueness and properties of equilibrium states are
important and popular topics in dynamical systems.
For example, see \cite{Bow75, Bow08, CT, DKU, IT, Ruelle}.

As indicated in \cite[Section 9.4]{Walters}, The topological
pressure $P(f,\cdot)$ for the system
$(X,f)$, as a functional on continuous potentials,
determines
all invariant probability measures

and their entropies. In this article, we would like to further
explore such connections and show its application to the
conjecture of Katok on intermediate entropies.

Denote
$$\spp(f,\phi):=\{P_\mu(f, \phi):\mu\in\cm_e(f)\}$$
and
$$\sh(f):=\spp(f,0)=\{h_\mu(f):\mu\in\cm_e(f)\}.$$
We say that $(X,f)$ has the \emph{intermediate entropy property}
if $\sh(f)\supset[0,h(f))$, i.e. for every $a\in[0,h(f))$,
there is an ergodic measure $\mu\in\cm_e(f)$
such that
$h_\mu(f)=a$.

\begin{conjecture}[Katok]
Let $(X,f)$ be a $C^2$ diffeomorphism on a compact Riemannian manifold.

Then
$(X,f)$ has
the intermediate entropy property.

\end{conjecture}

Partial results on Katok's conjecture have been obtained in
\cite{KM, Sun09, Sun10, Ures, Sun12, QS, GSW, KKK, LO,

	Sunintent,
Sunze, Sunct}. They represent two major approaches,
which are based on hyperbolic structures
and specification-like properties, respectively.

What we illustrate here is a new one, which shows that for USC systems,
certain results on equilibrium states
imply the intermediate entropy property.
This approach is inspired by

some facts in

\emph{ergodic optimization}.

Denote

$$\su(f):=\left\{\phi\in\cx:

\text{$(X,f,\phi)$
has  a unique equilibrium state
}\right\}.$$
For $\phi\in\cx$, denote by $\coo(f,\phi)$

the set of \emph{maximizing measures} for $\phi$, i.e.
invariant
measures $\mu\in\cm(f)$ that maximizes $\int\phi d\mu$.

\begin{theorem}\label{uesintent}
Let $(X,f)$ be a USC system.
Suppose that there is $\phi\in\cx$ such that:
\begin{enumerate}
\item $t\phi\in\su(f)$ for every $t\ge 0$.
\item $\coo(f,\phi)=\{\mu_\phi\}$ is a singleton and $h_{\mu_\phi}(f)=0$.
\end{enumerate}
Then $(X,f)$ has the intermediate entropy property.

\end{theorem}

In fact, the ergodic measures of intermediate entropies we obtain in
Theorem \ref{uesintent} are just the unique equilibrium states for
$t\phi$, whose metric entropy varies continuously with $t$.
When $\coo(f,\phi)=\{\mu_\phi\}$ is a singleton, $\mu_\phi$ is
the unique \emph{ground state}

and the \emph{zero temperature limit} of these equilibrium states.

We shall prove a more general result than Theorem \ref{uesintent}.
We can

relax the uniqueness of equilibrium states and

conclude on intermediate pressures.

For $\psi\in\cx

$, let $\sv(f,\psi)$ be the set of
all continuous potentials $\phi$ such that
$P_\mu(\psi)$ is a constant, denoted by $P^{\psi}(\phi)$,
for all $\mu\in\sE(\phi)$.
That is,
$$\sv(f,\psi):=\left\{\phi\in\cx:
\text{

$P_\mu(\psi)=P^{\psi}(\phi)$ for every

$\mu\in\cE(\phi)$}
\right\}.$$
In particular,
$\sv(f):=
\sv(f,0)$ is
the set of all potentials whose equilibrium states have
equal entropies.
By definition, we have
$\psi\in\sv(f,\psi)$

and $\su(f)\subset\sv(f,\psi)$ for every $\psi\in\cx$.

\begin{theorem}\label{intpres}
Let $(X,f)$ be a USC system.
Suppose that there are $\psi,\phi\in\cx$ and $\al\in\RR$ such that the following holds:
\begin{enumerate}
\item $\psi+t\phi\in\sv(f,\psi)$ for every $t\ge 0$.
\item $P_\mu(\psi)\le\al$ for every $\mu\in\coo(f,\phi)$.

\end{enumerate}
Then $\spp(f,\psi)\supset[\al, P(\psi)]$.
\end{theorem}

Theorem \ref{uesintent} and \ref{intpres} apply to the
Ma\~n\'e diffeomorphisms considered in \cite{CTmane} and
\cite{Sunct}.

The following theorem

completely verifies
 Katok's conjecture for such systems, which improves \cite[Corollary 1.2]{Sunct}.

See Section \ref{secmane} for details.

\begin{theorem}
Ma\~n\'e diffeomorphisms

have the intermediate entropy property.

\end{theorem}

\iffalse

\section{Preliminaries}

Readers are referred to \cite{Walters} for definitions and
basic properties of entropies and pressures.

Upper semi-continuity of the entropy map is usually related to
some sort of expansiveness.
More details on

expansiveness can be found in
\cite{Bow72, Misiu, Walters}.

\begin{lemma}
\begin{enumerate}
\item If $(X,f)$ is asymptotically entropy expansive, then it is a USC system
\cite{Misiu}.
\item If $f$ is a $C^\infty$ map on a compact manifold $X$, then
$(X,f)$ is aysmptotically entropy expansive, hence it is a USC system
\cite{Buzzi}.
\end{enumerate}
\end{lemma}

Let $(X,f)$ be a USC system.

For
$\phi\in C(X)$,

denote
by $P(\phi)=P(f,\phi)$ the topological pressure of $(X,f,\phi)$

and by
$P_\mu(\phi)=P_\mu(f,\phi):=h_\mu(f)+\int\phi d\mu$
the pressure
of $\mu\in\cm(X,f)$.
We usually omit $f$ for convenience.
The Variational Principle states that
\begin{equation}\label{vp}
P(\phi)=\sup\{P_\mu(\phi):\mu\in\cm(X,f)\}.
\end{equation}

We can observe that

\begin{equation}\label{preslip}
|P(\phi)-P(\psi)|\le\|\phi-\psi\|\text{ for any }\phi,\psi\in\cx.
\end{equation}
So $P(\cdot)=P(f,\cdot)$ is a (Lipschitz)
continuous function on $C(X)$.

\begin{proof}

For every $\nu\in\cm(X,f)$, we have
\begin{align*}
P_\nu(\phi')&=h_\nu(f)+\int\phi' d\nu
\\&\le h_\nu(f)+\int\phi d\nu+\left|\int\phi'd\nu-\int\phi d\nu\right|
\\&<P(\phi)+\ep.

\end{align*}
This implies that $P(\phi')-P(\phi)<\ep$. The proof of the other side
is analogous.
\end{proof}

Recall that a nonempty subset $K$ of $\cm(X,f)$ is called a \emph{face}
if every $\mu\in K$, almost every ergodic component of $\mu$
belongs to $K$.
In particular, a face must contain ergodic measures.

For any topological dynamical system $(X,f)$, $\coo(f,\phi)$ is always
a nonempty face for every $\phi\in\cx$.
When $(X,f)$ is a USC system, $\cE(\phi)$ is also a nonempty face for every $\phi\in C(X)$.

\begin{lemma}[{cf. \cite{Ruelle} or \cite{Walters}}]
Let $(X,f)$ be a USC system. Then
\begin{enumerate}
\item $P_\mu(\phi)<\infty$ for every $\phi\in\cx$.
\item
For every $\mu\in\cm(X,f)$,
$$h_\mu(f)=\inf\left\{P(\phi)-\int\phi d\mu:\phi\in\cx

\right\}$$
\item
$\mu\in\cE(\phi)$ if and only if
$P(\phi+\psi)\ge P(\phi)+\int\psi d\mu$ for every $\psi\in\cx$.

\item $\cE(\phi)$ is a nonempty compact {face}.
In particular, $\cE(\phi)\cap\cm_e(X,f)\ne\emptyset$.
\item $\su(f)$ is a residual subset of $\cx$.
\end{enumerate}
\end{lemma}

\fi

\section{Continuity of Equilibrium States}

Readers are also referred to \cite{Walters} for definitions and
basic properties of entropies and pressures.
The following lemma is a simple observation (cf. \cite[6.8]{Ruelle}).
\begin{lemma}
\label{prelipct}
For any system $(X,f)$ and any $\phi,\psi\in\cx$, we have
\begin{equation*}
|P(\phi)-P(\psi)|\le\|\phi-\psi\|\text{ for any }\phi,\psi\in\cx.
\end{equation*}
So $P(\cdot)$ is a (Lipschitz)
continuous function on $C(X)$.
\end{lemma}

We try to make our results as more applicable as possible.
Proposition \ref{limises} and \ref{zerotemp} are slight variations of
known results (cf. \cite[Theorem 4.2.11]{Kell} and \cite[Theorem 4.1]{Jenk}).
In particular, \eqref{munest} and \eqref{presclose}
(hence Proposition \ref{limises} and Proposition
\ref{zerotemp}) hold

when the measures are actually the equilibrium states
for the corresponding potentials, i.e.
$\mu_n\in\cE(\phi_n)$ or $\mu_n\in\cE(\psi+t_n\phi_n)$ for all $n$.

\begin{proposition}\label{limises}

Let $(X,f)$ be a USC system.

Let $\{\phi_n\}_{n=1}^\infty$ be a sequence of continuous potentials
such that $\|\phi_n-\phi\|\to0$.

Let $\{\mu_n\}_{n=1}^\infty$ be a sequence in $\cm(f)$ such that
$\mu_n\to\mu$

and
\begin{equation}\label{munest}
\lim_{n\to\infty}\left|P_{\mu_n}(\phi_n)-P(\phi_n)\right|=0.
\end{equation}
Then $\mu$ is an equilibrium state for $\phi$, i.e. $\mu\in\cE(\phi)$.

\end{proposition}

\begin{proof}

As $P(\cdot)$ is continuous, we have $P(\phi_n)\to P(\phi)$. So
\eqref{munest} is equivalent to the condition
\begin{equation}\label{munest2}
\lim_{n\to\infty}\left|P_{\mu_n}(\phi_n)-P(\phi)\right|=0.
\end{equation}
We fix a metric $D$ on $\cm(f)$ that induces the weak-$*$ topology.
For every $\ep>0$, as the entropy map is upper semi-continuous,
there is $\eta_\ep>0$ such that for every
$\nu\in B(\mu,\eta_\ep)$
we have

\begin{equation}\label{localest}
h_\nu(f)<h_\mu(f)+\ep\text{ and }
\left|\int\phi d\nu-\int\phi d\mu\right|<\ep.
\end{equation}

By \eqref{munest2},
for
every $\ep>0$,
there is $N$ such that
\begin{equation}\label{limest}
\|\phi_N-\phi\|<\ep, D(\mu_N,\mu)<\eta_\ep
\text{ and }P_{\mu_N}(\phi_N)>P(\phi)-\ep.
\end{equation}
Hence by \eqref{localest}, we have
\begin{equation}\label{intest}
h_\mu(f)>h_{\mu_N}(f)-\ep\text{ and }
\left|\int\phi d\mu_N-\int\phi d\mu\right|<\ep.
\end{equation}
By
\eqref{limest} and
\eqref{intest}, we have
\begin{align*}
P_\mu(\phi)&=h_\mu(f)+\int\phi d\mu
\\&> h_{\mu_N}(f)-\ep+\int\phi d\mu_N-
\left|\int\phi d\mu_N-\int\phi d\mu\right|
\\&> h_{\mu_N}(f)-\ep+\left(\int\phi_N d\mu_N-

\left|\int\phi_N d\mu_N-\int\phi d\mu_N\right|\right)-\ep
\\&>P_{\mu_N}(\phi_N)-\|\phi_N-\phi\|-2\ep

\\&>P(\phi)-4\ep.

\end{align*}
This implies that $P_\mu(\phi)\ge P(\phi)$ as it holds
for all $\ep>0$.

Hence $\mu$ is an equilibrium state for $\phi$.

\end{proof}

\iffalse
\begin{proposition}
Let $(X,f)$ be a USC system.
Suppose that there are $\psi,\phi\in\cx$, a sequence
$\{t_n\}_{n=1}^\infty$ of real numbers
and a sequence $\{\mu_n\}_{n=1}^\infty$ in
$\cm(X,f)$

such that:
\begin{enumerate}
\item $\lim_{n\to\infty}t_n=\infty$.
\item $\mu_n\in\sE(\psi+t_n\phi)$ for each $n$.
\item $\mu_n\to\mu$ in $\cm(X,f)$.
\end{enumerate}
Then we have $\mu$.
\end{proposition}
\fi

\begin{proposition}\label{zerotemp}
Let $(X,f)$ be any system and
$\psi,\phi\in\cx$.

Let $\{t_n\}_{n=1}^\infty$ be a sequence of real numbers
such that $t_n\to\infty$.

Let  $\{\mu_n\}_{n=1}^\infty$ be a sequence in
$\cm(f)$

such that $\mu_n\to\mu$ and
\begin{equation}\label{presclose}
\lim_{n\to\infty}\frac{1}{t_n}
\left|P_{\mu_n}(\psi+t_n\phi)-P(\psi+t_n\phi)\right|=0.
\end{equation}

Then $\mu$ is a maximizing measure for $\phi$, i.e. $\mu\in\coo(\phi)$.

\end{proposition}

\begin{proof}
For every $\ep>0$,
there is $\eta_\ep>0$ such that for every
$\nu\in B(\mu,\eta_\ep)$
we have
\begin{equation}\label{localest2}

\left|\int\psi d\nu-\int\psi d\mu\right|<\ep
\text{ and }
\left|\int\phi d\nu-\int\phi d\mu\right|<\ep.
\end{equation}

As $t_n\to\infty$, $\mu_n\to\mu$, by \eqref{presclose},
for
every $\ep>0$,
there is $N$ such that for all $n>N$ we have
\begin{equation}\label{limest2}

t_n\ep>h(f),
D(\mu_n,\mu)<\eta_\ep
\text{ and }
P(\psi+t_n\phi)-P_{\mu_n}(\psi+t_n\phi)<t_n\ep
\end{equation}

By \eqref{localest2} and \eqref{limest2},
for all $n>N$, we have
\begin{align}
\left|P_\mu(\psi)-P_{\mu_n}(\psi)\right|
\le&\left|h_\mu(f)-h_{\mu_n}(f)\right|
+\left|\int\psi d\mu-\int\psi d\mu_n\right| \notag
\\<& h(f)+\ep
<(t_n+1)\ep\label{pmunest1}
\end{align}
and hence
\begin{align}\label{pmunest2}
&P(\psi+t_n\phi)-P_{\mu}(\psi+t_n\phi) \notag
\\\le&P(\psi+t_n\phi)-P_{\mu_n}(\psi+t_n\phi)+
\left|P_\mu(\psi+t_n\phi)-P_{\mu_n}(\psi+t_n\phi)\right| \notag
\\<&
\left|P_\mu(\psi)-P_{\mu_n}(\psi)\right|
+t_n\left|\int\phi d\mu-\int\phi d\mu_n\right|
+t_n\ep \notag

\\<& (3t_n+1)\ep.
\end{align}

By \eqref{pmunest1} and \eqref{pmunest2},
for every $\nu\in\cm(f)$ and all $n>N$, we have
\begin{align}
&t_n\left(\int\phi d\mu-\int\phi d\nu\right)\notag
\\
=
&\left(P_\mu(\psi+t_n\phi)-P_\mu(\psi)\right)

-

\left(P_\nu(\psi+t_n\phi)-P_\nu(\psi)\right) \notag

\\>&\left(P(\psi+t_n\phi)-P_\nu(\psi+t_n\phi)\right) \notag

-\left(P(\psi+t_n\phi)-P_{\mu}(\psi+t_n\phi)\right)
\\&
-\left|P_\mu(\psi)-P_\nu(\psi)\right| \notag
\\>&-(4t_n+1)\ep. \label{tnest}
\end{align}
As $t_n\to\infty$, \eqref{tnest} implies that
\begin{equation}\label{phi4ep}
\int\phi d\mu\ge\int\phi d\nu-4\ep \text{ for every }\nu\in\cm(f).
\end{equation}
This implies that $\int\phi d\mu\ge\int\phi d\nu$ as \eqref{phi4ep} holds
for all $\ep>0$.
Hence $\mu$ is a maximizing measure for $\phi$.

\end{proof}

\iffalse

\begin{proposition}\label{thmcts}

Let $(X,f)$ be a USC system.
Then the map $\phi\mapsto\mu_\phi$ is
continuous
on $\su(f)$.
\end{theorem}

\begin{proof}

Take any $\phi\in\cA$, $\eta_0>0$.
Let $D$ be a metric on the space $\cm(X)$ of the probability measures
on $X$ that induces the weak-$*$ topology.

For every $\ep>0$, as the entropy map is upper semi-continuous,
there is $\eta_\ep\in(0,\eta_0)$ such that for every
$\nu\in B(\mu,\eta_\ep)$
we have:

\begin{equation}\label{localest2}
h_\nu(f)<h_\mu(f)+\ep\text{ and }
\left|\int\phi d\nu-\int\phi d\mu\right|<\ep.
\end{equation}

Suppose that $\phi_n\to\phi$ in $\su(f)$

and $\mu$ is a subsequential limit of $\{\mu_{\phi_n}\}$.

By Proposition \ref{limises},
$\mu$ is an equilibrium state for $\phi$. As
$\phi$ has a unique equilibrium state, we have
$\mu_{\phi_n}\to\mu$ and $\mu=\mu_\phi$.
Hence the map $\phi\mapsto\mu_\phi$ is continuous.

\end{proof}

For $\psi\in\cx

$, denote by $\sv(f,\psi)$ the set of
all continuous potentials $\phi$ such that
$P_\mu(\psi)$ is a constant, denoted by $P^{\psi}(\phi)$,
for all $\mu\in\sE(\phi)$.
That is,
$$\sv(f,\psi):=\left\{\phi\in\cx:
\text{

$P_\mu(\psi)=P_{\psi}(\phi)$ for every

$\mu\in\cE(\phi)$}
\right\}.$$
In particular,
$\sv(f):=\sv(f,0)$ is
the set of all potentials whose equilibrium states have
equal entropies.
By definition, we have
$\psi\in\sv(f,\psi)$

and $\su(f)\subset\sv(f,\psi)$ for every $\psi\in\cx$.

\begin{remark}

In particular, \eqref{munest} and \eqref{presclose}
(hence Proposition \ref{limises} and Proposition
\ref{zerotemp}) hold

when $\mu_n\in\cE(\phi_n)$ or $\mu_n\in\cE(\psi+t_n\phi_n)$ for all $n$.
\end{remark}

\fi

\section{Intermediate Pressures}

\begin{proposition}\label{entcts}
Let $(X,f)$ be a USC system and $\psi\in C(X)$.
Then the function $\phi\mapsto P^{\psi}(\phi)$ is
continuous
on $\sv(f,\psi)$.
\end{proposition}

\iffalse
\begin{remark}

In particular, Proposition \ref{entcts} holds

for $\psi=0$, which means that the
map $\phi\mapsto h_\mu(f)$ for $\mu\in\cE(\phi)$
is continuous on $\sv(f)$ (hence on $\su(f)$).

\end{remark}
\fi

\begin{proof}

Let $\{\phi_n\}_{n=1}^\infty$ be a sequence in $\sv(f,\psi)$
such that
$\phi_n\to\tilde\phi\in\sv(f,\psi)$.

For each $n$, take any $\mu_n\in\sE(\phi_n)$.

Let $\mu$ be the limit of a convergent
subsequence $\{\mu_{n_k}\}_{k=1}^\infty$.
As $\phi_{n_k}\to\tilde\phi$,
by Proposition \ref{limises}, we have $\mu\in\sE(\tilde\phi)$.

By Lemma \ref{prelipct},

we have
\begin{align*}
&\lim_{k\to\infty}\left|P^{\psi}(\phi_{n_k})-P^{\psi}(\tilde\phi)\right|
\\= &
\lim_{k\to\infty}\left|P_{\mu_{n_k}}(\psi)-P_{\mu}(\psi)\right|
\\\le &
\lim_{k\to\infty}\left(\left|P_{\mu_{n_k}}(\phi_{n_k})-P_{\mu}(\tilde\phi)\right|
+\left|\int(\phi_{n_k}-\psi)d_{\mu_{n_k}}-\int(\tilde\phi-\psi) d_{\mu}\right|\right)
\\\le&\lim_{k\to\infty}\left|P(\phi_{n_k})-P(\tilde\phi)\right|
+\lim_{k\to\infty}\left|\int(\phi_{n_k}-\tilde\phi) d_{\mu_{n_k}}\right|
\\&
+\lim_{k\to\infty}\left|\int(\tilde\phi-\psi) d_{\mu_{n_k}}-\int(\tilde\phi-\psi)d_{\mu}\right|
\\\le&\lim_{k\to\infty}\|\phi_{n_k}-\tilde\phi\|+\lim_{k\to\infty}\|\phi_{n_k}-\tilde\phi\|+0
\\=& 0.
\end{align*}

This implies that $P^\psi(\cdot)$ is a
continuous function on $\sv(f,\psi)$.
\end{proof}

\begin{corollary}\label{entctscor}
	Let $(X,f)$ be a USC system.
	Denote by $\mu_\phi$ the unique equilibrium state for each $\phi\in\su(f)$.
	Then the map $\phi\mapsto h_{\mu_\phi}(f)$ is
	continuous
	on $\su(f)$.
\end{corollary}

\begin{proof}
	Note that $\su(f)\in\sv(f)$ and $P^0(\phi)=P_{\mu_\phi}(0)=h_{\mu_\phi}(f)$
	for every $\phi\in\su(f)$.
	Apply Proposition \ref{entcts} for $\psi=0$.
\end{proof}

\iffalse
\begin{proposition}\label{entcts}
Let $(X,f)$ be a USC system.
Suppose that for $\phi\in\cx$ and $\al>0$ the following holds
\begin{enumerate}
\item $\psi+t\phi\in\sv(f,\psi)$ for every $t\ge0$.
\item
\end{enumerate}
Then the map $\phi\mapsto P_{\psi}(\phi)$ is
continuous
on $\sv(f,\psi)$.
\end{theorem}
\fi

\begin{proposition}\label{propmain}
Let $(X,f)$ be a USC system and $\psi\in\cx$. Suppose that there is a
continuous path
$\Phi:[0,T]\to\sv(f,\psi)$ and
$P_\psi(\Phi(0))<P_\psi(\Phi(T))$.
Then
\begin{equation}\label{intmed}
\spp(f,\psi)\supset[P^\psi(\Phi(0)),P^\psi(\Phi(T))]
\end{equation}

\end{proposition}

\begin{proof}
By Proposition \ref{entcts}, $P^\psi(\cdot)$ is continuous on
$\sv(f,\psi)$. Hence
$P^\psi\circ\Phi$ is continuous. Then \eqref{intmed} follows from
the Intermediate Value Theorem.
\end{proof}

Analogous to Corollary \ref{entctscor}, we have:

\begin{corollary}
Let $(X,f)$ be a USC system. Suppose that there is a
continuous path
$\Phi:[0,T]\to\su(f)$,
i.e. $\Phi(t)$ has a unique equilibrium state $\mu_t$ for each $t\in[0,T]$.

Assume that $h_{\mu_0}(f)\le h_{\mu_T}(f)$.
Then we have
\begin{equation*}
\sh(f)\supset[h_{\mu_0}(f),h_{\mu_T}(f)]
\end{equation*}
\end{corollary}

\begin{proof}[Proof of Theorem \ref{intpres}]
Take $\mu_n\in\cE(\psi+t\phi)$ for each $n\in\NN$.
As $\cm(f)$ is compact, there is a subsequence $\{t_k\}_{k=1}^\infty$
such that $t_k\to\infty$ and $\{\mu_{t_k}\}_{k=1}^\infty$ converges

to $\mu\in\cm(f)$.
By Proposition \ref{zerotemp}, $\mu$ is a maximizing measure for $\phi$
and hence $P_\mu(\psi)\le a$.
For any $b\in (a,P(\psi))$, as $\mu_{t_k}\to\mu$ and the entropy map
is upper semi-continuous,
there is $N$ such that
$P_{\mu_{t_N}}(\psi)<b$.
Note that $\Phi(t):=\psi+t\phi$ is a continuous path from $[0, t_N]$
to $\sv(f,\psi)$.
By Proposition \ref{propmain},
we have
\begin{equation}\label{pinc}
\spp(\psi)\supset(b,P(\psi)].
\end{equation}
Then $\spp(\psi)\supset[a, P(\psi)]$
because $P_\mu(\psi)\le a$ and
\eqref{pinc} holds for any $b\in (a,P(\psi))$.
\end{proof}

\begin{corollary}\label{intentropy}
Let $(X,f)$ be a USC system with positive topological entropy $h(f)>0$.

Suppose that there is a continuous potential $\phi$ such that
the following holds:

\begin{enumerate}
\item
$(X,f,t\phi)$ has a unique equilibrium state
for every $t\ge 0$.
\item
$(X,f,\phi)$ has a unique maximizing measure $\mu$

with $h_{\mu}(f)=0$.
\end{enumerate}
Then $(X,f)$ has the intermediate entropy property.

\end{corollary}

\begin{proof}
Note that for $t=0$, $\mu_0$ is just the measure of maximal entropy. Hence
$h_{\mu_0}(f)=h(f)$.
Apply Theorem \ref{intpres} for $\psi=0$.
\end{proof}

It is well known that every Axiom A system has unique equilibrium states
for H\"older potentials \cite{Bow75}. One can just pick a fixed point $p$ and take
$\phi(x):=-d(x,p)$ for every $x\in X$, which is a H\"older function
with the unique maximizing measure supported on $p$. In this case Corollary
\ref{intentropy} provides another proof that
every Axiom A system has the intermediate entropy property.

\iffalse

Theorem \ref{uesintent} is a direct corollary of the following proposition.
\begin{proposition}
Let $(X,f)$ be a USC system.
Suppose that there are $\phi\in\cx$ and $\al\in\RR$ such that the following holds:
\begin{enumerate}
\item $\psi+t\phi\in\sv(f,\psi)$ for every $t\ge 0$.
\item $P_\nu(\psi)\le\al$ for every $\nu\in\coo(\phi)$.

\end{enumerate}
Then $\spp(\psi)\supset[\al, P(\psi)]$.
\end{proposition}

\begin{proof}
We claim that
\begin{equation}\label{ptoinf}
\lim_{t\to\infty}P_\psi(\psi+t\phi)=\al.
\end{equation}
Suppose that \eqref{ptoinf} fails.
Then there must be a sequence $\{t_n\}$ and $\delta>0$ such that
$t_n\to\infty$ and
\begin{equation}\label{infdev}
\left|P_\psi(\psi+t_n\phi)-\al\right|>\delta\text{ for all }n\in\ZZ^+.
\end{equation}
Take $\mu_n\in\sE(\psi+t_n\phi)$ for each $n$.
Let $\mu$ be the limit of a convergent subsequence
$\{\mu_{n_k}\}$.
Then
\begin{align*}
&\lim_{k\to\infty}|P_\psi(\phi+t_{n_k}\phi)-\al|

\\\le &
\lim_{k\to\infty}\left|P_{\mu_{n_k}}(\psi)-P_{\mu}(\psi)\right|
\\\le &
\lim_{k\to\infty}\left(\left|P_{\mu_{n_k}}(\phi_{n_k})-P_{\mu}(\phi)\right|
+\left|\int(\phi_{n_k}-\psi)d_{\mu_{n_k}}-\int(\phi-\psi) d_{\mu}\right|\right)
\\\le&\lim_{k\to\infty}\left|P(\phi_{n_k})-P(\phi)\right|
+\lim_{k\to\infty}\left|\int(\phi_{n_k}-\phi) d_{\mu_{n_k}}\right|
\\&
+\lim_{k\to\infty}\left|\int(\phi-\psi) d_{\mu_{n_k}}-\int(\phi-\psi)d_{\mu}\right|
\\\le&2\lim_{k\to\infty}\|\phi_{n_k}-\phi\|
\\=& 0.
\end{align*}

\end{proof}

\begin{corollary}

\end{corollary}

\section{Application}\label{secapp}

For $\phi\in\cx$ and $\ep>0$,
denote
$$V(\phi,\ep):=\sup\left\{\left|\sum_{k=0}^{n-1}
\phi\left(f^k\left(x\right)\right)-\sum_{k=0}^{n-1}
\phi\left(f^k\left(y\right)\right)\right|:x\in X, n\in\ZZ^+, y\in B_n(x,\ep)\right\}.$$
Define the set of potentials satisfying \emph{Bowen property} as
$$BP(f):=\{\phi\in\cx:

\text{ there is $\ep_0>0$ such that }V(\phi,\ep_0)<\infty\}.$$
The seminal work \cite{Bow75} of Bowen shows that $(X,f,\phi)$ has a
unique equilibrium state if $(X,f)$ is an expansive
system with specification property and $\phi\in BP(f)$.
Theorem \ref{uesintent} applies to such systems. However, in this case
it only provides another insight as a much better result has been obtained
in \cite{Sunintent}.

Recently, Climenhaga and Thompson generalized Bowen's result
to systems with much weaker conditions on expansiveness
and specification \cite{CT}, which applies to certain
DA maps \cite{CFT, CTmane} with certain H\"older potentials.

In this note, we show that the unique equilibrium states vary continuously
if the entropy function is upper semi-continuous. This assumption
is a natural condition that guarantees existence of equilibrium states
for all continuous potentials.
Our results apply to the systems and potentials
discussed in the above references.

For convenience, we say that $(X,f)$ is a USC system if
the entropy function $\mu\mapsto h_\mu(f)$ is upper semi-continuous.

\begin{corollary}\label{intent}
Let $(X,f)$ be a USC system with positive topological entropy $h(f)>0$.

Suppose that there is a continuous potential $\phi$ such that
for every $t\in[0,1]$,
$(X,f,t\phi)$ has a unique equilibrium state $\mu_t$ and
$h_{\mu_1}(f)<h(f)$. Then
for every $\al\in[h_{\mu_1}(f),h(f)]$, there is $T=T_\al\in[0,1]$
such that
$h_{\mu_{T_\al}}(f)=\al$.
If in addition $(X,f,t\phi)$ has a unique equilibrium state
for all $t\ge 0$, $\lim_{t\to\infty}\mu_t=\mu$ and $h_{\mu}(f)=0$,
then $(X,f)$ has ergodic measures of arbitrary intermediate entropies.
\end{corollary}

\begin{theorem}\label{zerotempes}
Let $(X,f)$ be a USC system with positive topological entropy $h(f)>0$.

Suppose that there are $\phi\in C(X)$ and $\mu\in\cm(X)$
such that
\begin{enumerate}
\item
$t\phi\in\su(f)$, i.e.
$(X,f,t\phi)$ has a unique equilibrium state,
which is denoted by $\mu_t$,
for each $t\ge 0$.
\item
$\lim_{t\to\infty}\mu_t=\mu$ and $h_{\mu}(f)=0$.
\end{enumerate}
Then $(X,f)$ has the intermediate entropy property.

\end{theorem}

\begin{proof}
Note that $h_\mu(f)=0$ and
$$h_{\mu_0}(f)=P_{\mu_0}(f,0)=P(f,0)=h(f).$$
We just need to show that for every $a\in(0, h(f))$, there
is $\nu_a\in\cm(X,f)$ such that $h_{\nu_a}(f)=a$.

Take $b\in(0,a)$. As $h_\mu(f)=0$ and the entropy functions
is upper semi-continuous, there is $\eta>0$ such that
$$h_\nu(f)<b\text{ for every }\nu\in B(\mu,\eta).$$
As $\lim_{t\to\infty}\mu_t=\mu$, there is $T\in(0,\infty)$ such that
$\mu_T\in B(\mu,\eta)$, hence $h_{\mu_T}(f)<\eta$.
As $t\mapsto t\Phi$ is continuous, by Theorem \ref{}, there is $\nu_a$
with $h_{\nu_a}(f)=a$.
\end{proof}

\fi

\iffalse
\begin{corollary}
Let $(X,f)$ be a Ma\~n\'e diffeomorphism considered in \cite{CTmane} and
\cite{Sunct}. Then
$(X,f)$ has ergodic measures of arbitrary intermediate entropies.
\end{corollary}

\begin{proof}
Take a fixed point $q\in X$.
Let
$$\phi_q(x):=-d(x,q).$$

Denote by $\mu_q$ the Dirac measure on $q$.
Then $\int\phi_q d\mu_q>\int\phi_q d\nu$ for every
$\nu\in\cm(X,f)$.
As $\phi_q$ is Lipschitz, by \cite{CTmane}, $(X,f,t\phi_q)$
(when $r$ is sufficiently
small) has a unique equilibrium state $\mu_t$ for every $t\ge 0$.
But we must have $\mu_t\to\mu_q$.
Apply Corollary \ref{intent}.
\end{proof}
\fi

\section{Application to Ma\~n\'e Diffeomorphisms}
\label{secmane}
Following \cite{CTmane},
we consider the Ma\~n\'e family $\fm_{\rho,r}$,
which is a class of DA (derived from Anosov)
maps first introduced by Ma\~n\'e \cite{Ma78}. They are
$C^0$ perturbations of a hyperbolic toral automorphism $f_A:\btt^d\to\btt^d$,
which are partially hyperbolic  with 1-dimensional centers.
Let $q$ be a fixed point of $f_A$.
As described in \cite{CTmane}, for each $g\in\fm_{\rho,r}$ we assume that:
\begin{enumerate}

\item
$\rho>0$ such that the neighborhood $B(q,\rho)$ is  the support of the perturbation, i.e.
$g=f$ on $\btt^3\backslash B(q,\rho)$.

\item $r\in[0,1]$ such that if an orbit of $g$ spends a proportion at least $r$
of its time outside $B(q,\rho)$, then it contracts the vectors in the central direction.
\item The $C^0$ distance between $g$ and $f_A$ is sufficiently small,
i.e. there is a constant $\eta=\eta(f_A)$ depending only on $f_A$
such that $d_{C^0}(g,f_A)<\eta$.
In particular, this holds when $\rho$ is sufficiently small.
\end{enumerate}

For $\phi\in C(\bttd)$, $\rho, L>0$ and $r\in(0,1)$, denote
$$
\Xi(\rho,r,\phi, L):=(1-r)\sup_{B(q,\rho)}\phi+r(\sup_{\bttd}\phi+h(f_A)+L)+H(2r),
$$
where
$$
H(r):=-r\ln r-(1-r)\ln(1-r).$$
Denote
$$\sC_{\rho,r,g,L}:=\{\phi\in C(\bttd):P(g,\phi)>\Xi(\rho,r,\phi, L)\}$$
Let $\phi$ be an $\al$-H\"older potential on $\btt^d$ and
$$|\phi|_\al:=\sup\left\{\frac{|\phi(x)-\phi(y)|}{d(x,y)^\al}:x,y\in X,
x\ne y\right\}$$
be its H\"older semi-norm.
Denote
$$\sC^\al_M:=\{\phi\in C(\bttd):|\phi|_\al<M\}.$$

\begin{theorem}[{\cite[Theorem A and B]{CTmane}}]\label{ctm}
Let $g\in\fm_{\rho,r}$
and $\phi$ be an $\al$-H\"older continuous function
on $\btt^3$.
\begin{enumerate}
\item \label{bddgen}
There
is a constant $L=L(f_A)$

depending only on
$f_A$ such that
$(\btt^d,g,\phi)$ has a unique equilibrium state as long as
$\phi\in\sC_{\rho,r,g,L}$.

\item \label{bddhol}
There is a function
$M(\rho,r)$ such that
$M(\rho,r)\to\infty$ as $\rho,r\to 0$
and $(\bttd,g,\phi)$ has a unique equilibrium sate as long as
$\phi\in\sC^\al_{M(\rho,r)}$.

\end{enumerate}
\end{theorem}

Let $p$ be another fixed point of $f_A$ such that
$p\notin\overline{B(q,\rho)}$.

We can fix
$\Psi\in C(\bttd)$ such that
\begin{enumerate}
\item $\Psi(p)=0$.
\item $\Psi(x)=-1

$ for every $x\in B(q,\rho)$.
\item $\Psi(x)<0$ for every
$x\in\bttd\backslash
\{p\}$.

\item $\Psi$ is $\al$-H\"older.
\end{enumerate}

Denote by $\mu_p$ the Dirac measure on $p$.
Then $\mu_p$ is the unique maximizing measure for $\Psi$ and
$h_{\mu_p}(g)=0$.

\begin{lemma}\label{alltuni}
There are $\rho^*>0$ and $r^*\in(0,1)$ such that
for every $g\in\fm_{\rho^*,r^*}$, we have
$t\Psi\in\su(g)$ for all $t\ge 0$.
\end{lemma}

\begin{proof}

Denote $$\beta_r:=\frac{r(h(f_A)+L)+H(2r)}{1-r}.$$
Note that $\beta_r\to 0$ as $r\to0$.

By Theorem \ref{ctm}\eqref{bddhol},
there are

$\rho^*>0$ and $r^*\in(0,1)$ such that
$$
\beta_{r^*}|\Psi|_\al<M(\rho^*,r^*)
$$
This implies that

for every $g\in\fm_{\rho^*,r^*}$, we have
$$t\Psi\in\sC^\al_{M(\rho^*,r^*)}\subset\su(g)
\text{ for every }t\in[0,\beta_{r^*}].$$

For $t>\beta_{r^*}$,
as $h_{\mu_p}(g)=0$,

by the Variational Principle,
we have
\begin{align*}
\Xi(\rho^*,r^*,t\Psi, L)&=(1-r)(-t)+r(h(f_A)+L)+H(2r)
\\&<0
\\&=h_{\mu_p}(g)+\int (t\Psi)d\mu_p
\\&\le P(g,t\Psi).
\end{align*}

By Theorem \ref{ctm}\eqref{bddgen},
for every $g\in\fm_{\rho^*,r^*}$, we have
$$t\Psi\in\sC_{\rho^*,r^*,g,L}\subset\su(g)\text{ for every }t>\beta_{r^*}.$$
\end{proof}

\iffalse
\begin{lemma}\label{alltuni2}

Let $\psi$ be an $\al$-H\"older continuous function
on $\btt^3$.
There are $\rho^*>0$ and $r^*\in(0,1)$ such that
for every $g\in\fm_{\rho^*,r^*}$, we have
$\psi+t\Phi\in\su(g)$ for every $t\ge 0$.
\end{lemma}

\begin{proof}

Denote $$\beta_r:=\frac{r(h(f_A)+L)+H(2r)}{1-r}.$$

Note that $\beta_r\to 0$ as $r\to0$.

By Theorem \ref{ctm}\eqref{bddhol},
there are
$\rho^*,r^*>0$ such that
$$
\beta_{r^*}|\Phi|_\al<M(\rho^*,r^*)
$$
This implies that for every $g\in\fm_{\rho^*,r^*}$, we have
$$t\Phi\in\sC^\al_{M(\rho^*,r^*)}\subset\cU(g)
\text{ for every }t\in[0,\beta_{r^*}].$$

Denote by $\mu_p$ the Dirac measure on $p$.
For $t>\beta_{r^*}$, we have
\begin{align*}
\Xi(\rho^*,r^*,t\Phi, L)&=(1-r)(-t)+r(h(f_A)+L)+H(2r)
\\&<0
\\&=h_{\mu_p}(f)+\int (t\Phi)d\mu_p
\\&\le P(f,t\Phi).
\end{align*}
This implies that for every $g\in\fm_{\rho^*,r^*}$, we have
$$t\Phi\in\sC_{\rho^*,r^*,g,L}\subset\su(g)\text{ for }t>\beta_{r^*}.$$
\end{proof}
\fi

\begin{remark}
We may choose $\Psi$ such that $|\Psi|_\al$ is as small as possible
to achieve larger values of $\rho^*$ and $r^*$ as in Lemma \ref{alltuni}.

\end{remark}

\begin{theorem}
For every $g\in\fm_{\rho^*,r^*}$, the system $(\bttd,g)$
has the intermediate entropy property.
\end{theorem}

\begin{proof}
By \cite[Proposition 6]{CL}, every Ma\~n\'e diffeomorphism is
entropy expansive. So $(\bttd,g)$ is a USC system.
As $\mu_p$ is the unique maximizing measure for $\Psi$ and
$
h_{\mu_p}(g)=0$,

the
conclusion
follows from Lemma \ref{alltuni} and
Corollary \ref{intentropy}.

\end{proof}

\iffalse
\begin{proof}
Let $p$ be another fixed point of $f_A$.
Let
$$\psi(x):=-d(x,p).$$

Denote by $\mu_p$ the Dirac measure on $p$.
Then $\int\psi d\mu_p>\int\psi d\nu$ for every
$\nu\in\cm(X,f)$.
This implies that $\mu_t\to\mu_p$ as $t\to\infty$.

As $\phi_q$ is Lipschitz, by \cite{CTmane}, $(X,f,t\phi_q)$
(when $r$ is sufficiently
small) has a unique equilibrium state $\mu_t$ for every $t\ge 0$.
But we must have $\mu_t\to\mu_q$.
Apply Corollary \ref{intent}.
\end{proof}
\fi

\section*{Acknowledgments}
The author is supported by
National Natural Science Foundation of China
(No. 11571387) and
CUFE Young Elite Teacher Project (No. QYP1902).
The author learned about ergodic optimization
from Yiwei Zhang and Yun Yang, and
would like to thank them
for fruitful discussions.

config: arxiv


TITLE: Quotient of two smooth functions with same vanishing locus is $C^1$
QUESTION [0 upvotes]: Let $f$ and $g$ be smooth functions on a subset of $\mathbb{R}^{2n}$, $n > 1$, and suppose that $dg \neq 0$. Say that $f$ vanishes on the vanishing locus of $g$. It is claimed that $f/g$ is $C^1$, but I can't quite show why this is the case.  When you differentiate you get
$$ df/g - (f/g^2) dg $$
and while in all the examples I have done there is cancellation, I can't see why this has to hold in general.
Thanks for your help.

REPLY [1 votes]: The idea is that since $dg\neq 0$ on the vanishing locus of $g$, $g$ vanishes linearly on it. But $f$ can only vanish at those points either in a linear way or higher. In the first case, $f$ is locally proportional to $g$ ang $f/g$ is finite and well behaved there. In the second, $f/g$ vanishes but is still well behaved.  In a sentence: $g$ is an infinitesimal of the lowest possible order there.


TITLE: What makes a condition unary vs. n-ary (n>1)?
QUESTION [1 upvotes]: For any two disjoint sets $A$ and $B$, a set $W$ is a connection of $A$ with $B$ if

$Z\in W\implies (\exists x\in A)(\exists y\in B)[Z=\{x,y\}]$
$(\forall x\in A)(\exists !y\in B)[\{x,y\}\in W]$
$(\forall y\in B)(\exists !x\in A)[\{x,y\}\in W]$

I know that each of the conditions 1-3 are definite conditions (and indeed being a connection in general is a definite condition). But I'm having trouble telling when I should consider something an $n$-ary condition $(n>1)$ vs a unary condition.
I want to use the Axiom of Separation (Zermelo axioms) which requires a definite condition to be unary. The condition that I want to use is that of a set being a connection. But, to me, that would be a ternary condition, say $P(A,B,W)$. So I would have to "decompose" this ternary condition into unary conditions and apply the separation axiom multiple times, no?

REPLY [1 votes]: Axiom schema of Separation, in its more general form says that :

If $\phi(x)$ is a formula (in which $z$ does not occur free) then $$( \forall w_1 ... w_n ) ( \forall y ) ( \exists z ) ( \forall x ) ( x \in z \leftrightarrow ( x \in y \wedge \phi (x, w_1, ... w_n) ) )$$

or, in words, given any set $y$, there is a set $z$ such that, given any set $x$, $x$ is a member of $z$ if and only if $x$ is a member of $y$ and $\phi$ holds for $x$, where $w_1, ... w_n (n \ge 0 )$ are the parameters.
So, using the formula $P(A,B,W)$, you have a formula with $W$ free and $A$ and $B$ as parameters. Then you can use it in the Axiom schema, obtaining that :

$( \forall w_1 ) ( \forall w_2 ) ( \forall y ) ( \exists z ) ( \forall x ) ( x \in z \leftrightarrow ( x \in y \wedge P(w_1, w_2, x) ) )$.


TITLE: Imaginary number and absolute value integral - Fourier transform
QUESTION [2 upvotes]: I came across this integral problem:
$$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$
Now I know how to integrate simple absolute value functions like:
$\int_{-2}^{4}|x-2| dx$, we just find the 'break points', which in this case is $x=2$ so we would integrate $\int_{-2}^2 (x-2) dx + \int_{2}^{4}(2-x)dx$.
But in  that first equation we have an imaginary number. How do I deal with it?

REPLY [1 votes]: $$\int_{-\infty}^{\infty} e^{-|x| + xi\xi} \ dx = \int_{-\infty}^0 e^{x(1+i\xi)} \ dx + \int_0^{\infty}e^{-x(1-i\xi)} \ dx$$
And then my friend pointed out the following:
$e^{x+ix\xi}=e^xe^{ix\xi}=e^x (\cos(x\xi)+i\sin(x\xi))\leq e^x (1+i)$
Now:
$e^x (1+i)$ - so we can ignore the $(1+i)$ as it is bounded and we only care about $e^x$ which goes to zero as $x \rightarrow -\infty$
therefore:
$$\int_{-\infty}^0 e^{x(1+i\xi)} \ dx + \int_0^{\infty}e^{-x(1-i\xi)} = \frac{1}{1+i\xi} + \frac{1}{1-i\xi}=\frac{2}{1+\xi^2}$$


TITLE: Jech 3rd Edition Section 12 page 162 Models of Set Theory
QUESTION [2 upvotes]: Jech page 162 states :
Let Form denote the set of all formulas of the language {$\in$}. As with any actual (metamathematical) natural number we can associate the corresponding element of N (i.e. the smallest inductive set), we can similarly associate with any given formula of set theory the corresponding element of the set Form. To make the distinction, if $\varphi$ is a formula, let $\ulcorner \varphi \urcorner    $ denote the corresponding element of Form.
If M is a set and E is a binary relation on M and if $a_1,..,a_n$ are elements of M, then
(12.16)$\hspace{2cm} \varphi^M{^,}^E(a_1,...,a_n)\leftrightarrow(M,E)\models\ulcorner \varphi \urcorner[a_1,...,a_n]$
as can be easily verified. Thus in the case when M is a set and $\varphi$ a particular (metamathematical) formula, we shall not make a distinction between the two meanings of the symbol $\models$. We note however that the left-hand side of (12.16) (relativization) is not defined for $\varphi$ $\in$ Form, and the right-hand side (satisfaction) is not defined if M is a proper class.
Would it be possible to ask two questions :
1) As Jech works in ZFC metatheory, is Form a set in the metatheory defined by a metamathematical ZFC formula with its variables ranging over the formulae $\varphi$ coded as metamathematical ZFC sets  $\ulcorner \varphi \urcorner$.
and
2) Why in (12.16) does $\ulcorner \varphi \urcorner$ appear on the right hand side, and not just $\varphi$, with (12.16) being a metamathematical ZFC formula, or is (12.16) actually a metamathematical ZFC formula thats using coded formula within the coded logic & model theory? (apologies this isnt too clear - hence the question - I tried to use https://mathoverflow.net/questions/23060/set-theory-and-model-theory/23077#23077). If anyone knows of a book that describes in detail how metamathematical ZFC can be used to code logic & model theory that would be good to know.

REPLY [1 votes]: I'll answer your second question first: I don't really know of a book that discusses exactly what is going on in this case in depth. But the ideas of coding used here are the same as those in Godel's incompleteness theorem. Recall that $ZFC$ is formally a countable language. So that kind of coding makes sense. This can be done with PRA for example, which is much weaker than set theory.
As to the first question; I believe what is going on is the following (hopefully someone will point it out if I'm wrong); We work within a meta theory, such as PRA (Primitive Recursive Arithmetic). The exact meta theory used is of interest to certain people, but is not really necessary for the purposes of this question.
Now we have way of talking about what it means for $ZFC$ to prove something. We develop model theory inside of ZFC the usual way. Now Jech is saying: $ZFC\vdash({\varphi}^{(M,E)}\leftrightarrow{(M,E)\models{\ulcorner\varphi\urcorner}})$. Here inside of $ZFC$ we have developed the formal theory of (a copy of) ZFC. So we assume that $\ulcorner\varphi\urcorner$ is associated in some canonical way to $\varphi$ (The formula $\varphi$ is actually represented by a code (a unique natural number) in the meta theory. Now $ZFC$ also has access to a version of the natural numbers, then we are implicitly assuming that the code in the meta theory is the "same as the" code in the formal copy of $ZFC$ in $ZFC$).


TITLE: Logarithm Question $\log_{3}9x+\log_{3}x=4$
QUESTION [0 upvotes]: Edit: Mistake was at $\log_{3}9x^2=4$ solution shown below
Confused on this question, not sure what I did wrong here.
$$\boxed{\log_{3}9x+\log_{3}x=4}$$
$$\log_{3}9x+\log_{3}x=4$$
$$\log_{3}9x\cdot x=4$$
$$\log_{3}9x^2=4$$
$$\dfrac{2\log_{3}9x=4}{2}$$
$$\log_{3}9x=2$$
$$3^2=9x$$
$$9=9x$$
$$x=1$$

$$\log_{3}9x^2=4$$
$$3^4=9x^2$$
$$\dfrac{81=9x^2}{9}$$
$$\sqrt{x^2=9}$$
$$\boxed{x=3, x\ne-3}$$

REPLY [3 votes]: A slightly different approach
$\log_{3}9x+\log_{3}x=4\iff \log_39+\log_3x+\log_3x=4\iff 2+2\log_3x=4\iff$
$\iff 2\log_3x=2\iff \log_3x=1\iff x=3$


TITLE: Meaning of "a mapping factors over another"?
QUESTION [14 upvotes]: I was wondering what "a mapping factors over another mapping" generally means? Does it have something to do with commutative diagram in category theory?
I have seen this usage in different situations, and would like to give examples, but cannot recall them one by one except the most recent one from Fiber bundle:

Mappings which factor over the projection map are known as bundle
  maps.

Thanks and regards!

REPLY [11 votes]: As was pointed out in the comments:
Given a morphism $f:A \to C$ and a morphism $g: A \to B$ then $f$ is said to factor over $g$ if there exists $h: B \to C$ such that $hg = f$. Note that $h$ is unique as soon as $g$ is an epimorphism.
In the case of bundles $\pi_{E}: E \to M$ and $\pi_F: F \to N$ then a map $\varphi: E \to F$ is a bundle map if there exists $f:M \to N$ such that $\pi_{F} \varphi = \varphi \pi_{E}$, as is noted further down on the wikipedia page you linked to. So $\varphi : E \to F$ is a bundle map if and only if $\pi_{F} \varphi$ factors over $\pi_{E}$ via a (necessarily unique) map $f: M \to N$, as the bundle projections are assumed to be surjective. Note that this means in particular that $\varphi$ maps fibers to fibers.
Similarly, there is the notion of factoring through (you need to scroll down a little).
I'm mostly using this for the situation $f: A \to C$ and $h: B \to C$ then $f$ factors through $h$ if there is $g: A \to B$ such that $f = hg$. If $h$ happens to be a monomorphism then $g$ is unique (if it exists).
However, the distinction I'm making in this post are far from universally accepted.
You'll also find $f:A \to C$ factors through $B$ if there are maps $g: A \to B$ and $h: B \to C$ such that $f = hg$ and so on, $f$ factors through $g$, $f$ factors through $h$ in this same situation.
Basically, all it means is that $f$ can be decomposed as $f = hg$ in some way that should be obvious from the context. To repeat, if either $h$ is a monomorphism then $g$ is unique and if $g$ is an epimorphism then $h$ is unique.

REPLY [3 votes]: The most general, broader description of your question is that you have two sets $A,B$ and maps $f,g$ into a common third space $C$, i.e., you have ; $f:A\rightarrow C$, and you have $g\colon B\rightarrow C$, and then you want to see if there exists a map $h$ between $A$ and $C$, so that "the diagram commutes" (if you ever try to do algebraic topology you will see this everywhere; in analysis, almost everywhere ; ) , i.e., if you follow the arrows and do the composition from $A$ to $B$ to $C$ , composing $g\circ h$, you will get the same result as if you go along the arrow from $A$ to $C$ alone. It is difficult to generalize because the results will depend on the structure of $A,B$ and $C$ given.
In the case of bundles you mention, a bundle map associated with a bunlde $\pi\colon E\rightarrow B$ is a map that sends an element $b$ in the base to its fiber under $\pi$, so that, if you compose back the fiber element, you get $b$ back, i.e., $\pi \circ f(b)=b$ e.g., if your fiber is a vector space, your bundle map will send $b$ to an element in the vector space, so that the composition with $\pi$ will project back down to $b$.