text:
\begin{document}
	\title{A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations}

	\author[*]{
		Chunrong Feng}
	\author [$\dag$,*] {Baoyou Qu}
	\author[*]{Huaizhong Zhao}
	\affil[*]{Department of Mathematical Sciences, Loughborough
		University, LE11 3TU, UK}
	\affil[$\dag$] {Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China}

	\affil[ ]{C.Feng@lboro.ac.uk, qu@mail.sdu.edu.cn, H.Zhao@lboro.ac.uk}
	\date{}

\maketitle

\begin{abstract}
	This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincar\'{e} section or completely outside a Poincar\'{e} section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give a number of  examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.
	\medskip

	\noindent
	{\bf Keywords:} ergodicity, periodic measure, random dynamical system, periodic path, Markov semigroup, PS-ergodic, sublinear expectation.

\end{abstract}

\section{Introduction}
  Ergodic theory is one of the most important observations in mathematics made in the last century with significance in many areas of physics such as statistical physics (c.f. \cite{Birkhoff}, \cite{Neumann1}, \cite{Neumann2}, \cite{Walters}). Concerning the spreading and irreducibility nature of dynamical systems, ergodicity is a fundamentally natural phenomenon to many stochastic systems (c.f. \cite{Da Prato}, \cite{Doob}, \cite{Durrett}, \cite{Hasminskii}). The theory was substantially developed for (weakly) mixing invariant measures/stationary processes in the stationary regime. Many useful results were obtained especially in the Markov stochastic dynamical system case.

  However, the classical theory excludes periodic cases, which is even the case in the theory of Markov chains. The ergodic theory of periodic random dynamical system was observed recently in \cite{F.Z2}. The concept of periodic measure was introduced and its ``equivalence" with random periodic processes was established. Moreover, the average of a periodic measure over one period is an invariant measure from which the ergodicity can be studied. It is defined as the ergodicity of a measure preserving canonical dynamical systems lifted from the invariant measure on the phase space. From the ergodic theory of periodic random dynamical systems in \cite{F.Z2}, the distinction between the stationary regime and random periodic regime is characterised by the spectral structure of Markov semigroups or their infinitesimal generators. In the stationary regime, Koopman-von Neumann theorem tells us that 0 is a simple and unique eigenvalue of the generator on the imaginary axis (c.f. \cite{Da Prato}), while in the periodic regime, the infinitesimal generator has infinitely many equally placed eigenvalues (including 0), which are simple, and no other eigenvalues, on the imaginary axis (\cite{F.Z2}).

  In this paper, we continue the study on the ergodicity of periodic measures and obtain some new results. First we study random periodic paths on a separable Banach space and associated periodic measures $\mu_s=\delta_{Y(s, \theta_{-s}\omega)}\times dP$ on the product space $(\bar{\Omega}, \bar{\mathcal{F}})=(\Omega\times \mathbb{X}, \mathcal{F}\otimes \mathcal{B}(\mathbb{X}))$. Then $\{\mu_s\}_{s\geq 0}$ is a periodic measure with the skew product $(\bar{\Theta}_t)_{t\geq 0}$. We prove in the first part of the paper that $(\Omega, \mathcal{F}, P, (\theta_{n\tau})_{n\geq 0})$ is ergodic if and only if $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_s, (\bar{\Theta}_{n\tau})_{n\geq 0})$ is ergodic, i.e. the dynamical system $(\bar{\Omega}, \bar{\mathcal{F}}, \{\mu_{s}\}_{s\in \mathbb{R}}, (\bar{\Theta}_{t})_{t\geq 0})$ is PS-ergodic. Note here there is no need of any other conditions on its random periodic paths $Y$ apart from the existence and the definition of random periodic paths.

  The metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_{n\tau})_{n\geq 0})$ being ergodic is stronger than the statement that $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\geq 0})$ is ergodic. They are not normally equivalent. We will give an example that $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\geq 0})$ is ergodic, but the discrete metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_{n\tau})_{n\geq 0})$ is not ergodic. However, we will prove in this paper, for the canonical Wiener process and the Brownian shift, both the discrete dynamical systems and continuous time dynamical systems are ergodic. The result of the discrete dynamical systems of Wiener space is new. This means that our results can apply to
 stochastic differential equations and stochastic partial differential equations driven by Wiener processes. Suggested by fundamental results of \cite{Arnold}, \cite{Elworthy}, \cite{Flandoli}, \cite{Kunita}, \cite{Meyer} and \cite{Mohammed}, these equations can generate random dynamical systems, of which the noise metric dynamical system over a group of discrete time is ergodic according to our results here.

  For the Markov random dynamical systems, the random periodic paths give rise to periodic measures $(\rho_s)_{s\in \mathbb{R}}$ on the state space $\mathbb{X}$. For each $s$, $\rho_s$ is an invariant measure with respect to discrete semigroup $P(n\tau), n\in \mathbb{N}$. We will give a necessary and sufficient condition for the periodic measure $(\rho_s)_{s\in \mathbb{R}}$ being PS-ergodic (i.e. for each $s\in \mathbb{R}$, $\rho_s$ is ergodic as an invariant measure with respect to $P(n\tau), n\in \mathbb{N}$), which says for any invariant set $\Gamma$ such that $P_{\tau}\I_{\Gamma}=\I_{\Gamma}$ $\rho_s-a.s.$, the section $L_s^{\omega}:=\{Y(s+k\tau, \omega), k\in \mathbb{Z}\}$ satisfies $L_s^{\omega}\subset \Gamma$ or $L_s^{\omega}\cap \Gamma=\emptyset$, $P-a.s.$. However, it is not known whether or not this result is true in the stationary case.

  Sublinear expectation is used to model uncertainty and ambiguity of probabilities such as subjective probabilities due to heterogeneity of expectation formation process (c.f. \cite{Artzner}, \cite{Delbaen}, \cite{Peng}). An ergodic theory of sublinear expectation was developed recently by \cite{F.Z1}. In the second part of this paper, we construct an ergodic sublinear expectation from an ergodic periodic measure as an upper expectation for the first time in literature. We prove that if a periodic measure is ergodic, then the generated sublinear expectation, which is invariant with respect to the skew product dynamical system or the Markov semigroup, is ergodic. As for the Birkhoff type ergodic theorem, i.e. the law of large number, we obtain the convergence in the quasi-sure sense when we apply the ergodic theory of upper expectations, whilst we can only obtain the convergence in the almost-sure sense by the ergodic theory of periodic measures (\cite{F.Z2}). This provides one of the justifications why we see construction of upper expectation and the investigation of its ergodicity, can provide useful new information. The point of view of upper expectations from periodic measures could also be interesting to the study of finance or coherent risk measure.

  \section{Ergodicity of skew product dynamical systems: necessary and sufficient conditions}
  Consider a random dynamical system $\Phi: \mathbb{R^+}\times \Omega\times \mathbb{X} \rightarrow \mathbb{X}$ over a metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_t)_{t\in \mathbb{R}})$ on a separable Banach space $\mathbb{X}$. It is a measurable mapping and almost surely $\Phi_0=id$ and $\Phi(t+s, \omega)=\Phi(t, \theta_s\omega)\circ \Phi(s, \omega)$ for any $t,s\geq 0$. The map $\theta: \mathbb{R}\times \Omega \rightarrow \Omega$ is measurable with respect to $(\mathcal{B}(\mathbb{R})\otimes\mathcal{F}, \mathcal{F} )$ such that $\theta_t\circ\theta_s=\theta_{t+s}, t, s\in \mathbb{R}$  and preserves the measure $P$, i.e. $\theta_sP=P$. Random dynamical systems can be generated by stochastic differential equations (\cite{Arnold}, \cite{Elworthy}, \cite{Kunita}, \cite{Meyer}), stochastic partial differential equations (\cite{Flandoli}, \cite{Garrido}, \cite{Mohammed}) and Markov chains (\cite{Kifer}).

  Let us recall the definition of random periodic paths (\cite{FZZ}, \cite{F.Z3}, \cite{F.Z2}, \cite{ZZ}).

  \begin{definition}
  	A random periodic path of period $\tau$ of the random dynamical system $\Phi : \mathbb{R}^+ \times \Omega \times \mathbb{X} \rightarrow  \mathbb{X}$ is an $\mathcal{F}$-measurable map $Y:\mathbb{R} \times \Omega \rightarrow \mathbb{X}$ such that for almost all $\omega \in \Omega$,
  	\begin{equation}
  	\label{RPP}
  	\Phi(t, \theta_{s}\omega)Y(s, \omega)=Y(t+s, \omega),    Y(s+\tau, \omega)=Y(s, \theta_{\tau}\omega).
  	\end{equation}
  	It is called a random periodic path with the minimal period $\tau$ if $\tau >0$ is the smallest number such that (\ref{RPP}) holds.
  	It is a stationary path of $\Phi$ if $Y(s, \theta_{-s}\omega)=Y(0, \omega)=:Y_{0}(\omega)$ for all $s \in \mathbb{R}, \omega \in \Omega$ i.e. $Y_{0}: \Omega \rightarrow \mathbb{X}$ is a stationary path if for almost all $\omega \in \Omega$,
  	\begin{equation}
  	\Phi (t, \omega)Y_{0}(\omega)=Y_0(\theta_{t}\omega), \text{ for all } t \in \mathbb{R}^+.
  	\end{equation}
  \end{definition}

  Consider a standard product measurable space $(\bar{\Omega}, \bar{\mathcal{F}})=(\Omega \times \mathbb{X}, \mathcal{F}\otimes \mathcal{B}(\mathbb{X}))$ and the skew-product of the metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in \mathbb{R}})$ and the cocycle $\Phi(t, \omega)$ on $\mathbb{X}$,
  $\bar{\Theta}_{t}: \bar{\Omega}\rightarrow \bar{\Omega}$
  \begin{equation}
  \bar{\Theta}_{t}(\bar{\omega})=(\theta_{t}\omega, \Phi(t,\omega)x),  \text{ for all } \bar{\omega}=(\omega, x)\in \bar{\Omega}, t \in \mathbb{R}^+.
  \end{equation}
  Recall
  $$\mathcal{P}_{P}(\Omega \times \mathbb{X}):=\{\mu: \text{probability measure on } (\Omega \times \mathbb{X}, \mathcal{F}\otimes \mathcal{B}(\mathbb{X})) \text{ with marginal } P \text{ on } (\Omega, \mathcal{F})\}$$
  and
  $$\mathcal{P}(\mathbb{X})=\{\rho: \text{ probability measure on } (\mathbb{X}, \mathcal{B}(\mathbb{X}))\}.$$

  The following definition was given in \cite{F.Z2}.

  \begin{definition}
  	A map $\mu: \mathbb{R}\rightarrow \mathcal{P}_{P}(\Omega \times \mathbb{X})$ is called a periodic probability measure of period $\tau$ on $(\Omega \times \mathbb{X}, \mathcal{F}\otimes \mathcal{B}(\mathbb{X}))$ for the random dynamical system $\Phi$ if
  	\begin{equation}
  	\label{PM}
  	\mu_{s+\tau}=\mu_{s} \text{ and } \bar{\Theta}_{t}\mu_{s}=\mu_{t+s}.
  	\end{equation}
  	It is called a periodic measure with minimal period $\tau>0$ if $\tau$ is the smallest number such that (\ref{PM}) holds. It is an invariant measure if it also satisfies $\mu_{s}=\mu_{0}$ for any $s \in \mathbb{R}$, i.e. $\mu_{0}$ is an invariant measure of $\Phi$ if $\mu_{0}\in \mathcal{P}_{P}(\Omega \times \mathbb{X})$ and
  	\begin{equation}
  	\bar{\Theta}_{t}\mu_{0}=\mu_{0}, \text{ for all } t \in \mathbb{R}^+.
  	\end{equation}
  \end{definition}

  \begin{theorem}
  	\label{RPPM}
  	(\cite{F.Z2}) If a random dynamical system $\Phi: \mathbb{R}^+\times \Omega \times \mathbb{X}\rightarrow \mathbb{X}$ has a random periodic path $Y: \mathbb{R}\times \Omega\rightarrow \mathbb{X}$, it has a periodic measure on $(\Omega\times \mathbb{X}, \mathcal{F}\otimes\mathcal{B}(\mathbb{X}))$ $\mu: \mathbb{R}\rightarrow \mathcal{P}_{P}(\Omega \times \mathbb{X})$ given by
  	\begin{equation}
  	\label{PMRDS}
  	\mu_{s}(A)=\int_{\Omega}\delta_{Y(s, \omega)}(A_{\theta_{s}\omega})P(d\omega),
  	\end{equation}
  	where $A_{\omega}$ is the $\omega$-section of $A$. Moreover, the time average of the periodic measure defined by
  	\begin{equation}
  	\bar{\mu}=\frac{1}{\tau}\int_{0}^{\tau}\mu_{s}ds
  	\end{equation}
  	is an invariant measure of $\Phi$ whose random factorisation is supported by $L^{\omega}$.

  \end{theorem}

  \begin{remark}
  	For a periodic path Y, it is easy to see that the factorization of $\mu_{s}$ defined in Theorem \ref{RPPM} is
  	\begin{equation}
  	(\mu_{s})_{\omega}=\delta_{Y(s, \theta_{-s}\omega)}
  	\end{equation}
  	and
  	\begin{equation}
  	(\mu_{s+\tau})_{\omega}=(\mu_{s})_{\omega}, \quad \Phi(t, \omega)(\mu_{s})_{\omega}=(\mu_{t+s})_{\omega}.
  	\end{equation}
  \end{remark}

  In this section, we always assume the following condition and use the construction of periodic measure given in (\ref{PMRDS}).\\
  \\
  \textbf{Condition P}. \textit{There exists a random periodic path $Y$ with period $\tau$ for the random dynamical system $\Phi$}.

  Thoughout the paper, we adopt the standard definition of ergodicity of a measure preserving dynamical system, i.e. any invariant set of the dynamical system has either full measure or zero measure.

  \subsection{Periodic measures}
  \begin{theorem}
  	\label{ET}
  	Assume Condition P. If the metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_{\tau}^{n})_{n\geq 0})$ is ergodic, then the skew product dynamical systems $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^{n})_{n\geq 0})$ for each $s\in \mathbb{R}$ and $(\bar{\Omega}, \bar{\mathcal{F}}, \bar{\mu}, \\
  	(\bar{\Theta}_{t})_{t\geq 0})$ are ergodic.
  \end{theorem}
  \begin{proof}
  	First, note by Theorem \ref{RPPM},  that $\mu_{s}$ defined in (\ref{PMRDS})  is a periodic measure on $(\bar{\Omega}, \bar{\mathcal{F}})$, so $\bar{\Theta}_{\tau}$ preserves measures $\mu_{s}$ for each $s\in \mathbb{R}$ and $(\bar{\Theta}_{t})_{t\geq 0}$ preserves the measure $\bar{\mu}$.

  	Next, we will show that the dynamical system $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for any fixed $s\in \mathbb{R}$. By definition of ergodicity, we need to show that for any $A\in \bar{\mathcal{F}}$ with $\bar{\Theta}_{\tau}^{-1}A=A$, either $\mu_{s}(A)=0$ or 1. Define $A_{s}:=\{\omega: (\omega, Y(s,\theta_{-s}\omega))\in A\}$, then
  	\begin{equation}
  	\label{key1}
  	\begin{split}
  	\mu_{s}(A)
  	&=\int_{\Omega}\I_{A_{\omega}}(Y(s, \theta_{-s}\omega))P(d\omega)\\
  	&=P(\{\omega: (\omega, Y(s, \theta_{-s}\omega))\in A\})\\
  	&=P(A_{s}).\\
  	\end{split}
  	\end{equation}
  	Moreover
  	\begin{equation*}
  	\begin{split}
  	\theta_{\tau}^{-1}A_{s}&=\{\omega: \theta_{\tau}\omega\in A_{s}\}\\
  	&=\{\omega: (\theta_{\tau}\omega, Y(s, \theta_{-s}\theta_{\tau}\omega))\in A\}\\
  	&=\{\omega: \bar{\Theta}_{\tau}(\omega, Y(s-\tau, \theta_{-(s-\tau)}\omega))\in A\}\\
  	&=\{\omega: (\omega, Y(s-\tau, \theta_{-(s-\tau)}\omega))\in \bar{\Theta}_{\tau}^{-1}A\}\\
  	&=\{\omega: (\omega, Y(s-\tau, \theta_{-(s-\tau)}\omega))\in A\}\\
  	&=\{\omega: (\omega, Y(s, \theta_{-s}\omega))\in A\}\\
  	&=A_{s},\\
  	\end{split}
  	\end{equation*}
  	for all $s\in \mathbb{R}$. Thus $A_s$ is an invariant set with respect to $\theta_{\tau}$.

  	Since $(\Omega, \mathcal{F}, P, (\theta_{\tau}^{n})_{n\geq 0})$ is ergodic, then we have $P(A_{s})=0$ or $P(A_{s})=1$. Thus $\mu_{s}(A)=P(A_s)=0$ or 1 from (\ref{key1}).

  	Let us show that the dynamical system $(\bar{\Omega}, \bar{\mathcal{F}}, \bar{\mu}, (\bar{\Theta}_t)_{t \geq 0})$ is ergodic, i.e. for any $A\in \bar{\mathcal{F}}$ with $\bar{\Theta}_{t}^{-1}A=A$ for all $t \in \mathbb{R}^+$, we need to prove that $\bar{\mu}(A)=0$ or 1. For such $A$, by what we just proved, we know that $\mu_{s}(A)=0$ or 1 for all $s\in \mathbb{R}$. On the other hand, since
  	\begin{equation}
  	\begin{split}
  	\theta_{t}^{-1}A_{s}&=\{\omega: \theta_{t}\omega\in A_{s}\}\\
  	&=\{\omega: (\theta_{t}\omega, Y(s, \theta_{-s}\theta_{t}\omega))\in A\}\\
  	&=\{\omega: \bar{\Theta}_{t}(\omega, Y(s-t, \theta_{-(s-t)}\omega))\in A\}\\
  	&=\{\omega: (\omega, Y(s-t, \theta_{-(s-t)}\omega))\in \bar{\Theta}_{t}^{-1}A\}\\
  	&=\{\omega: (\omega, Y(s-t, \theta_{-(s-t)}\omega))\in A\}\\
  	&=A_{s-t},\\
  	\end{split}
  	\end{equation}
  	 we have $\mu_{s}(A)=P(A_{s})=P(\theta_{t}^{-1}A_{s})=P(A_{s-t})=\mu_{s-t}(A)$. This means $\mu_{s}(A)=\mu_{0}(A)$. Hence $\bar{\mu}(A)=\frac{1}{\tau}\int_{0}^{\tau}\mu_{s}(A)ds=\mu_{0}(A)=0$ or 1.
  \end{proof}
The following theorem shows that the converse of Theorem \ref{ET} also holds.
\begin{theorem}
	\label{Converse ET}
	Assume Condition P. If $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for some $s\in \mathbb{R}$, then $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n\geq 0})$ is ergodic.
\end{theorem}

\begin{proof}
	For any $F\in \mathcal{F}$ with $\theta_{\tau}^{-1}F=F$. Let $\bar{F}=F\times \mathbb{X} \in \bar{\mathcal{F}}$, then we have
	\begin{equation}
	 \label{key4}
		\begin{split}
		\bar{\Theta}_{\tau}^{-1}\bar{F}&=\{(\omega, x): \bar{\Theta}_{\tau}(\omega, x)\in \bar{F}\}\\
		     &=\{(\omega, x): (\theta_{\tau}\omega, \Phi(\tau, \omega)x)\in F\times \mathbb{X}\}\\
		     &=\{(\omega, x): \theta_{\tau}\omega \in F\}\\
		     &=(\theta_{\tau}^{-1}F)\times \mathbb{X} = F\times \mathbb{X}=\bar{F}.\\
		\end{split}
	\end{equation}
	And by (\ref{key1}), we also have
	$$\mu_{s}(\bar{F})=P(\bar{F}_s) \text{ for all } s\in \mathbb{R},$$
	where $\bar{F}_s=\{\omega: (\omega, Y(s, \theta_{-s}\omega))\in \bar{F}\}=\{\omega: (\omega, Y(s, \theta_{-s}\omega))\in F\times \mathbb{X}\}=F.$
	Thus
	\begin{equation}
	\label{key2}
	\mu_{s}(\bar{F})=P(F) \text{ for all } s\in \mathbb{R}.
	\end{equation}
    Since $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 1})$ for some $s\in \mathbb{R}$ is ergodic, by (\ref{key4}) and (\ref{key2}) we have
    $$P(F)=\mu_{s}(\bar{F})=0 \text{ or } 1.$$
\end{proof}

\begin{remark}
	\label{re1}
	(i). From Theorem \ref{ET} and Theorem \ref{Converse ET}, we can conclude that $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for some $s\in \mathbb{R}$ implies that $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for all $s\in \mathbb{R}$. This conclusion does not seem to be true in the phase space case, which we will consider in the next section.

	(ii). It is easy to check that if the dynamical system $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n \geq 0})$ is ergodic, then $(\Omega, \mathcal{F}, P, (\theta_t)_{t \geq 0})$ will be ergodic. The converse is not true in general. A counterexample can be found in Section \ref{Example2}.

	(iii). It is well known that, for a canonical Wiener space $(\Omega, \mathcal{F}, P)$, the corresponding canonical dynamical system $(\Omega, \mathcal{F}, P, (\theta_t)_{t \geq 0})$ is ergodic, where $\theta_t$ is the Brownian shift. In Section \ref{Example3} we will prove the discrete dynamical system $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n \geq 0})$ is also ergodic. Thus our results in this paper can apply to stochastic differential equations and stochastic partial differential equations driven by Brownian motions.
\end{remark}

Next we will give these two examples of metric dynamical systems.

\subsubsection{Example: a metric dynamical system on torus
}
\label{Example2}
Now we consider $\tilde{\Omega}=[0, 1)\times [0, 1)$ and $\tilde{\mathcal{F}}:=\mathcal{B}(\tilde{\Omega})$. Define the map $\tilde{\theta}^{\alpha}: \mathbb{R}\times \tilde{\Omega} \rightarrow \tilde{\Omega}$ by
$$\tilde{\theta}^{\alpha}_t(r, x):=((r+t) \ \text{mod } 1, (x+t\alpha)\  \text{mod } 1) \text{ for all } t\in \mathbb{R}, (r, x)\in \tilde{\Omega},$$
where $\alpha$ is a fixed positive irrational number. Then it is easy to check that $(\tilde{\Omega}, \tilde{\mathcal{F}}, (\tilde{\theta}^{\alpha}_t)_{t\in \mathbb{R}})$ is a dynamical system. Let $L$ be the Lebesgue measure on $[0, 1)$ and $\mathcal{P}(\tilde{\Omega})$ be the set of probability measures on $(\tilde{\Omega}, \tilde{\mathcal{F}})$. Define $\mu: \mathbb{R} \rightarrow \mathcal{P}(\tilde{\Omega})$ by
$$\mu_{s}:= \delta_{\{s\ \text{mod }1\}}\times L,$$
then $\mu$ is a periodic probability measure with period 1 on $(\tilde{\Omega}, \tilde{\mathcal{F}})$. Let $\bar{\mu}:= \int_{0}^{1}\mu_{s}ds$.

\begin{proposition}
	\label{prop1}
	The dynamical systems $(\tilde{\Omega}, \tilde{\mathcal{F}}, \mu_{s}, ((\tilde{\theta}^{\alpha}_1)^n)_{n\in \mathbb{N}}) \text{ for all } s\in \mathbb{R}$ and $(\tilde{\Omega}, \tilde{\mathcal{F}}, \bar{\mu}, \\
	(\tilde{\theta}^{\alpha}_t)_{t\in \mathbb{R}})$ are ergodic.
\end{proposition}

\begin{proof}
	Let $\Omega:=[0,1), \mathcal{F}:=\mathcal{B}(\Omega)$ and $\theta_{\alpha}: \Omega \rightarrow \Omega, \theta_{\alpha}(x)=(x+\alpha)\ \text{mod } 1$. It is well known that the dynamical system $(\Omega, \mathcal{F}, L, (\theta_{\alpha}^n)_{n \in \mathbb{N}})$ with an irrational number $\alpha$ is ergodic.

	For any $s\in \mathbb{R}$ and $\tilde{A}\in \tilde{\mathcal{F}}$ with $(\tilde{\theta}^{\alpha}_1)^{-1}\tilde{A}=\tilde{A}$. Define $\tilde{A}_{r}:=\{x: (r, x)\in \tilde{A}\}$ for all $r\in [0, 1)$, then
	\begin{equation}
	\label{key5}
	\mu_{s}(\tilde{A})=\int_{\tilde{\Omega}}I_{\tilde{A}}(r,x)\delta_{\{s\ \text{mod }1\}}(dr)L(dx)
	=\int_{0}^{1}I_{\tilde{A}}(s\ \text{mod }1, x)L(dx)
	=L(\tilde{A}_{(s\ \text{mod }1)}).
	\end{equation}
	and
	\begin{equation}
	\label{key6}
	\begin{split}
	\theta_{\alpha}^{-1}\tilde{A}_{(s\ \text{mod }1)}&=\{x: \theta_{\alpha}x\in \tilde{A}_{(s\ \text{mod }1)}\}\\
	&=\{x: (s\ \text{mod 1}, (x+\alpha)\ \text{mod }1)\in \tilde{A}\}\\
	&=\{x: \tilde{\theta}^{\alpha}_1(s\ \text{mod }1, x)\in \tilde{A}\}\\
	&=\{x: (s\ \text{mod }1, x)\in (\tilde{\theta}^{\alpha}_1)^{-1}\tilde{A}=\tilde{A}\}\\
	&=\tilde{A}_{(s\ \text{mod }1)}.\\
	\end{split}
	\end{equation}
	Applying the ergodicity of the dynamical system $(\Omega, \mathcal{F}, L, (\theta_{\alpha}^n)_{n\in \mathbb{N}})$ and (\ref{key5}), (\ref{key6}), we have
	$$\mu_{s}(\tilde{A})=L(\tilde{A}_{(s\ \text{mod }1)})=0 \text{ or } 1.$$
	This means the dynamical systems $(\tilde{\Omega}, \tilde{\mathcal{F}}, \mu_{s}, ((\tilde{\theta}^{\alpha}_1)^n)_{n\in \mathbb{N}})$ for all $s\in \mathbb{R}$ are ergodic.

	Again for any $\tilde{A} \in \tilde{\mathcal{F}}$ with $(\tilde{\theta}^{\alpha}_t)^{-1}\tilde{A}=\tilde{A}$ for all $t\in \mathbb{R}$, then $(\tilde{\theta}^{\alpha}_1)^{-1}\tilde{A}=\tilde{A}$ and $\mu_0(\tilde{A})=0$ or 1. Since
	$$\tilde{A}=\bigcup_{s\in [0,1)}(\{s\}\times \tilde{A}_s)$$
	and
	\begin{equation*}
	\begin{split}
	(\tilde{\theta}^{\alpha}_t)^{-1}\tilde{A}
	&=\bigcup_{r\in [0,1)}(\tilde{\theta}^{\alpha}_t)^{-1}(\{r\}\times \tilde{A}_r)\\
	&=\bigcup_{r\in [0,1)}\{(s, x): s\in [0,1), \tilde{\theta}^{\alpha}_t(s,x)\in \{r\}\times \tilde{A}_r\}\\
	&=\bigcup_{r\in [0,1)}\{(s, x): s\in [0,1), ((s+t)\ \text{mod }1, (x+t\alpha)\ \text{mod }1)\in \{r\}\times \tilde{A}_r\}\\
	&=\bigcup_{r\in [0,1)}\{(s, x): s\in [0,1), (s+t)\ \text{mod }1=r, \theta_{t\alpha}x\in \tilde{A}_r\}\\
	&=\bigcup_{r\in [0,1)}\{(s, x): s\in [0,1), (s+t)\ \text{mod }1=r, \theta_{t\alpha}x\in \tilde{A}_{((s+t)\ \text{mod }1)}\}\\
	&=\bigcup_{s\in [0,1)}\{(s, x): \theta_{t\alpha}x\in \tilde{A}_{((s+t)\ \text{mod }1)}\}\\
	&=\bigcup_{s\in [0,1)}(\{s\}\times \theta_{t\alpha}^{-1}\tilde{A}_{((s+t)\ \text{mod }1)}),\\
	\end{split}
	\end{equation*}
	so $\bigcup_{s\in [0,1)}(\{s\}\times \theta_{t\alpha}^{-1}\tilde{A}_{((s+t) \ \text{mod }1)})=\bigcup_{s\in [0,1)}(\{s\}\times \tilde{A}_s)$ and hence $\theta_{t\alpha}^{-1}\tilde{A}_{((s+t) \ \text{mod }1)}=\tilde{A}_s$ for all $t\in \mathbb{R}$. In particular, $\theta_{t\alpha}^{-1}\tilde{A}_t=\tilde{A}_0$ for all $t\in [0, 1)$. Therefore
	\begin{equation}
	\label{eq4}
	\mu_{s}(\tilde{A})=L(\tilde{A}_s)=L(\theta_{s\alpha}^{-1}\tilde{A}_s)=L(\tilde{A}_0)=\mu_0(\tilde{A})
	\text{ for all } s\in [0,1).
	\end{equation}
	It turns out that $\bar{\mu}(\tilde{A})=\int_{0}^{1}\mu_{s}(\tilde{A})ds=\mu_0(\tilde{A})=0$ or 1, which means the dynamical system $(\tilde{\Omega}, \tilde{\mathcal{F}}, \bar{\mu}, (\tilde{\theta}^{\alpha}_t)_{t\in \mathbb{R}})$ is ergodic.
\end{proof}

\begin{remark}
	It is easy to see that the dynamical system $(\tilde{\Omega}, \tilde{\mathcal{F}}, \bar{\mu}, ((\tilde{\theta}^{\alpha}_1)^n)_{n\in \mathbb{N}})$ is not ergodic considering the first coordinate of the mapping. This provides an example that the continuous time dynamical system is ergodic, but its discretization may not be.
\end{remark}

\subsubsection{Example: the metric dynamical system of Brownian motions}
\label{Example3}
A standard Brownian motion or Wiener process $(W_t)_{t \in \mathbb{T}}$ ($\mathbb{T}=\mathbb{R^+}$ (one-sided time) or $\mathbb{T}=\mathbb{R}$ (two-sided time)) in $\mathbb{R}^m$ is a process with $W_0=0$ and stationary independent increments satisfying $W_t-W_s \sim \mathcal{N}(0, |t-s|I)$. The corresponding measure $\mathbb{P}$ on $(\Omega, \mathcal{F})$, where $\Omega=\mathcal{C}_0(\mathbb{T}, \mathbb{R}^m)$ and $\mathcal{F}$ is the Borel $\sigma$-algebra on $\Omega$, is called Wiener measure, the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is called Wiener space. The corresponding canonical metric dynamical system $\Sigma=(\Omega, \mathcal{F}, \mathbb{P}, (\theta_t)_{t \in \mathbb{T}})$ describes Brownian motion or (Gaussian) white noise as a metric dynamical system of random dynamical systems generated by stochastic differential equations or stochastic partial differential equations driven by Brownian motions.

Let $\Sigma=(\Omega, \mathcal{F}, \mathbb{P}, (\theta_t)_{t \in \mathbb{T}})$ be one of the canonical dynamical system introduced above, with the canonical filtration $\mathcal{F}_s^t:=\sigma(W_u-W_v, s\leq u,v \leq t), s\leq t.$ The following notations are standard (c.f. \cite{Arnold}),
$$\mathcal{T}^{\infty}:=\bigcap_{t \in \mathbb{T}}\mathcal{F}_t^{\infty},$$
and for two-sided time
$$\mathcal{T}_{-\infty}:=\bigcap_{t \in \mathbb{T}}\mathcal{F}_{-\infty}^t,$$
as the tail $\sigma$-algebras ($\mathcal{T}_{-\infty}$: remote past, $\mathcal{T}^{\infty}$: remote future). Set $$\mathcal{I}:=\{A\in \mathcal{F}: \theta_t^{-1}A=A, \text{ for all } t \in \mathbb{T}\},$$
and for a given $\tau>0$, define
$$\mathcal{I_{\tau}}:=\{A\in \mathcal{F}: \theta_{\tau}^{-1}A=A\}.$$
We say $\mathcal{A} \subset \mathcal{B}$ mod $\mathbb{P}$ if for each $A \in \mathcal{A}$, there is a $B \in \mathcal{B}$ with $\mathbb{P}(A\bigtriangleup B)=0$.

\begin{proposition}
	\label{nkey2}
	Assume that $\mathcal{I}, \mathcal{I_{\tau}}, \mathcal{T}^{\infty}$ and $\mathcal{T}_{-\infty}$ defined as above. Then
	\begin{enumerate}[(i)]
		\item if $\mathbb{T}$ is one-sided ($\mathbb{T}=\mathbb{R^+}$): $\mathcal{I} \subset \mathcal{I_{\tau}} \subset \mathcal{T}^{\infty}$.
		\item if $\mathbb{T}$ is two-sided ($\mathbb{T}=\mathbb{R}$): $\mathcal{I} \subset \mathcal{I_{\tau}} \subset \mathcal{T}^{\infty}$ mod $\mathbb{P}$, and $\mathcal{I} \subset \mathcal{T}_{-\infty}$ mod $\mathbb{P}$.
	\end{enumerate}
\end{proposition}

\begin{proof}
	Obviously, by the definitions of $\mathcal{I}, \mathcal{I_{\tau}}$, we have $\mathcal{I} \subset \mathcal{I_{\tau}}$.

	$(i)$ We just need to prove that $\mathcal{I_{\tau}} \subset \mathcal{T}^{\infty}$. For any $A \in \mathcal{I_{\tau}}$, since
	$$\mathcal{F}=\mathcal{B}(\Omega)=\sigma(W_t, t \in \mathbb{R^+}) =\sigma(W_u-W_v, 0\leq u, v <\infty)=\mathcal{F}_0^{\infty}$$
	and
	$$\theta_{\tau}^{-1}\mathcal{F}_t^{\infty}=\mathcal{F}_{t+\tau}^{\infty} \text{ for all } t\in \mathbb{R^+},$$
	then
	$$A=\theta_{\tau}^{-1}A \in \theta_{\tau}^{-1}\mathcal{F}_0^{\infty}=\mathcal{F}_{\tau}^{\infty}.$$
	By induction, we have $A\in \mathcal{F}_{n\tau}^{\infty}$ for all $n\in \mathbb{N}$. Then $A\in \bigcap_{n\in \mathbb{N}}\mathcal{F}_{n\tau}^{\infty}$. But
	$$\mathcal{T}^{\infty}=\bigcap_{t\in \mathbb{R^+}}\mathcal{F}_t^{\infty}=\bigcap_{n\in \mathbb{N}}\mathcal{F}_{n\tau}^{\infty},$$
	thus $A\in \mathcal{T}^{\infty}$. This means $\mathcal{I_{\tau}}\subset \mathcal{T}^{\infty}$.\\

	$(ii)$ Without loss of generality, we just need to prove $\mathcal{I_{\tau}}\subset \mathcal{T}^{\infty}$ mod $\mathbb{P}$. Similarly we can prove $\mathcal{I_{\tau}}\subset \mathcal{T}_{-\infty}$ mod $\mathbb{P}$. To prove the desired result, for any $A\in \mathcal{I_{\tau}}$, we set
	$$A_n:=\{\omega\in \Omega: \text{ there exists } \omega' \in A \text{ such that } \omega(t)=\omega'(t) \text{ for all } t \in [-n\tau, \infty)\}$$
	for all $n \in \mathbb{N}$. We can conclude that $A_n\in \mathcal{F}_{-n\tau}^{\infty}$ and $A_n\supset A_{n+1} \supset A$. Assume $A_n\downarrow \tilde{A}$, then $A\subset \tilde{A}$ and $\mathbb{P}(A)\leq \mathbb{P}(\tilde{A})$. But since
	\begin{equation*}
		\begin{split}
		\mathcal{F}&=\sigma(W_t, t\in \mathbb{R})\\
		&=\{A|\text{ there exist } \{t_n\}_{n=1}^{\infty}\subset \mathbb{R}, B\in \mathcal{B}((\mathbb{R}^{m})^{\mathbb{N}}) \text{ such that }
		A=\{\omega: \{\omega(t_n)\}_{n=1}^{\infty}\in B\} \},
		\end{split}
	\end{equation*}
	so for a given $A\in \mathcal{I}_{\tau}$, there exist a sequence $\{t_n\}_{n=1}^{\infty}$ and a set $B\in \mathcal{B}((\mathbb{R}^{m})^{\mathbb{N}})$ such that $A=\{\omega: \{\omega(t_n)\}_{n=1}^{\infty}\in B\}$. Let
	$$B^n=\{\Pi_nx=\{x_r\}_{r=1}^{n}: x=\{x_r\}_{r=1}^{\infty}\in B \}$$
	be the projection of $B$ from $(\mathbb{R}^{m})^{\mathbb{N}}$ to $(\mathbb{R}^m)^n$ and $$B_n=\{\omega:(\omega(t_1), \omega(t_2), \cdots, \omega(t_n))\in B^n\}.$$
	Then we know that $\mathbb{P}(A)=\lim_{n\rightarrow \infty}\mathbb{P}(B_n)$ from construction of finite dimensional distribution of Wiener measure. By the definition of $\tilde{A}$ we have $\tilde{A}\subset B_n$ for all $n$, then $\mathbb{P}(\tilde{A})\leq \lim_{n\rightarrow \infty}\mathbb{P}(B_n)=\mathbb{P}(A)$. Since we know that $\mathbb{P}(A)\leq \mathbb{P}(\tilde{A})$, so we conclude $\mathbb{P}(\tilde{A})=\mathbb{P}(A)$ .

	Next we prove the following claim.\\
	Claim $(*)$: For all $n\geq 2$, we have $\theta_{\tau}^{-1}A_n \supset A_{n-1} \supset A_n$.

	Proof of Claim $(*)$: The claim $A_{n-1}\supset A_n$ is obvious. Now for any $\omega \in A_{n-1}$, there exists $\omega' \in A$ such that $\omega(t)=\omega'(t)$ for all $t \in [-(n-1)\tau, \infty)$, then $\theta_{\tau}\omega(s)=\omega(s+\tau)-\omega(\tau)=\omega'(s+\tau)-\omega'(\tau)=\theta_{\tau}\omega'(s)$ for all $s \in [-n\tau, \infty)$. Since $A$ is $\theta_{\tau}$-invariant set and $\omega'\in A$, so $\theta_{\tau}\omega' \in A$. Thus $\theta_{\tau}\omega \in A_n$ and $\omega \in \theta_{\tau}^{-1}A_n$. This means Claim $(*)$ holds.\\

	Now we continue our proof. Since $\theta_{\tau}$ preserves probability $\mathbb{P}$, then by Claim $(*)$, we have
	$$\mathbb{P}(A_n)=\mathbb{P}(\theta_{\tau}^{-1}A_n)\geq \mathbb{P}(A_{n-1})\geq \mathbb{P}(A_n).$$
	It turns out that
	$$\mathbb{P}(A_n)=\mathbb{P}(\tilde{A})=\mathbb{P}(A) \text{ for all } n\in \mathbb{N}.$$
	Let $B_k=\theta_{k\tau}^{-1}A_2$. By Claim $(*)$, $\theta_{\tau}^{-1}A_2\supset A_2,$ i.e. $\theta_{\tau}A_2 \subset A_2$. So for any $\omega \in B_k=\theta_{k\tau}^{-1}A_2$, then $\theta_{k\tau}\omega \in A_2$, thus $\theta_{\tau}(\theta_{k\tau}\omega)=\theta_{(k+1)\tau}\omega \in A_2$, therefore $\omega \in \theta_{(k+1)\tau}^{-1}A_2=B_{k+1}$. This means $B_k\subset B_{k+1}$. And also we have that $$\mathbb{P}(B_k)=\mathbb{P}(\theta_{k\tau}^{-1}A_2)=\mathbb{P}(A_2)=\mathbb{P}(A_n)=\mathbb{P}(\tilde{A})=\mathbb{P}(A),$$
	for all $k, n \in \mathbb{N}$. Since $A_2\in \mathcal{F}_{-2\tau}^{\infty}$, then $B_k=\theta_{k\tau}^{-1}A_2 \in \mathcal{F}_{(-2+k)\tau}^{\infty}$. Let
	$$\tilde{B}=\lim_{k\rightarrow \infty}B_k=\bigcup_{k\in \mathbb{N}}B_k.$$
	Then
	$$\tilde{B}\in \bigcap_{k\in \mathbb{N}}\mathcal{F}_{(-2+k)\tau}^{\infty}=\mathcal{T}^{\infty}$$
	and
	$$\tilde{B}\supset B_k \supset A_n \supset \tilde{A} \supset A$$
	and
	$$\mathbb{P}(\tilde{B})=\mathbb{P}(B_k)=\mathbb{P}(A_n)=\mathbb{P}(\tilde{A})=\mathbb{P}(A),$$
	for all $k,n \in \mathbb{N}$.

	Then for any $A\in \mathcal{I_{\tau}}$, there exist $\tilde{B}\in \mathcal{T}^{\infty}$ such that $\mathbb{P}(A\bigtriangleup \tilde{B})=\mathbb{P}(\tilde{B}\setminus A)=\mathbb{P}(\tilde{B})-\mathbb{P}(A)=0$, i.e. $\mathcal{I_{\tau}} \subset \mathcal{T}^{\infty}$ mod $\mathbb{P}$.
\end{proof}

\begin{theorem}
	\label{lastTH}
	The canonical dynamical systems driven by Brownian motion $\Sigma=(\Omega, \mathcal{F}, \mathbb{P},\\
	 (\theta_t)_{t \in \mathbb{T}})$ $(\mathbb{T}=\mathbb{R^+}$ or $\mathbb{R})$ and their discrete dynamical systems $\Sigma^{\tau}=(\Omega, \mathcal{F}, \mathbb{P}, (\theta_{\tau}^n)_{n\geq 0})$ are ergodic.
\end{theorem}

\begin{proof}
	By Proposition \ref{nkey2}, if $\mathcal{T}^{\infty}$ is trival mod $\mathbb{P}$, $\Sigma$ and $\Sigma^{\tau}$ are ergodic. Since these canonical dynamical systems $\Sigma, \Sigma^{\tau}$ driven by a standard Brownian motion, then the tail $\sigma$-algebra $\mathcal{T}^{\infty}$ is trival mod $\mathbb{P}$ by Kolmogorov's zero-one law.
\end{proof}

\begin{remark}
	(i) It is noted that the ergodicity of $\Sigma$ in Theorem \ref{lastTH} was known in literature (c.f. \cite{Arnold}). The main purpose of the Theorem is to prove $\Sigma^{\tau}$ is ergodic in the discrete case. This result is new. But the continuous case $\Sigma$ being ergodic is proved as a byproduct of the techniques we build here.

	(ii) Proposition \ref{nkey2} and Theorem \ref{lastTH} hold also for Wiener process on a seperable Hilbert space where the Wiener measure was given in \cite{Da Prato1}. The proofs are exactly the same.
\end{remark}

\subsection{Invariant measures}
The following result is well-known:
\begin{theorem}
	\label{SIM}
	If a random dynamical system $\Phi: \mathbb{R}^+\times \Omega\times \mathbb{X} \rightarrow \mathbb{X}$ has a stationary path $Y: \Omega\rightarrow \mathbb{X}$, it has an invariant measure of $\Phi$ on $(\bar{\Omega}, \bar{\mathcal{F}})$, $\mu \in \mathcal{P}_{P}(\Omega\times \mathbb{X})$ defined by
	\begin{equation}
	\label{IMmu}
		\mu(A)=\int_{\Omega}\delta_{Y(\omega)}(A_{\omega})P(d\omega),
	\end{equation}
	where $A_{\omega}$ is the $\omega$-section of $A$ and $\mu_{\omega}=\delta_{Y(\omega)}$.
\end{theorem}

\begin{theorem}
	\label{SPET}
	Assume that the random dynamical system $\Phi$ has a stationary path $Y$. Then the metric dynamical system $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\geq 0})$ is ergodic if and only if the skew product dynamical system $(\bar{\Omega}, \bar{\mathcal{F}}, \mu, (\bar{\Theta}_t)_{t\geq 0})$ is ergodic, where $\mu$ is defined by (\ref{IMmu}).
\end{theorem}

\begin{proof}
	First prove the ``$\Rightarrow$" part. Assume $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\geq 0})$ is ergodic.
	Since $\mu$ is an invariant measure on $(\bar{\Omega}, \bar{\mathcal{F}})$, so $\bar{\Theta}_{t}$ preserves the measure $\mu$. To prove $(\bar{\Omega}, \bar{\mathcal{F}}, \mu, (\bar{\Theta}_t)_{t\geq 0})$ is ergodic, we need to prove that for any $A\in \bar{\mathcal{F}}$ with $\bar{\Theta}_{t}^{-1}A=A \text{ for all } t\geq 0$, $\mu(A)=0$ or 1. Set $A_{0}:=\{\omega: (\omega, Y(\omega))\in A\}$, then for any $t \geq 0$, we have
	\begin{equation*}
	\begin{split}
	\theta_{t}^{-1}A_{0}&=\{\omega: \theta_{t}\omega \in A_0\}\\
	                             &=\{\omega: (\theta_{t}\omega, Y(\theta_{t}\omega))\in A\}\\
	                             &=\{\omega: \bar{\Theta}_{t}(\omega, Y(\omega))\in A\}\\
	                             &=\{\omega: (\omega, Y(\omega))\in \bar{\Theta}_{t}^{-1}A\}\\
	                             &=\{\omega: (\omega, Y(\omega))\in A\}\\
	                             &=A_{0}.\\
	\end{split}
	\end{equation*}
	This means that $A_{0}$ is an invariant set with respect to $(\theta_{t})_{t\geq 0}$. Note that
	$$\mu(A)=\int_{\Omega}\delta_{Y(\omega)}(A_{\omega})P(d\omega)=P(\{\omega:(\omega, Y(\omega))\in A\})=P(A_0).$$
	By the ergodicity of $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\geq 0})$, we have $P(A_{0})=0$ or 1. Therefore $\mu(A)=0$ or 1.\\

	Now we prove the ``$\Leftarrow$" part. Assume $(\bar{\Omega}, \bar{\mathcal{F}}, \mu, (\bar{\Theta}_t)_{t\geq 0})$ is ergodic. For any $F\in \mathcal{F}$ with $\theta_{t}^{-1}F=F$ for all $t\geq 0$, we consider $\bar{F}=F\times\mathbb{X}\in \bar{\mathcal{F}}$, then for any $t\geq 0$
	\begin{equation*}
		\begin{split}
		\bar{\Theta}_{t}^{-1}\bar{F}&=\{(\omega, x): \bar{\Theta}_{t}(\omega, x)\in \bar{F}\}\\
		&=\{(\omega, x): (\theta_{t}\omega, \Phi(t, \omega)x)\in F\times \mathbb{X}\}\\
		&=\{(\omega, x): \theta_{t}\omega \in F\}\\
		&=(\theta_{t}^{-1}F)\times \mathbb{X} = F\times \mathbb{X}=\bar{F}.\\
		\end{split}
	\end{equation*}
	Since $$\mu(\bar{F})=P(\{\omega: (\omega, Y(\omega))\in F\times \mathbb{X}\})=P(F).$$
	By the ergdicity of $(\bar{\Omega}, \bar{\mathcal{F}}, \mu, (\bar{\Theta})_{t\geq 0})$, we have
	$$P(F)=\mu(\bar{F})=0 \text{ or }1.$$
\end{proof}

\section{Ergodicity of canonical Markovian systems from periodic measures: necessary and sufficient conditions}
Now we consider a Markovian cocycle random dynamical system $\Phi$ on a filtered dynamical system $(\Omega, \mathcal{F}, P, (\theta_{t})_{t \in \mathbb{R}}, (\mathcal{F}_s^t)_{s \leq t})$, i.e. $\mathcal{F}_s^t\subset \mathcal{F}$, assuming for any $s,t, u\in \mathbb{R}, s\leq t,$ $\theta_{u}^{-1}\mathcal{F}_s^t=\mathcal{F}_{s+u}^{t+u}$ and for any $t\in \mathbb{R}^+, \Phi(t,\cdot)$ is measurable with respect to $\mathcal{F}_0^t$. We also assume the random periodic path $Y(s)$ is adapted, that is to say that for each $s\in \mathbb{R}, Y(s,\cdot)$ is measurable with respect to $\mathcal{F}_{-\infty}^s:=\vee_{r\leq s}\mathcal{F}_r^s$.

Denote the transition probability of Markovian process $\Phi(t,\cdot)x$ on the polish space $\mathbb{X}$ with Borel $\sigma$-field $\mathcal{B}(\mathbb{X})$ by (c.f. \cite{Arnold}, \cite{Da Prato})
$$P(t,x,\Gamma)=P(\{\omega: \Phi(t, \omega)x\in \Gamma\}), \quad t\in \mathbb{R}^+, \quad \Gamma\in \mathcal{B}(\mathbb{X}).$$

Denote $L_{b}(\mathbb{X})$ be the set of all real-valued bounded Borel measurable functions defined on $\mathbb{X}$ and $\mathcal{P}(\mathbb{X})$ be the set of all probability measures defined on $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$. For any $t \geq 0$ and $\rho \in \mathcal{P}(\mathbb{X})$ we set
$$P_{t}^*\rho(\Gamma)=\int_{\mathbb{X}}P(t,x, \Gamma)\rho(dx), \quad \Gamma \in \mathcal{B}(\mathbb{X}),$$
and for any $\varphi \in L_b(\mathbb{X})$, define
\begin{equation}
   \label{Markov semigroup}
	(P_t\varphi)(x)=\int_{\mathbb{X}}P(t, x, dy)\varphi(y), \quad x\in \mathbb{X},
\end{equation}
as a semigroup from $L_b(\mathbb{X})$ to $L_b(\mathbb{X})$.

\begin{definition}
	(\cite{F.Z2}) A measure function $\rho.: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{X})$ is called a periodic measure of period $\tau$ on $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$ for the Markovian semigroup $P_{t}$ if it satisfies
	\begin{equation}
	\label{rho}
		P_{t}^*\rho_{s}=\rho_{s+t} \text{ and } \rho_{s+\tau}=\rho_{s}, \text{ for all } s\in \mathbb{R}, t\in \mathbb{R}^+.
	\end{equation}
	It is called a periodic measure with minimal period $\tau$ if $\tau >0$ is the smallest number such that (\ref{rho}) holds. It is called an invariant measure if it satisfies $\rho_{s}=\rho_{0}$ for all $s\in \mathbb{R}, i.e.$ $ \rho_{0}$ is an invariant measure for the Markovian semigroup $P_{t}$ if
	\begin{equation}
		P_{t}^*\rho_{0}=\rho_{0}, \text{ for all } t\in \mathbb{R}^+.
	\end{equation}
\end{definition}

With a given Markovian semigroup $P_t, t\geq 0$, and an invariant measure $\rho\in \mathcal{P}(\mathbb{X})$, we will associate now, in the following unique way, a dynamical system $(\Omega^*, \mathcal{F}^*, (\theta^*_t)_{t\in \mathbb{R}}, \mathbb{P}^{\mu})$ on the space $\Omega^*=\mathbb{X}^{\mathbb{R}}$ of all $\mathbb{X}$-valued functions.

Define $\Omega^*=\mathbb{X}^\mathbb{R}$, the space of all $\mathbb{X}$-valued functions on $\mathbb{R}$, $\mathcal{F}^*$ is the smallest $\sigma$-algebra containing all cylindrical sets of $\Omega^*$. And the shift $\theta^*: \mathbb{R}\times \Omega^*\rightarrow \Omega^*$ defined by $(\theta_{t}^*\omega^*)(s)=\omega^*(t+s)$, for all $\omega^*\in \Omega^*.$ For an arbitrary invariant measure $\rho \in \mathcal{P}(\mathbb{X})$ and an arbitrary finite set $I=\{t_1,t_2,\cdots,t_n\}, t_1<t_2<\cdots<t_n,$ we can define a probability $\mathbb{P}^{\rho}_I$ on $(\mathbb{X}^I, \mathcal{B}(\mathbb{X}^I))$ by the formula
$$\mathbb{P}_I^{\rho}(\Gamma)= \int_{\mathbb{X}}\rho(dx_1)\int_{\mathbb{X}}P_{t_2-t_1}(x_1,dx_2)\cdots  \int_{\mathbb{X}}P_{t_n-t_{n-1}}(x_{n-1},dx_n) I_{\Gamma}(x_1,x_2,\cdots,x_n), \quad \Gamma\in \mathcal{B}(\mathbb{X}^I).$$
By the Kolmogorov extension theorem, there exists a unique probability measure $\mathbb{P}^{\rho}$ on $(\Omega^*, \mathcal{F}^*)$ such that
$$\mathbb{P}^{\rho}(\{\omega^*:(\omega^*(t_1), \omega^*(t_2), \cdots, \omega^*(t_n))\in \Gamma\}) = \mathbb{P}^{\rho}_I(\Gamma), \quad \Gamma\in \mathcal{B}(\mathbb{X}^I).$$
From \cite{Da Prato}, the transformations $\theta^*_t, t\in \mathbb{R}$ preserves the measure $\mathbb{P}^{\rho}$, and the quadruplet $S^{\rho}=(\Omega^*, \mathcal{F}^*, (\theta^*_t)_{t\geq 0}, \mathbb{P}^{\rho})$ defines a dynamical system, called the \textit{canonical dynamical system} associated with $P_t, t\geq 0, \rho$ and $\theta^*_t$.
\begin{definition}
	(\cite{Da Prato}) The invariant measure $\rho$ is said to be ergodic with respect to the Markovian semigroup $P_t, t\geq 0$, if its associated canonical dynamical system $S^{\rho}=(\Omega^*, \mathcal{F}^*, (\theta^*_t)_{t\geq 0},\\
	 \mathbb{P}^{\rho})$ is ergodic.
\end{definition}

\begin{definition}
	(\cite{F.Z2}) The $\tau$-periodic measure $\{\rho_{s}\}_{s\in \mathbb{R}}$ is said to be PS-ergodic if for each $s\in [0, \tau), \rho_s$ as the invariant measure of the $\tau$-mesh discrete Markovian semigroup $\{P_{k\tau}\}_{k\in \mathbb{N}}$, at integral multiples of the period on the Poincar\'{e} section, is ergodic.
\end{definition}

In \cite{F.Z2}, we have the following theorem.
\begin{theorem}
	\label{PM2}
	Assume the Markovian cocycle $\Phi: \mathbb{R}^+\times \Omega\times \mathbb{X} \rightarrow \mathbb{X}$ has an adapted random periodic path $Y: \mathbb{R}\times \Omega\rightarrow \mathbb{X}$. Then the measure function $\rho.: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{X})$ defined by
	\begin{equation}
	\label{nkey1}
		\rho_{s}:=\mathbb{E}(\mu_{s}).=\mathbb{E}\delta_{Y(s, \theta_{-s}\cdot)}=\mathbb{E}\delta_{Y(s, \cdot)},
	\end{equation}
	which is the law of the random periodic path $Y$, is a periodic measure of the semigroup $P_t$ on $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$. Its time average $\bar{\rho}$ over a time interval of exactly one period defined by
	$$\bar{\rho}=\frac{1}{\tau}\int_{0}^{\tau}\rho_sds,$$
	is an invariant measure and satisfies that for any $\Gamma \in \mathcal{B}(\mathbb{X}), t\in \mathbb{R},$
	\begin{equation*}
	\begin{split}
	\bar{\rho}(\Gamma)&=\mathbb{E}(\frac{1}{\tau}\{s\in [0, \tau): Y(s,\cdot)\in \Gamma\})\\
	                             &=\mathbb{E}(\frac{1}{\tau}\{s\in [t, t+\tau): Y(s,\cdot)\in \Gamma\}).\\
	\end{split}
	\end{equation*}
\end{theorem}

\subsection{Ergodic canonical dynamical systems generated from periodic measure}
Given a random periodic path $Y$ of the Markovian cocycle $\Phi$, for any given $s\in \mathbb{R}$, define
$$L_s^{\omega}:=\{Y(s+k\tau, \omega): k\in \mathbb{Z}\},$$
and the $\rho_s$-invariant set $\mathcal{I}_s^{\tau}$ associated with the discrete Markovian semigroup $(P_{\tau}^k)_{k\in \mathbb{Z}},$
$$\mathcal{I}_s^{\tau}:=\{\Gamma\in \mathcal{B}(\mathbb{X}): P_{\tau}I_{\Gamma}=I_{\Gamma}, \rho_{s}-a.s.\}.$$
For these two sets, we consider\\
\textbf{Condition A.}  \textit{For any $s\in \mathbb{R}$, $\Gamma\in \mathcal{I}_s^{\tau}$ and $P$-almost all $\omega\in \Omega$, either $L_s^{\omega}\cap \Gamma=\emptyset$ or $L_s^{\omega}\subseteq \Gamma$.}

\begin{theorem}
	\label{NewET}
	Assume that the random periodic path $Y$ satisfies Condition A and the periodic measure $\rho.: \mathbb{R}\rightarrow \mathcal{P}(\mathbb{X})$ is given in Theorem \ref{PM2}. Then if the dynamical system $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n\geq 1})$ is ergodic, the $\tau$-periodic measure $\{\rho_s\}_{s\in \mathbb{R}}$ defined in (\ref{nkey1}) is PS-ergodic, and hence $\bar{\rho}$ is ergodic.
\end{theorem}
\begin{proof}
	Since $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n\geq 1})$ is ergodic, by Theorem \ref{ET} we know that $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 1})$ is ergodic. By definition of PS-ergodic, we will show that $\rho_{s}$ is ergodic associated with the discrete Markovian semigroup $\{P_{\tau}^k\}_{k\in \mathbb{N}}$. Equivalently, we need to show that for any $\Gamma\in \mathcal{B}(\mathbb{X})$, if $P_{\tau}I_{\Gamma}=I_{\Gamma}, \rho_{s}-a.s.$, then $\rho_{s}(\Gamma)=0$ or 1.

	For any $A\in \bar{\mathcal{F}}$ and $\Gamma\in \mathcal{B}(\mathbb{X})$, since $\mu_{s}(A)=P(\{\omega: (\omega, Y(s, \theta_{-s}\omega))\in A\})$ and $\rho_{s}(\Gamma)=P(\{\omega: Y(s, \theta_{-s}\omega)\in \Gamma\})$, then $\rho_{s}(\Gamma)=\mu_{s}(\Omega\times \Gamma)$.
	Next we consider the subset $\Omega\times \Gamma$ in $\bar{\Omega}$.
	Note
	\begin{equation*}
	\begin{split}
	\bar{\Theta}_{\tau}^{-1}(\Omega\times \Gamma)
	      &=\{(\omega, x): \bar{\Theta}_{\tau}(\omega, x)\in \Omega\times \Gamma\}\\
	      &=\{(\omega, x): (\theta_{\tau}\omega, \Phi(\tau, \omega)x)\in \Omega\times \Gamma\}\\
	      &=\{(\omega, x): \Phi(\tau, \omega)x \in \Gamma\}.\\
	\end{split}
	\end{equation*}
	Define $\Omega_{\tau, \Gamma}^x:=\{\omega: \Phi(\tau, \omega)x\in \Gamma\}$, then we have
	\begin{equation}
	\label{eq1}
    \begin{split}
    \bar{\Theta}_{\tau}^{-1}(\Omega\times \Gamma)
          =\bigcup_{x\in \mathbb{X}}(\Omega_{\tau, \Gamma}^x\times \{x\})
          =\left(\bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right)\bigcup \left(\bigcup_{x\notin \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right).
    \end{split}
    \end{equation}
    It is easy to see that $\bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})$ is a subset of $\Omega\times \Gamma$, and
    \begin{equation*}
    \begin{split}
    \mu_{s}\left(\bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right)
         &=\int_{\Omega\times \mathbb{X}}I_{\cup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})}(\omega, x)\delta_{Y(s, \theta_{-s}\omega)}(dx)P(d\omega)\\
         &=\int_{\Omega}I_{\cup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})}(\omega, Y(s,\theta_{-s}\omega))P(d\omega)\\
         &=P\left(\{\omega: (\omega, Y(s,\theta_{-s}\omega))\in\bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\}\right)\\
         &=P\left(\{\omega: Y(s,\theta_{-s}\omega)\in \Gamma, \omega \in \Omega_{\tau, \Gamma}^{Y(s, \theta_{-s}\omega)}\}\right)\\
         &=P\left(\{\omega: Y(s,\theta_{-s}\omega)\in \Gamma, \Phi(\tau, \omega)Y(s,\theta_{-s}\omega)\in \Gamma\}\right)\\
         &=P\left(\{\omega: Y(s,\theta_{-s}\omega)\in \Gamma, Y(s+\tau,\theta_{-s}\omega)\in \Gamma\}\right).\\
    \end{split}
    \end{equation*}
    As $\Gamma\in \mathcal{I}_s^{\tau}$, and Condition A holds, we have
    $$P\left(\{\omega: Y(s,\theta_{-s}\omega)\in \Gamma, Y(s+\tau,\theta_{-s}\omega)\in \Gamma\}\right)=P\left(\{\omega: Y(s,\theta_{-s}\omega)\in \Gamma\}\right).$$
    So
    \begin{equation*}
    \begin{split}
    \mu_{s}\left(\bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right)
         =P\left(\{\omega: Y(s,\theta_{-s}\omega)\in \Gamma\}\right)
         =\rho_{s}(\Gamma)
         =\mu_{s}(\Omega\times \Gamma).
    \end{split}
    \end{equation*}
    Hence
    \begin{equation}
    \label{*}
    	\mu_{s}\left((\Omega\times \Gamma)\setminus \bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right)=0.
    \end{equation}
    Similarly, for the set $\bigcup_{x\notin \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})$,
    \begin{equation}
    \begin{split}
    \label{**}
    \mu_{s}\left(\bigcup_{x\notin \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right)
    &=P\left(\{\omega: (\omega, Y(s,\theta_{-s}\omega))\in\bigcup_{x\notin \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\}\right)\\
    &=P\left(\{\omega: Y(s,\theta_{-s}\omega)\notin \Gamma, \omega \in \Omega_{\tau, \Gamma}^{Y(s, \theta_{-s}\omega)}\}\right)\\
    &=P\left(\{\omega: Y(s,\theta_{-s}\omega)\notin \Gamma, Y(s+\tau,\theta_{-s}\omega)\in \Gamma\}\right)\\
    &=0.\\
    \end{split}
    \end{equation}
    The last equality due to Condition A. Since it follows from (\ref{eq1}) that
    \begin{equation*}
    \begin{split}
     (\Omega\times \Gamma) \bigtriangleup \bar{\Theta}_{\tau}^{-1}(\Omega\times \Gamma)
        &=\left((\Omega\times \Gamma)\setminus \bar{\Theta}_{\tau}^{-1}(\Omega\times \Gamma)\right)\cup \left(\bar{\Theta}_{\tau}^{-1}(\Omega\times \Gamma)\setminus (\Omega\times \Gamma)\right)\\
        &\subseteq \left((\Omega\times \Gamma)\setminus \bigcup_{x\in \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right)\bigcup \left(\bigcup_{x\notin \Gamma}(\Omega_{\tau, \Gamma}^x\times \{x\})\right),\\
    \end{split}
    \end{equation*}
    then from (\ref{*}) and (\ref{**}), we have
    $$\mu_{s}((\Omega\times \Gamma) \bigtriangleup \bar{\Theta}_{\tau}^{-1}(\Omega\times \Gamma))=0.$$
    Applying equivalent condition of the ergodicity of the dynamical system $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 1})$ (c.f. \cite{Walters}), we have
    $$\rho_{s}(\Gamma)=\mu_{s}(\Omega\times \Gamma)=0 \text{ or } 1.$$
    So  $\rho_{s}$ is ergodic associated with the discrete Markovian semigroup $\{P_{\tau}^k\}_{k\in \mathbb{Z}}$ for all $s\in \mathbb{R}$ and therefore $\{\rho_{s}\}_{s\in \mathbb{R}}$ is PS-ergodic. Finally by Theorem 1.3.6 in \cite{F.Z2}, we know that $\bar{\rho}$ is ergodic.
\end{proof}

\begin{theorem}
	\label{Add1}
	Assume Condition P. If the $\tau$-periodic measure $\{\rho_{s}\}_{s\in \mathbb{R}}$ defined in (\ref{nkey1}) is PS-ergodic, then for any given $s\in \mathbb{R}, \Gamma \in \mathcal{I}_s^{\tau}$, we have either for $P$-almost all $\omega$, $L_s^{\omega}\cap \Gamma=\emptyset$ or for $P$-almost all $\omega$, $L_s^{\omega}\subseteq \Gamma$.
\end{theorem}

\begin{proof}
	Since $\{\rho_{s}\}_{s\in \mathbb{R}}$ is PS-ergodic, then for any given $s\in \mathbb{R}$ and $\Gamma\in \mathcal{I}_s^{\tau}$, we have $\rho_{s}(\Gamma)=0$ or 1. By Theorem \ref{PM2}, we have $\rho_s(\Gamma)=P(\{\omega: Y(s, \omega)\in \Gamma\})$.

	Let $\Lambda=\{\omega: Y(s, \omega)\in \Gamma\}$ and $\tilde{\Lambda}=\bigcup_{k\in \mathbb{Z}}\theta_{k\tau}^{-1}\Lambda$. It is easy to see that for any $\omega \in \tilde{\Lambda}^c$, $\theta_{k\tau}\omega \in \tilde{\Lambda}^c$ for all $k\in \mathbb{Z}$. Thus for any $\omega \in \tilde{\Lambda}^c$
	\begin{equation*}
		\begin{split}
		L_s^{\omega}&=\{Y(s+k\tau, \omega): k\in \mathbb{Z}\}\\
		                    &=\{Y(s, \theta_{k\tau}\omega): k\in \mathbb{Z}\}\\
		                    &\subseteq \bigcup_{\omega\in \tilde{\Lambda}^c}\{Y(s, \omega)\}.\\
		\end{split}
	\end{equation*}
	By the definition of $\Lambda$ and $\tilde{\Lambda}$, we have
	$$\bigcup_{\omega\in \tilde{\Lambda}^c}\{Y(s, \omega)\} \subseteq \bigcup_{\omega\in \Lambda^c}\{Y(s, \omega)\} \subseteq \Gamma^c.$$
	This means $L_s^{\omega}\cap \Gamma=\emptyset$ for all $\omega \in \tilde{\Lambda}^c$, i.e. $\tilde{\Lambda}^c \subset \{\omega: L_s^{\omega}\cap \Gamma = \emptyset\}$. However, if $\rho_s(\Gamma)=0$, then $P(\Lambda)=\rho_s(\Gamma)=0$. It follows that $P(\tilde{\Lambda})=0$ and $P(\tilde{\Lambda}^c)=1$. Finally $P(\{\omega: L_s^{\omega}\cap \Gamma=\emptyset\})=1$.

	Similarly, let $\Sigma=\{\omega: Y(s, \omega)\notin \Gamma\}$ and $\tilde{\Sigma}=\bigcup_{k\in \mathbb{Z}}\theta_{k\tau}^{-1}\Sigma$. It is also easy to see that for all $\omega \in \tilde{\Sigma}^c$, then $\theta_{k\tau}\omega \in \tilde{\Sigma}^c$ for all $k\in \mathbb{Z}$ and
	$$L_s^{\omega}\subseteq \bigcup_{\omega \in \tilde{\Sigma}^c}\{Y(s, \omega)\} \subseteq \bigcup_{\omega \in \Sigma^c}\{Y(s,\omega)\} \subseteq \Gamma.$$
	This means $L_s^{\omega}\subseteq \Gamma$ for all $\omega \in \bar{\Sigma}^c$, i.e. $\tilde{\Sigma}\subset \{\omega: L_s^{\omega}\subseteq \Gamma\}$. However, when $\rho_{s}(\Gamma)=1$, $P(\Sigma)=1-P(\Sigma^c)=1-\rho_s(\Gamma)=0$. Thus $P(\tilde{\Sigma})=0$ and $P(\tilde{\Sigma}^c)=1$. It then follows that $P(\{\omega: L_s^{\omega}\subseteq \Gamma\})=1$.
\end{proof}

\begin{remark}
	\label{remark}
	The result of Theorem \ref{Add1} is stronger than Condition A. This can be seen that as the statement that either for P-almost all $\omega$, $L_s^{\omega}\cap \Gamma=\emptyset$ or for P-almost all $\omega$, $L_s^{\omega}\subseteq \Gamma$ means either there exists $\Omega_{1}\subset \Omega$ with $P(\Omega_{1})=1$ such that when $\omega\in\Omega_{1}$, $L_s^{\omega}\cap \Gamma=\emptyset$ or there exists $\Omega_{2}\subset \Omega$ with $P(\Omega_{2})=1$ such that when $\omega\in\Omega_{2}$, $L_s^{\omega}\subseteq \Gamma$. In the first case, let $\Omega_0=\{\omega: L_s^{\omega}\cap \Gamma=\emptyset\}$. Then $\Omega_0\supset\Omega_{1}$, so $P(\Omega_0)=1$. It is obvious that when $\omega\in\Omega_0, L_s^{\omega}\cap \Gamma=\emptyset$. In the second case, let $\Omega_0=\{\omega: L_s^{\omega}\subseteq \Gamma\}$. Then $\Omega_0\supset\Omega_{2}$, so $P(\Omega_0)=1$. It is obvious that when $\omega\in\Omega_0, L_s^{\omega}\subseteq \Gamma$. In both cases, there exists $\Omega_0$ with $P(\Omega_0)=1$ such that when $\omega\in\Omega_0$, either $L_s^{\omega}\cap \Gamma=\emptyset$ or $L_s^{\omega}\subseteq \Gamma$.
\end{remark}

\begin{corollary}
	If the dynamical system $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n\geq 1})$ is ergodic, then Condition A and the $\tau$-periodic measure $\{\rho_{s}\}_{s\in \mathbb{R}}$ being PS-ergodic are equivalent. In this case, the statement that P-a.s. either $L_s^{\omega}\cap \Gamma = \emptyset$ or $L_s^{\omega}\subseteq \Gamma$ and the statement that either P-a.s. $L_s^{\omega}\cap \Gamma = \emptyset$ or P-a.s. $L_s^{\omega}\subseteq \Gamma$ are equivalent.
\end{corollary}

\begin{proof}
	This corollary can be easily obtained from Theorem \ref{NewET}, Theorem \ref{Add1} and Remark \ref{remark}.
\end{proof}

\subsection{Ergodicity of canonical dynamical system generated from invariant measure: sufficient condition}
Next we consider a stationary path $Y: \Omega\rightarrow \mathbb{X}$ of the Markovian cocycle $\Phi$ and the invariant measure $\mu$ defined in (\ref{IMmu}). Then we recall that the measure $\rho\in\mathcal{P}(\mathbb{X})$ defined by
 \begin{equation}
 \label{eq3}
 	\rho:=\mathbb{E}(\mu).=\mathbb{E}\delta_{Y(\cdot)}.
 \end{equation}
which is the law of the stationary path $Y$, is an invariant measure and satisfies that $\rho(\Gamma)=P(\{\omega: Y(\omega)\in \Gamma\})$, for any $\Gamma \in \mathcal{B}(\mathbb{X}), t\in \mathbb{R}$.

Now we define
$$L^{\omega}:=\{Y(\theta_{s}\omega): s\in \mathbb{R}\},$$
and the $\rho$ invariant set associated with the Markovian semigroup $(P_t)_{t\geq 0}$
$$\mathcal{I}:=\{\Gamma\in \mathcal{B}(\mathbb{X}): P_tI_{\Gamma}=I_{\Gamma}, \rho-a.s.\}.$$
\\
For these two sets, we consider\\
\textbf{Condition $\text{A}^{\prime}$.}  \textit{For any $\Gamma\in \mathcal{I}$ and $P$-almost all $\omega\in \Omega$, $L^{\omega}\cap \Gamma=\emptyset$ or $L^{\omega}\subseteq \Gamma$.}

\begin{theorem}
	\label{SME}
	Assume that the stationary path $Y$ satisfies Condition $A^{\prime}$ and the invariant measure $\rho \in \mathcal{P}(\mathbb{X})$ given in (\ref{eq3}). If the dynamical system $(\Omega, \mathcal{F}, P, (\theta_t)_{t\in \mathbb{R}})$ is ergodic, then $\rho$ is ergodic.
\end{theorem}

To prove Theorem \ref{SME} we need the following lemma, which is of interests on its own right. We present it as a metric dynamical system. But it does not have to link with a random dynamical system. It is true for a continuous dynamical system of a probability space (measure space).

\begin{lemma}
	\label{lemma}
	Assume that $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in \mathbb{R}})$ is a dynamical system, then the following three statements are equivalent:
	\begin{enumerate}[(i)]
	   \item $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in \mathbb{R}})$ is ergodic.
	   \item If $A \in \mathcal{F}$ and $P\left((\bigcup_{t\geq T}\theta_t^{-1}A)\bigtriangleup A\right)=0$ for any $T\in \mathbb{R}$, then $P(A)=0$ or 1.
	   \item If $A \in \mathcal{F}$ and $P\left(\bigcup_{t\in \mathbb{R}}(\theta_t^{-1}A\bigtriangleup A)\right)=0$, then $P(A)=0$ or 1.
\end{enumerate}
\end{lemma}

\begin{proof}
	$(i)\Rightarrow (ii)$. Assume $A\in \mathcal{F}$ and $P\left((\bigcup_{t\geq T}\theta_t^{-1}A)\bigtriangleup A\right)=0$ for any $T\in \mathbb{R}$. Define
	$$A_{\infty}:=\bigcap_{T\in \mathbb{R}}\bigcup_{t\geq T}\theta_t^{-1}A.$$
	Then it is easy to see that for all $s\in \mathbb{R}$,
	$$\theta_s^{-1}A_{\infty}=\bigcap_{T\in \mathbb{R}}\bigcup_{t\geq T+s}\theta_t^{-1}A=A_{\infty}.$$
	Thus $A_{\infty}$ is an invariant set. By the ergodicity assumption, we have
	\begin{equation}
		\label{AIM}
		P(A_{\infty})=0 \text{ or } P(A_{\infty})=1.
	\end{equation}
	For any $T \in \mathbb{R}$, since we have
	$$P\left((\bigcup_{t\geq T}\theta_t^{-1}A)\bigtriangleup A\right)=0,$$
	then
	$$P\left((\bigcup_{t\geq T}\theta_t^{-1}A)\setminus A\right)=0
	\text{ and }
	P\left(A\setminus \bigcup_{t\geq T}\theta_t^{-1}A\right)=0.$$
	But note as $T \rightarrow \infty$, we have
	$$(\bigcup_{t\geq T}\theta_t^{-1}A)\setminus A\downarrow A_{\infty}\setminus A
	\text{ and }
	A\setminus \bigcup_{t\geq T}\theta_t^{-1}A\uparrow A\setminus A_{\infty},$$
	By continuity of measure, we have
	$$P(A_{\infty}\setminus A)=0
	\text{ and }
	P(A\setminus A_{\infty})=0.$$
	Now recall (\ref{AIM}). Consider the case $P(A_{\infty})=0$, then
	$$P(A)=P(A)-P(A_{\infty})\leq P(A\setminus A_{\infty})=0.$$
	Now consider the case that $P(A_{\infty})=1$, then
	$$1-P(A)=P(A_{\infty})-P(A)\leq P(A_{\infty}\setminus A)=0.$$
	Thus the assertion $(ii)$ is proved.

	$(ii)\Rightarrow (iii)$. Assume $A \in \mathcal{F}$ and $P\left(\bigcup_{t\in \mathbb{R}}(\theta_t^{-1}A\bigtriangleup A)\right)=0$. For any $T\in \mathbb{R}$, since
	$$(\bigcup_{t\geq T}\theta_t^{-1}A)\bigtriangleup A\subseteq \bigcup_{t\geq T}(\theta_t^{-1}A\bigtriangleup A),$$
	then we have
	$$P\left((\bigcup_{t\geq T}\theta_t^{-1}A)\bigtriangleup A\right)\leq P\left( \bigcup_{t\geq T}(\theta_t^{-1}A\bigtriangleup A)\right)\leq P\left( \bigcup_{t\in \mathbb{R}}(\theta_t^{-1}A\bigtriangleup A)\right)=0.$$
	By the assertion $(ii)$, we know that
	$$P(A)=0 \text{ or } 1.$$
	Therefore the assertion $(iii)$ is proved.

	$(iii)\Rightarrow (i)$. Assume $A\in \mathcal{F}$ and $\theta_t^{-1}A=A$ for all $t\in \mathbb{R}$. Then we have
	$$\bigcup_{t\in \mathbb{R}}(\theta_t^{-1}A \bigtriangleup A)=\emptyset.$$
	Hence
	$$P\left(\bigcup_{t\in \mathbb{R}}(\theta_t^{-1}A \bigtriangleup A)\right)=0.$$
	By assertion $(iii)$, we have
	$$P(A)=0 \text{ or } 1.$$
	Thus the assertion $(i)$ is proved.
\end{proof}

Next we  will give the proof of Theorem \ref{SME}.\\
\\
	\textbf{Proof of Theorem \ref{SME}}.
	Since $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in \mathbb{R}})$ is ergodic, by Theorem \ref{SPET} we know that $(\bar{\Omega}, \bar{\mathcal{F}}, \mu, (\bar{\Theta}_t)_{t\in\mathbb{R}})$ is ergodic. Next we just need to show that for any $\Gamma\in \mathcal{B}(\mathbb{X})$, if $P_tI_{\Gamma}=I_{\Gamma}, \text{ for all } t\geq 0, \rho-a.s.$, then $\rho(\Gamma)=0$ or 1.

	Similar to the proof of Theorem \ref{NewET},  for any $A\in \bar{\mathcal{F}}$ and $\Gamma\in \mathcal{B}(\mathbb{X})$, since $\mu(A)=P(\{\omega: (\omega, Y(\omega))\in A\})$ and $\rho(\Gamma)=P(\{\omega: Y( \omega)\in \Gamma\})$, then $\rho(\Gamma)=\mu(\Omega\times \Gamma)$.
	Next we consider the subset $\Omega\times \Gamma$ in $\bar{\Omega}$.
	Note
	\begin{equation*}
	\begin{split}
	\bar{\Theta}_{t}^{-1}(\Omega\times \Gamma)
	&=\{(\omega, x): \bar{\Theta}_{t}(\omega, x)\in \Omega\times \Gamma\}\\
	&=\{(\omega, x): (\theta_{t}\omega, \Phi(t, \omega)x)\in \Omega\times \Gamma\}\\
	&=\{(\omega, x): \omega \in \Omega, \Phi(t, \omega)x \in \Gamma\},\\
	\end{split}
	\end{equation*}
	Define $\Omega_{t, \Gamma}^x:=\{\omega: \Phi(t, \omega)x\in \Gamma\}$. Then we have
	\begin{equation*}
	\begin{split}
	\bar{\Theta}_{t}^{-1}(\Omega\times \Gamma)
	&=\bigcup_{x\in \mathbb{X}}(\Omega_{t, \Gamma}^x\times \{x\})\\
	&=\left(\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)\bigcup \left(\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right).\\
	\end{split}
	\end{equation*}
	Now for any $T \in \mathbb{R}$,
	$$\bigcup_{t\geq T}\bar{\Theta}_{t}^{-1}(\Omega\times \Gamma)=\left(\bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)\bigcup \left(\bigcup_{t\geq T}\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right).$$
	Again, it is easy to see that $\bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})$ is a subset of $\Omega\times \Gamma$, and
	\begin{equation*}
	\begin{split}
	\mu\left(\bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)
	&=\int_{\Omega\times \mathbb{X}}I_{\cup_{t\geq T}\cup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})}(\omega, x)\delta_{Y(\omega)}(dx)P(d\omega)\\
	&=\int_{\Omega}I_{\cup_{t\geq T}\cup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})}(\omega, Y(\omega))P(d\omega)\\
	&=P\left(\{\omega: (\omega, Y(\omega))\in\bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\}\right)\\
	&=P\left(\{\omega: Y(\omega)\in \Gamma \text{ and } \omega \in \Omega_{t_0, \Gamma}^{Y(\omega)} \text{ for some } t_0\geq T\}\right)\\
	&=P\left(\{\omega: Y(\omega)\in \Gamma \text{ and }\Phi(t_0, \omega)Y(\omega)\in \Gamma \text{ for some } t_0\geq T\}\right)\\
	&=P\left(\{\omega: Y(\omega)\in \Gamma \text{ and } Y(\theta_{t_0}\omega)\in \Gamma \text{ for some } t_0\geq T\}\right).\\
	\end{split}
	\end{equation*}
	As $\Gamma\in \mathcal{I}$, and Condition $A^{\prime}$ holds, we have
	$$P\left(\{\omega: Y(\omega)\in \Gamma \text{ and } Y(\theta_{t_0}\omega)\in \Gamma \text{ for some } t_0\geq T\}\right)=P\left(\{\omega: Y(\omega)\in \Gamma\}\right),$$
	then
	\begin{equation*}
	\begin{split}
	\mu\left(\bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)
	=P\left(\{\omega: Y(\omega)\in \Gamma\}\right)
	=\rho(\Gamma)
	=\mu(\Omega\times \Gamma).
	\end{split}
	\end{equation*}
	Hence
	\begin{equation}
	\label{***}
		\mu\left((\Omega\times \Gamma)\setminus \bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)=0.
    \end{equation}
	Similarly, for the set $\bigcup_{t\geq T}\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})$,
	\begin{equation*}
	\begin{split}
	\mu\left(\bigcup_{t\geq T}\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)
	&=P\left(\{\omega: (\omega, Y(\omega))\in\bigcup_{t\geq T}\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\}\right)\\
	&=P\left(\{\omega: Y(\omega)\notin \Gamma \text{ and }\omega \in \Omega_{t_0, \Gamma}^{Y(\omega)} \text{ for some } t_0\geq T\}\right)\\
	&=P\left(\{\omega: Y(\omega)\notin \Gamma \text{ and }Y(\theta_{t_0}\omega)\in \Gamma \text{ for some } t_0\geq T\}\right),\\
	\end{split}
	\end{equation*}
	and from Condition $A^{\prime}$, we have
	\begin{equation}
	\label{****}
		\mu\left(\bigcup_{t\geq T}\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right)=P\left(\{\omega: Y(\omega)\notin \Gamma \text{ and } Y(\theta_t\omega)\in \Gamma \text{ for some } t_0\geq T\}\right)=0.
	\end{equation}
	Now note that
	\begin{equation*}
	\begin{split}
	 &\left(\bigcup_{t\geq T}\bar{\Theta}_t^{-1}(\Omega\times \Gamma)\right) \bigtriangleup (\Omega\times \Gamma)\\
	=&\left(\left(\bigcup_{t\geq T}\bar{\Theta}_t^{-1}(\Omega\times \Gamma)\right)\setminus (\Omega\times \Gamma)\right) \bigcup  \left((\Omega\times \Gamma)\setminus \bigcup_{t\geq T}\bar{\Theta}_t^{-1}(\Omega\times \Gamma)\right)\\
	\subseteq& \left(\bigcup_{t\geq T}\bigcup_{x\notin \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right) \bigcup \left((\Omega\times \Gamma)\setminus \bigcup_{t\geq T}\bigcup_{x\in \Gamma}(\Omega_{t, \Gamma}^x\times \{x\})\right).\\
	\end{split}
	\end{equation*}
	Then for all $T\in \mathbb{R}$, from (\ref{***}) and (\ref{****}), we have
	$$\mu\left( \left(\bigcup_{t\geq T}\bar{\Theta}_t^{-1}(\Omega\times \Gamma)\right) \bigtriangleup (\Omega\times \Gamma)\right)=0.$$
	Applying the ergodicity of the dynamical system $(\bar{\Omega}, \bar{\mathcal{F}}, \mu, (\bar{\Theta}_t)_{t \in \mathbb{R}})$ and Lemma \ref{lemma}, we have
	$$\rho(\Gamma)=\mu(\Omega\times \Gamma)=0 \text{ or } 1.$$
	Therefore, $\rho$ is ergodic. $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \square$

	\section{Sublinear dynamical systems from periodic measures}
	In this section, we also assume Condition P. We will give a construction of upper expectations via the periodic measures $\mu.$ and $\rho.$ defined in (\ref{PMRDS}) and (\ref{nkey1}) respectively. Then we can study the ergodicity of the sublinear expectation dynamical system and sublinear canonical dynamical system generated by the upper expectations and Markov semigroup $P_t$ defined in (\ref{Markov semigroup}).

	\subsection{Ergodic sublinear dynamical system on upper expectation space}
	Consider a sublinear expectation $\bar{\mathbb{E}}$ on $(\bar{\Omega}, L_{b}(\bar{\Omega}))$ defined by
	\begin{equation}
	\bar{\mathbb{E}}[\varphi]=\sup_{s\in [0,\tau)}\mathbb{E}_{\mu_{s}}[\varphi], \text{ for all } \varphi \in L_{b}(\bar{\Omega}),
	\end{equation}
	where $L_{b}(\bar{\Omega}):=\{\varphi: \bar{\Omega}\rightarrow \mathbb{R}| \varphi \text{ is measurable and bounded} \}$, $\mu_s$ is the periodic measure defined in (\ref{PMRDS}), and
	\begin{equation*}
	\begin{split}
	\mathbb{E}_{\mu_{s}}[\varphi]
	&:=\int_{\Omega\times \mathbb{X}}\varphi(\omega, x)(\mu_{s})_{\omega}(dx)P(d\omega)\\
	&=\int_{\Omega\times \mathbb{X}}\varphi(\omega, x)\delta_{Y(s, \theta_{-s}\omega)}(dx)P(d\omega)\\
	&=\int_{\Omega}\varphi(\omega, Y(s, \theta_{-s}\omega))P(d\omega).\\
	\end{split}
	\end{equation*}
	Recall (\cite{F.Z1})
	\begin{equation}
	\label{key3}
	\bar{\Theta}_t\bar{\mathbb{E}}[\varphi(\cdot)]=\bar{\mathbb{E}}[\varphi(\bar{\Theta}_t\cdot)].
	\end{equation}

	\begin{proposition}
		\label{ISE}
		Assume Condition P. The skew product $(\bar{\Theta}_{t})_{t\geq 0}$ preserves the sublinear expectation $\bar{\mathbb{E}}$.
	\end{proposition}
	\begin{proof}
		We just need to prove $\bar{\Theta}_{t}\bar{\mathbb{E}}[\varphi]=\bar{\mathbb{E}}[\varphi], \text{ for all } \varphi \in L_{b}(\bar{\Omega})$. To see the proof, for any $s\in \mathbb{R}$, by definition of $\mathbb{E}_{\mu_{s}}$, we have
		\begin{equation*}
		\begin{split}
		\bar{\Theta}_{t}\mathbb{E}_{\mu_{s}}[\varphi]
		&=\mathbb{E}_{\mu_{s}}[\varphi\circ \bar{\Theta}_{t}]\\
		&=\int_{\Omega\times \mathbb{X}}\varphi(\bar{\Theta}_{t}(\omega, x))\delta_{Y(s, \theta_{-s}\omega)}(dx)P(d\omega)\\
		&=\int_{\Omega\times \mathbb{X}}\varphi(\theta_{t}\omega,\Phi(t, \omega) x)\delta_{Y(s, \theta_{-s}\omega)}(dx)P(d\omega)\\
		&=\int_{\Omega}\varphi(\theta_{t}\omega, \Phi(t, \omega)Y(s, \theta_{-s}\omega))P(d\omega)\\
		&=\int_{\Omega}\varphi(\theta_{t}\omega, Y(t+s, \theta_{-s}\omega))P(d\omega)\\
		&=\int_{\Omega}\varphi(\theta_{t}\omega, Y(t+s, \theta_{-(t+s)}\theta_{t}\omega))P(d\omega)\\
		&=\int_{\Omega}\varphi(\omega, Y(t+s, \theta_{-(t+s)}\omega))P(d\omega)\\
		&=\mathbb{E}_{\mu_{t+s}}[\varphi].\\
		\end{split}
		\end{equation*}
		Then by definition of $\bar{\mathbb{E}}$, the periodic property of $\mu.$ and (\ref{key3}), we have
		\begin{equation*}
		\begin{split}
		\bar{\Theta}_{t}\bar{\mathbb{E}}[\varphi]
		=\sup_{s\in [0,\tau)}\bar{\Theta}_{t}\mathbb{E}_{\mu_{s}}[\varphi]
		=\sup_{s\in [0,\tau)}\mathbb{E}_{\mu_{t+s}}[\varphi]
		=\sup_{s\in [0,\tau)}\mathbb{E}_{\mu_{s}}[\varphi]
		=\bar{\mathbb{E}}[\varphi].
		\end{split}
		\end{equation*}
	\end{proof}

	Recall the definition of ergodicity of a sublinear expectation dynamical system.
	\begin{definition}
		\label{D1}
		(\cite{F.Z1}) Let $(\hat{\Omega}, \hat{\mathcal{D}}, \hat{\mathbb{E}})$ be a sublinear expectation space and the transformations $(\hat{\theta}_t)_{t\in \mathbb{R}}$ preserves expectation $\hat{\mathbb{E}}$. We say that the sublinear expectation dynamical system $(\hat{\Omega}, \hat{\mathcal{D}}, \hat{\mathbb{E}}, (\hat{\theta}_t)_{t\in \mathbb{R}})$ is ergodic if for any $B\in \hat{\mathcal{F}}$ with $\hat{\theta}_t^{-1}B=B, \text{ for all } t\in \mathbb{R}$, then $\hat{\mathbb{E}}[I_B]=0$ or $\hat{\mathbb{E}}[I_{B^c}]=0$.
	\end{definition}
	Consider the dynamical system $(\bar{\Omega}, L_{b}(\bar{\Omega}), \bar{\mathbb{E}}, (\bar{\Theta}_{t})_{t\geq 0})$ defined as above, we have
	\begin{theorem}
		\label{ENES}
		Assume Condition P. If $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for some $s\in\mathbb{R}$, then $(\bar{\Omega}, L_{b}(\bar{\Omega}), \bar{\mathbb{E}}, (\bar{\Theta}_{t})_{t\geq 0})$ is an ergodic sublinear expectation dynamical system.
	\end{theorem}

	\begin{proof}
		Firstly, we know that $\bar{\Theta}_{t}$ preserves the sublinear expectation $\bar{\mathbb{E}}$ from Proposition \ref{ISE}. For any $A \in \bar{\mathcal{F}}$ with $\bar{\Theta}_{t}^{-1}A=A, \text{ for all } t\geq 0$, by Definition \ref{D1}, we need to prove $V(A)=0$ or $V(A^{c})=0$ where $V(A):=\sup_{s\in [0,\tau)}\mu_{s}(A)$. Since $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for some $s\in\mathbb{R}$, by Theorem \ref{Converse ET} we know that $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n\geq 0})$ is ergodic. Consider the proof of Theorem \ref{ET} we have $\mu_s(A)=\mu_0(A)$ for all $s\in\mathbb{R}$ and $\mu_0(A)=0$ or 1. Then

		$$V(A)=\sup_{s\in [0,\tau)}\mu_{s}(A)=\mu_{0}(A)=0$$
		or
		$$V(A^c)=\sup_{s\in [0,\tau)}\mu_{s}(A^c)=\mu_{0}(A^c)=1-\mu_{0}(A)=0.$$
	\end{proof}

	We say a statement holds $\bar{\mathbb{E}}-q.s.$ if the statement is true on $A^c$ where $\bar{\mathbb{E}}\chi_A=0$.

	\begin{proposition}
		Assume Condition P and $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^n)_{n\geq 0})$ is ergodic for some $s\in\mathbb{R}$, Then for any $\xi \in L_{b}(\bar{\Omega})$, we have
		$$\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}U_{t}\xi dt=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{s}}[\xi]ds, \quad \bar{\mathbb{E}}-q.s.$$
		where $(U_{t}\xi)(\omega, x)=\xi(\bar{\Theta}_{t}(\omega, x))$.
	\end{proposition}

	\begin{proof}
		By (i) of Remark \ref{re1}, we know that $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^{n})_{n\geq 1})$ is ergodic for each $s\in \mathbb{R}$. For any $\xi\in L_b^0(\bar{\Omega})$, without any loss of generality, we can assume that $\xi\geq 0, \bar{\mathbb{E}}-q.s.$. Let $\xi_{\tau}:=\frac{1}{\tau}\int_{0}^{\tau}U_{t}\xi dt$, then
		$$\frac{1}{n}\sum_{k=0}^{n-1}U_{\tau}^{k}\xi_{\tau}=\frac{1}{n\tau}\int_{0}^{n\tau}U_{t}\xi dt.$$
		Applying Birkhoff's ergodic theorem (c.f. \cite{Da Prato}) for $\xi_{\tau}$ on $(\bar{\Omega}, \bar{\mathcal{F}}, \mu_{s}, (\bar{\Theta}_{\tau}^{n})_{n\geq 1})$, we have
		$$\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=0}^{n-1}U_{\tau}^{k}\xi_{\tau}=\mathbb{E}_{\mu_{s}}[\xi_{\tau}], \quad \mu_{s}-a.s..$$
		For any arbitrary $T>0$, let $n_{T}=[\frac{T}{\tau}]$ be the maximal nonnegative integer less than or equal to $\frac{T}{\tau}$. Then $n_{T}\tau\leq T<(n_{T}+1)\tau$ and
		$$\frac{n_{T}}{n_{T}+1}\cdot\frac{1}{n_{T}\tau}\int_{0}^{n_{T}\tau}U_{t}\xi dt\leq \frac{1}{T}\int_{0}^{T}U_{t}\xi dt\leq \frac{n_{T}+1}{n_{T}}\cdot\frac{1}{(n_{T}+1)\tau}\int_{0}^{(n_{T}+1)\tau}U_{t}\xi dt.$$
		Thus
		$$\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}U_{t}\xi dt=\mathbb{E}_{\mu_{s}}[\xi_{\tau}] \quad \mu_{s}-a.s..$$
		However,
		\begin{equation*}
		\begin{split}
		\mathbb{E}_{\mu_{s}}[\xi_{\tau}]&=\mathbb{E}_{\mu_{s}}[\frac{1}{\tau}\int_{0}^{\tau}U_{t}\xi dt]\\
		&=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{s}}[U_{t}\xi] dt\\
		&=\frac{1}{\tau}\int_{0}^{\tau}\bar{\Theta}_{t}\mathbb{E}_{\mu_{s}}[\xi] dt\\
		&=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{s+t}}[\xi] dt\\
		&=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{t}}[\xi] dt,\\
		\end{split}
		\end{equation*}
		thus
		$$\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}U_{t}\xi dt=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{t}}[\xi] dt \quad \mu_{s}-a.s..$$
		Let $A_{s}$ be the $\mu_{s}$-null set such that
		$$\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}U_{t}\xi dt=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{t}}[\xi] dt, \quad \text{on }A_{s}^{c}.$$
		Let $A=\cap_{s \in [0, \tau)}A_{s}$, then $V(A)=\sup_{s\in [0,\tau)}\mu_{s}(A)=0$ and
		$$\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}U_{t}\xi dt=\frac{1}{\tau}\int_{0}^{\tau}\mathbb{E}_{\mu_{t}}[\xi] dt \quad \text{on } A^c.$$
	\end{proof}

    From the Condition P and the assumption that the periodic measure $\{\mu_{s}\}_{s\in\mathbb{R}}$ on the production space is PS-ergodic, we obtained the Birkhoff's law of large numbers with the convergence in the quasi-sure sense. This result is stronger than the Birkhoff ergodic type theorem in the almost sure sense that we can obtain from the ergodic theory of periodic measure (\cite{F.Z2}). This justifies the study of the construction of invariant sublinear expectations from periodic measures.

	Next we also give two examples of ergodic sublinear dynamical system.
	\subsubsection{Example: an ergodic sublinear dynamical system with discrete time}
	\label{Example1}
	We consider $\Omega_{1}=[0,1), \theta_{\alpha}: \Omega_{1}\rightarrow \Omega_{1}, \theta_{\alpha}(x)=(x+\alpha) \text{ mod }1$. It is well known that the dynamical system $(\Omega_{1}, \mathcal{B}(\Omega_{1}), P_{1}, (\theta_{\alpha}^{n})_{n\geq 1})$ is ergodic when $\alpha$ is irrational. Here $P_{1}$ is the Lebesgue measure on $\Omega_{1}$.\\

	Next, we consider $\hat{\Omega}=[0,2)=[0,1)\cup [1,2)=\Omega_{1}\cup \Omega_{2}$, where $\Omega_{2}=[1,2)$. Define $\hat{\theta}_{\alpha}: \hat{\Omega}\rightarrow \hat{\Omega}$ by
	\begin{equation*}
	\hat{\theta}_{\alpha}(x)=
	\begin{cases}
	\theta_{\alpha}(x)+1, & \text{$x\in \Omega_{1}$},\\
	x-1,                         & \text{$x\in  \Omega_{2}$}.
	\end{cases}
	\end{equation*}
	Then we know that $\hat{\theta}_{\alpha}^2$ maps $\Omega_{i}$ into $\Omega_{i}$ for $i=1,2$. Let $P_{2}$ be the Lebesgue measure on $\Omega_{2}$, then $(\Omega_{i}, \mathcal{B}(\Omega_{i}), P_{i}, (\theta_{\alpha}^{2n}|_{\Omega_{i}})_{n\geq 1}), i=1,2$ are ergodic dynamical systems.\\

	Define
	$$\bar{P}_{i}(A):=P_{i}(A\cap \Omega_{i}), \text{ for any } A \in \mathcal{B}(\hat{\Omega}),$$
	and
	$$\hat{\mathbb{E}}[X]=\mathbb{E}_{\bar{P}_{1}}[X]\vee\mathbb{E}_{\bar{P}_{2}}[X], \text{ for all }  X \in L^{1}(\hat{\Omega}).$$
	Then for any $\xi \in L^1(\hat{\Omega})$, we have
	\begin{equation*}
	\begin{split}
	\hat{\theta}_{\alpha}\mathbb{E}_{\bar{P}_1}[\xi]
	&=\mathbb{E}_{\bar{P}_1}[\xi\circ \hat{\theta}_{\alpha}]\\
	&=\int_{\hat{\Omega}}\xi(\hat{\theta}_{\alpha}(x))\bar{P}_1(dx)\\
	&=\int_{\Omega_1}\xi(\hat{\theta}_{\alpha}(x))P_1(dx)\\
	&=\int_{\Omega_2}\xi(y)P_2(dy)\\
	&=\int_{\hat{\Omega}}\xi(y)\bar{P}_2(dy)\\
	&=\mathbb{E}_{\bar{P}_2}[\xi].\\
	\end{split}
	\end{equation*}
	Similarly,
	$$\hat{\theta}_{\alpha}\mathbb{E}_{\bar{P}_2}[\xi]=\mathbb{E}_{\bar{P}_1}[\xi].$$
	Thus $\hat{\theta}_{\alpha}\hat{\mathbb{E}}[\xi]= \mathbb{E}_{\bar{P}_1}[\xi\circ\hat{\theta}_{\alpha}]\vee\mathbb{E}_{\bar{P}_2}[\xi\circ\hat{\theta}_{\alpha}] =\mathbb{E}_{\bar{P}_2}[\xi]\vee\mathbb{E}_{\bar{P}_1}[\xi] = \hat{\mathbb{E}}[\xi]$, which means $\hat{\theta}_{\alpha}$ preserves the sublinear expectation $\hat{\mathbb{E}}$. Then $(\hat{\Omega}, \mathcal{B}(\hat{\Omega}), (\hat{\theta}_{\alpha}^n)_{n\geq 1}, \hat{\mathbb{E}})$ is a sublinear dynamical system.

	Let $\bar{P}=\frac{1}{2}(\bar{P}_1+\bar{P}_2)$. Then $\bar{P}$ is an invariant measure with respect to $\hat{\theta}_{\alpha}$ and ergodic as it is PS-ergodic by Theorem 1.3.6 in \cite{F.Z2}. Moreover, $\hat{\theta}_{\alpha}$ is also ergodic under the sublinear upper expectation setting that we observe in this section. Here we give a straightforward proof of this result in this special case.

	\begin{proposition}
		The sublinear dynamical system $\hat{S}=(\hat{\Omega}, \mathcal{B}(\hat{\Omega}), (\hat{\theta}_{\alpha}^n)_{n\geq 1}, \hat{\mathbb{E}})$ is ergodic while $\alpha$ is irrational.
	\end{proposition}

	\begin{proof}
		For any $A \in \mathcal{B}(\hat{\Omega})$ with $\hat{\theta}^{-1}_{\alpha}A=A$, we need to show that
		$$\hat{\mathbb{E}}[I_A]=\bar{P}_1(A)\vee\bar{P}_2(A)=0 \text{ or } \hat{\mathbb{E}}[I_{A^c}]=\bar{P}_1(A^c)\vee\bar{P}_2(A^c)=0.$$
		By $\hat{\theta}^{-1}_{\alpha}A=A$ we have $(\hat{\theta}^2_{\alpha})^{-1}A=A$, which means
		$$\{(\hat{\theta}^2_{\alpha})^{-1}(A\cap\Omega_1)\}\cup\{(\hat{\theta}^2_{\alpha})^{-1}(A\cap\Omega_2)\}=(A\cap \Omega_1)\cup(A\cap\Omega_2).$$
		Since $\hat{\theta}^2_{\alpha}$ maps $\Omega_{i}$ into $\Omega_{i},i=1,2,$ then $$(\hat{\theta}^2_{\alpha})^{-1}(A\cap\Omega_i)=A\cap\Omega_i.$$
		As we already know, $(\Omega_{i}, \mathcal{B}(\Omega_{i}), ((\hat{\theta}^2_{\alpha})^n)_{n\geq 1}, P_i)$ are ergodic while $\alpha$ is irrational, Then
		$$P_i(A\cap\Omega_{i})=0 \text{ or } 1, i=1,2.$$
		Since $\hat{\theta}_{\alpha}(A\cap\Omega_{1})=A\cap\Omega_{2}$, then we have $$\bar{P}_2(A)=P_2(A\cap\Omega_{2})=P_2(\hat{\theta}_{\alpha}(A\cap\Omega_{1}))=P_1(A\cap\Omega_{1})=\bar{P}_1(A).$$
		Thus
		$$\hat{\mathbb{E}}[I_A]=\bar{P}_1(A)\vee\bar{P}_2(A)=0
		\text{ or }
		\hat{\mathbb{E}}[I_A^c]=\bar{P}_1(A^c)\vee\bar{P}_2(A^c)=0.$$
		This means the sublinear dynamical system $\hat{S}=(\hat{\Omega}, \mathcal{B}(\hat{\Omega}), (\hat{\theta}_{\alpha}^n)_{n\geq 1}, \hat{\mathbb{E}})$ is ergodic.
	\end{proof}

	\subsubsection{Example: Ergodic sublinear dynamical system on torus}
	We consider the same dynamical system $(\tilde{\Omega}, \tilde{\mathcal{F}}, (\tilde{\theta}^{\alpha}_t)_{t\in \mathbb{R}})$ on torus and periodic probability measure $\mu_s, s\in \mathbb{R},$ as in Section \ref{Example2}. Now we define the upper expectation $\tilde{\mathbb{E}}$ on $L_b^0(\tilde{\Omega})$ by
	$$\tilde{\mathbb{E}}[\varphi]:= \sup_{s\in [0,1)}\mathbb{E}_{\mu_{s}}[\varphi], \quad \text{ for all } \varphi \in L_b^0(\tilde{\Omega})$$.

	\begin{proposition}
		The sublinear dynamical system $(\tilde{\Omega}, L_b^0(\tilde{\Omega}), \tilde{\mathbb{E}}, (\tilde{\theta}^{\alpha}_t)_{t\in \mathbb{R}}))$ is ergodic.
	\end{proposition}

	\begin{proof}
	  The proof is similar to that of Theorem \ref{ENES}. But in Theorem \ref{ENES}, we assumed that is a random periodic path. But its proof does not depend on this assumption. The key is $\mu_{s}(\tilde{A})=\mu_{0}(\tilde{A})$ for any invariant set $\tilde{A}$. This identify for the current case
	  is proved by a different method, see Proposition \ref{prop1} and (\ref{eq4}).
	\end{proof}

\subsection{Ergodicity of sublinear canonical dynamical systems with respect to a Markovian semigroup}
Next we will give the definitions of sublinear Markovian systems and their ergodicity (\cite{F.Z1}, \cite{Peng1}).
\begin{definition}
	(\cite{Peng1}) We say that $T: \mathbb{R}^+\times L_{b}(\mathbb{X})\rightarrow L_{b}(\mathbb{X})$ is a sublinear Markovian semigroup if:
	\begin{enumerate}[(i)]
		\item For each fixed $(t,x)\in \mathbb{R}^+\times\mathbb{X}$, $T_{t}[\varphi](x)$ is a sublinear expectation defined on $L_{b}(\mathbb{X})$.
		\item $T_0[\varphi](x)=\varphi(x)$, for each $\varphi\in L_{b}(\mathbb{X})$.
		\item $T_{t}[\varphi]$ satisfies the following Chapman semigroup formula:
		$$T_{t}\circ T_{s}[\varphi]=T_{t+s}[\varphi], \text{ for any }t,s\geq 0.$$
	\end{enumerate}
	Here $L_{b}(\mathbb{X})$ is the $\cal B(\mathbb X)$-measurable real-valued function defined on $\mathbb X$ such that $\sup_{x\in \mathbb X}|\phi(x)|<\infty$.
\end{definition}

\begin{remark}
	The Markovian semigroup $P_t, t\geq 0$ given in (\ref{Markov semigroup}) is also a sublinear Markovian semigroup.
\end{remark}

\begin{definition}
	(\cite{F.Z1}) A nonlinear expectation $\tilde{T}: L_{b}(\mathbb{X})\rightarrow \mathbb{R}$ is said to be an invariant nonlinear expectation under a sublinear Markovian semigroup $T_{t}, t\geq 0$ if it  satisfies
	$$T_{t}\tilde{T}=\tilde{T} \text{ for all } t\geq 0.$$
	Here $T_{t}\tilde{T}[\varphi]:=\tilde{T}[T_{t}\varphi].$
\end{definition}

Similarly as in the invariant measure case, denote by $\Omega^*=\mathbb{X}^\mathbb{R}$ the space of all $\mathbb{X}$-valued functions on $\mathbb{R}$, by $\mathcal{F}^*$ the smallest $\sigma$-algebra containing all cylindrical sets of $\Omega^*$, the shift $\theta^*: \mathbb{R}\times \Omega^*\rightarrow \Omega^*$ is defined by $(\theta_{t}^*\omega^*)(s)=\omega^*(t+s)$, for all $\omega^*\in \Omega^*.$
Set
\begin{equation*}
	\begin{split}
	&L_0(\mathcal{F}^*)\\
	&:=\{\xi| \text{there exists } n \geq 1, t_1,\cdots,t_n\in \mathbb{R}, \varphi\in L_b(\mathbb{X}^n) \text{ such that } \xi(\omega^*)=\varphi(\omega^*(t_1),\cdots,\omega^*(t_n))\}.
	\end{split}
\end{equation*}
It is clear that $L_0(\mathcal{F}^*)$ is a linear subspace of $L_b(\mathcal{F}^*)$.

For any $\varphi\in L_b(\mathbb{X}^n)$ and $t_1<t_2<\cdots<t_n$, we define $\varphi_i\in L_b(\mathbb{X}^{n-i}), i=1,2,\cdots,n$ as follows:
\begin{equation*}
\begin{split}
\varphi_1(x_1,x_2,\cdots,x_{n-1})&:=T_{t_n-t_{n-1}}[\varphi(x_1,x_2,\cdots,x_{n-1},\cdot)](x_{n-1})\\
\varphi_2(x_1,x_2,\cdots,x_{n-2})&:=T_{t_{n-1}-t_{n-2}}[\varphi_1(x_1,x_2,\cdots,x_{n-2},\cdot)](x_{n-2})\\
&\vdots\\
\varphi_{n-1}(x_1)&:=T_{t_{2}-t_{1}}[\varphi_{n-2}(x_1,\cdot)](x_{1})\\
\end{split}
\end{equation*}
For any $T_{t}, t\geq 0$ invariant sublinear expectation $\tilde{T}$, we define the sublinear expectation $\mathbb{E}^{\tilde{T}}$ on $L_0(\mathcal{F}^*)$ by
$$\mathbb{E}^{\tilde{T}}[\xi]=\tilde{T}[\varphi_{n-1}(\cdot)], \text{ where } \xi(\omega^*)=\varphi(\omega^*(t_1),\omega^*(t_2),\cdots,\omega^*(t_n)).$$
Then we can extend $L_0(\mathcal{F}^*)$ into $L_0^p(\Omega^*)$ under the norm $(\mathbb{E}^{\tilde{T}}[|\cdot|^p])^{1/p}, p\geq 1$. Define the space
\begin{equation*}
\begin{split}
&Lip_{b,cyl}(\Omega^*)\\
&:=\{\xi| \text{there exists } n \geq 1, t_1,\cdots,t_n\in \mathbb{R}, \varphi\in C_{b,lip}(\mathbb{X}^n) \text{ such that } \xi(\omega^*)=\varphi(\omega^*(t_1),\cdots,\omega^*(t_n))\}.
\end{split}
\end{equation*}
and $L_G^p(\Omega^*)$ the completion of $Lip_{b,cyl}(\Omega^*)$ under the norm $(\mathbb{E}^{\tilde{T}}[|\cdot|^p])^{1/p}, p\geq 1$. From \cite{F.Z1}, $\theta^*_t$ preserves the expectation $\mathbb{E}^{\tilde{T}}$, and $S^{\tilde{T}}=(\Omega^*, L_G^2(\Omega^*), (\theta^*_t)_{t\in \mathbb{R}}, \mathbb{E}^{\tilde{T}})$ defines a dynamical system, called \textit{canonical dynamical system} associated with $T_t, t\geq 0, \tilde{T},$ and $\theta^*_t$.

\begin{definition}
	(\cite{F.Z1}) The invariant expectation $\tilde{T}$ is said to be ergodic with respect to the sublinear Markovian semigroup $T_t, t\geq 0$, if its associated canonical dynamical system $S^{\tilde{T}}=(\Omega^*, L_G^2(\Omega^*), (\theta^*_t)_{t\in \mathbb{R}}, \mathbb{E}^{\tilde{T}})$ on the sublinear expectation space is ergodic.
\end{definition}
Next we will consider the ergodic property of a sublinear expectation $\tilde{T}$ on $(\mathbb{X}, L_b(\mathbb{X}))$ defined by
\begin{equation}
\label{eq2}
\tilde{T}[\varphi]=\sup_{s\in [0,\tau)}\mathbb{E}_{\rho_{s}}[\varphi], \quad \text{ for all } \varphi\in L_b(\mathbb{X}),
\end{equation}
where $\mathbb{E}_{\rho_{s}}[\varphi]:=\int_{\mathbb{X}}\varphi(x)\rho_{s}(dx).$

First we have the following property of $\tilde{T}$.
\begin{proposition}
	The sublinear expectation $\tilde{T}$ is an invariant expectation under the Markovian semigroup $P_t, t\geq 0.$
\end{proposition}
\begin{proof}
	We need to prove that $P_t\tilde{T}[\varphi]=\tilde{T}[\varphi], \text{ for all } t\geq 0, \varphi\in L_b(\mathbb{X}).$ Since $\rho.$ is a periodic measure of the Markovian semigroup $P_t$, we have
	\begin{equation}
	\label{PE}
	\begin{split}
	P_t\mathbb{E}_{\rho_s}[\varphi]&=\mathbb{E}_{\rho_s}[P_t\varphi]\\
	&=\int_{\mathbb{X}}P_t\varphi(x)\rho_s(dx)\\
	&=\int_{\mathbb{X}}\int_{\mathbb{X}}P_t(x,dy)\varphi(y)\rho_s(dx)\\
	&=\int_{\mathbb{X}}\int_{\mathbb{X}}P_t(x,dy)\rho_s(dx)\varphi(y)\\
	&=\int_{\mathbb{X}}\varphi(y)\rho_{s+t}(dy)\\
	&=\mathbb{E}_{\rho_{s+t}}[\varphi],\\
	\end{split}
	\end{equation}
	therefore by (\ref{eq2}), (\ref{PE}) and the periodicity of $\rho.$, we have that
	\begin{equation*}
	\begin{split}
	P_t\tilde{T}[\varphi]=\tilde{T}[P_t\varphi]
	=\sup_{s\in [0,\tau)}\mathbb{E}_{\rho_s}[P_t\varphi]
	=\sup_{s\in [0,\tau)}\mathbb{E}_{\rho_{s+t}}[\varphi]
	=\sup_{s\in [0,\tau)}\mathbb{E}_{\rho_s}[\varphi]
	=\tilde{T}[\varphi].
	\end{split}
	\end{equation*}
\end{proof}

\begin{theorem}
	Under the assumption in Theorem \ref{NewET}, if $(\Omega, \mathcal{F}, P, (\theta_{\tau}^n)_{n\geq 1})$ is ergodic, then $\tilde{T}$ is ergodic with respect to the Markovian semigroup $P_t, t\geq 0$.
\end{theorem}
\begin{proof}
	By Theorem 3.25 in \cite{F.Z1},	we need to show that for any $\varphi\in L_b(\mathbb{X})$, if $P_t\varphi=\varphi$ (since $P_t$ is a linear operator, then $P_t(-\varphi)=-\varphi$) and $\varphi^2$ has no mean uncertainty under $\tilde{T}$, then $\varphi$ is constant, $\tilde{T}$-q.s.\\
	Firstly by Theorem \ref{NewET}, we know that $\{\rho_{s}\}_{s\in\mathbb{R}}$ is PS-ergodic. Then by $P_{\tau}\varphi=\varphi$, we have $\varphi$ is constant, $\rho_{s}$-a.s. for all $s\in\mathbb{R}$. Let $l_s$ be that constant, i.e. $\varphi=l_s, \rho_{s}$-a.s..\\
	Since $P_t\varphi=\varphi$ for all $t\geq 0$ and by  (\ref{PE}), we have
	\begin{equation*}
	\begin{split}
	l_{s+t}=\mathbb{E}_{\rho_{s+t}}[\varphi]
	=\mathbb{E}_{\rho_{s}}[P_t\varphi]
	=\mathbb{E}_{\rho_{s}}[\varphi]
	=l_s.
	\end{split}
	\end{equation*}
	So $l_s\equiv l_0$ and $\varphi=l_0, \rho_{s}-a.s.$ for all $s\in \mathbb{R}$. Let $A:=\{x: \varphi(x)\neq l_0\}$, then $\rho_{s}(A)=0, \text{ for all } s\in \mathbb{R}$. Thus $\tilde{T}[I_A]=\sup_{s\in [0,\tau)}\rho_{s}(A)=0$, which means $\varphi=l_0, \tilde{T}-q.s.$.
\end{proof}

    \section*{Acknowledgements}
    We acknowledge the financial support of the Royal Society Newton Advanced Fellowship grant NA150344.


TITLE: How can you calculate $\int_{-\infty}^\infty\frac{dx}{x^2+1}$ using the residue at $-i$?
QUESTION [0 upvotes]: Instead of using traditional upper-half semicircle, I have tried to calculate this integral using lower-half and have obtained $-\pi$, that is the opposite to the actual value.
Can anyone show me how to calculate it?

REPLY [0 votes]: Almost certainly Robert Lee is right about your error. The residue is the result of integrating around a closed curve in the counter-clockwise direction. If you traverse the curve in the clockwise direction, you get the opposite value.
If you use the upper half-circle and travel counter-clockwise, then the part of the curve on the real line is integrated from negative to positive, and taking the limit gives $\int_{-\infty}^\infty \frac{dx}{x^2 + 1}$. But if you use the lower half-circle, you can either go counter-clockwise, in which case the portion on the real line is integrated from positive to negative, so taking the limit gives $\int_{\infty}^{-\infty} \frac{dx}{x^2 + 1}$, or you can travel clockwise, in which case you do get $\int_{-\infty}^\infty$, but it is equal to the opposite of the residue, instead of the residue itself.
Either way, there is an additional negation you missed performing.


TITLE: Confidence Intervals
QUESTION [1 upvotes]: How can we calculate $90\%$ confidence intervals for the variance and standard deviation of the concentration of artificial food colouring in different lots of colouring, given a random sample of $12$ lots, which yielded the following concentration values?
$0.013,0.016,0.009,0.018,0.014,0.010,0.013,0.014,0.012,0.014,0.015,0.010$
What assumptions are made?

REPLY [1 votes]: If you assume that the concentration is randomly distributed with mean $\mu$ and standa deviation $\sigma$, then you can use the same method as in this example.


TITLE: Show that a recursive sequence formula is correct
QUESTION [0 upvotes]: Given the Fibonacci numbers, denoted by {Fn}, with $$F_1=1, F_2=1$$and $$F_{n+1}=F_n+F_{n-1}$$Let another another sequence be $$a_{n}=\frac{F_{2n-5}}{F_{2n-3}}$$ for $${n\ge3}$$
Now show that $$a_{n+1}=\frac{1}{3-a_n}$$
Is this question as simple as plugging values into a(n) and a(n+1) and showing they are equal? Or is there a proper proof to show this recursive formula?

REPLY [1 votes]: We have to prove $a_{n+1}(3-a_n)=1$ or:
$$ \frac{F_{2n-3}}{F_{2n-1}}\cdot\frac{3F_{2n-3}-F_{2n-5}}{F_{2n-3}} = 1$$
or:
$$ 3 F_{2n-3} =  F_{2n-5} + F_{2n-1}. \tag{1}$$
The characteristic polynomial of the Fibonacci sequence is $x^2-x-1$, with roots $\varphi$ and $\bar{\varphi}$, hence the characteristic polynomial of the sequence $\{F_{2n}\}_{n\geq 0}$ or $\{F_{2n+1}\}_{n\geq 0}$ has roots $\varphi^2$ and $\bar{\varphi}^2$, so it is $x^2-3x+1$ and $(1)$ is proved.


TITLE: Proving convergence of a sum
QUESTION [0 upvotes]: The question is:
Show that the sum $\sum_{n=2}^{\infty} \sqrt{n^4+2n^2}-1-n^2$ converge.
The previous
assignment showed that $|\sqrt{1+2x}-1-x| \leq C |x|^2 \quad |x| \leq 1/4$.
Its hinted that you need to use this information, but I´m not sure how to proceed.

REPLY [0 votes]: $$\sum_{n=2}^{\infty} \sqrt{n^4+2n^2}-1-n^2=\\\sum_{n=2}^{\infty} (\sqrt{n^4+2n^2}-1-n^2)\dfrac{\sqrt{n^4+2n^2}+1+n^2}{\sqrt{n^4+2n^2}+1+n^2}=\\
\sum_{n=2}^{\infty} \dfrac{n^4+2n^2-n^4-2n^2-1}{\sqrt{n^4+2n^2}+1+n^2}=\\
\sum_{n=2}^{\infty} \dfrac{-1}{\sqrt{n^4+2n^2}+1+n^2}\\ \sim \sum_{n=2}^{\infty} \dfrac{-1}{\sqrt{n^4}+n^2}\sim \sum_{n=2}^{\infty} \dfrac{-1}{2n^2}$$


TITLE: Are these integrals $\int_{0}^\infty f(t)dt$, $\int_{-\infty}^0 f(-t)dt$ equal?
QUESTION [1 upvotes]: Question about changing an integral bounds due to a reflection.

Suppose you have an integral across $\Bbb R^{+}$ such that $$I=\int_{0}^\infty f(t)dt$$
Would $I$ be the same if I applied a reflection across the y-axis such that $t\to-t$ and integrated along $\Bbb R^{-}$ such that$$I=\int_{-\infty}^0 f(-t)dt$$
Is this ok?

REPLY [1 votes]: By the change of variable,
$$
u=-t, \qquad dt=-du, \qquad 0 \rightarrow 0,\qquad \infty \rightarrow -\infty,
$$ one gets
$$
I=\int_{0}^\infty f(t)dt=\int_{0}^{-\infty} f(-u)(-du)=\int_{-\infty}^0 f(-u)du,
$$ as announced.