## Tag Archives: Birkhoff Ergodic Theorem

### Ergodic decomposition

Let $X$ be a compact metric space and $f:X\to X$ be a homeomorphism. Let $\mathcal{M}(f)$ be the set of $f$-invariant measures and $\mathcal{E}(f)$ be the set of $f$-invariant ergodic measures.

Let $\mu$ be an invariant measure. A distribution $\tau$ on $\mathcal{E}(f)$ is said to be the ergodic decomposition of $\mu$ if for each continuous function $\phi:X\to\mathbb{R}$, the following holds:

$\int_{\mathcal{E}(f)}(\int_X \phi d\nu) d\tau(\nu)=\int_X \phi d\mu$.

The following approach is attributed to R. Mane.
Let $G_\nu$ be the set of generic points of $\nu$ and $G=\bigcup_{\nu\in \mathcal{E}(f)}G_\nu$. Then $G$ is a Borel subset of $X$ and of full probability. Consider the map $\beta:G\to \mathcal{E}(f),x\mapsto\nu_x=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}$.

Proposition: The map $\beta:G\to \mathcal{E}(f)$ is Borel.

So the pushforwad $\tau=\beta_*(\mu)$ is a Borel distribution on $\mathcal{E}(f)$. Moreover for each continuous map $\Phi:\mathcal{M}(X)\to \mathbb{R}$, we have
$\int_{\mathcal{E}(f)}\Phi(\nu)d\tau(\nu)=\int_G\Phi(\beta(x))d\mu(x)$.

In particular letting $\Phi(\nu)=\int_X \phi d\nu$, we have $\Phi(\beta(x))=\int\phi d\nu_x=\phi^*(x)$ and

$\int_{\mathcal{E}(f)}(\int_X \phi d\nu)d\tau(\nu)=\int_G \phi^*(x)d\mu(x)=\int_X\phi(x)d\mu(x)$, where $\phi^*$ is the Birkhoff average and the last equality follows from Brikhoff ergodic theorem. So $\tau=\beta_*(\mu)$ is the ergodic decomposition of $\mu$.

### Some definitions

5. Let $f:X\to X$ be a homeomorphism and $\mu$ be an invariant ergodic measure. Consider a bounded function $\phi$ with $\mu(\phi)=0$ and the induced cocycle $\phi_n$.
Then by Birkhoff ergodicity Theorem, we know $\frac{\phi_n}{n}\to0$, a.e.. A related statement is that, for a.e. $x$, $\displaystyle |\phi_n(x)|\le\|\phi\|_0$ infinitely often.

Proof. Let $a=\|\phi\|_0$ and $X_{n,a}=\{x:\phi_n(x)\in [-a,a]\}$, $X_a=\bigcap_{k\ge1}\bigcup_{n\ge k}X_{n,a}$. Clearly $X_a$ is invariant and hence has measure either zero or one. Suppose $\mu(X_a)=0$. Let $Y_{+}$ be the set of points with $\phi_n(x)\ge a$ for all large enough $n$. Similarly we define $Y_-$. Clearly they are disjoint and both are invariant. Then by the choice $a$, we see $\mu(Y_+\cup Y_-)=1$. By ergodic assumption, we can assume $\mu(Y_+)=1$. So $\displaystyle 0=\mu(\phi_n)\ge\mu(\liminf_{n\to\infty}\phi_n)\ge a>0$, which is absurd.

Moreover we have the following dichotomy:
– either $\phi$ is a coboundary: $\phi(x)=h(x)-h(fx)$, $\mu$-a.e.,

– or $\sup_{n\ge0}\phi_n(x)=+\infty$ and $\inf_{n\ge0}\phi_n(x)=-\infty$ $\mu$-a.e..

Proof. Let $h(x)=\sup_{n\ge1}\phi_n(x)$ (measurable). Let’s assume $E=\{x:h(x)<+\infty\}$ has positive measure. Clearly $E$ is invariant and hence full measure. In particular $h$ is well defined a.e.. Let $g(x)=h(x)-\sup_{n\ge2}\phi_n(x)$. Clearly $g(x)\ge0$.

Note that $\phi_{k+1}(x)=\phi_k(fx)+\phi(x)$ for all $k\ge1$. So $\sup_{k\ge1}\phi_{k+1}(x)=\sup_{k\ge1}\phi_k(fx)+\phi(x)$, or equivalently, $h(x)-g(x)=h(fx)+\phi(x)$, or $\phi(x)+g(x)=h(x)-h(fx)$. So we need to show $g(x)=0$, $\mu$-a.e.. A sufficient condition is $\sum_{n\ge0} g_n(x)<\infty$, a.e..

Note that for $\mu$-a.e. $x$, there exists $n_k\to\infty$ with $\phi_{n_k}(x)\in[-a,a]$ and $h(f^{n_k}x)$ bounded, and hence $g_{n_k}(x)+\phi_{n_k}(x)=h(x)-h(f^{n_k}x)$ stays bounded.

It seems that $\{\phi_n(x):n\ge0\}$ is dense in $\mathbb{R}$ $\mu$-a.e. in the second alternative.

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1. Let $f:X\to X$ be a topological dynamical system. Birkhoff Pointwise Ergodic Theorem states that, there exists a full measure set $G\subset X$ ($\mu(G^c)=0$ for all $f$-invariant measure $\mu$) such that the limit $\nu_x=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}$ exists and is $f$-invariant and ergodic for each $x\in G$. Then consider the measure (be careful) $\nu=\int_G \nu_x d\mu(x)$. It is an $f$-invariant probability measure with that:
$\nu(A)=\int_G\nu_x(A)d\mu(x)=\int_G\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}(A)d\mu(x)$
$\:=\int_G\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{A}(f^kx)d\mu(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\int_G\delta_{A}(f^kx)d\mu(x)$
$=\mu(A)$ for each measurable subset $A\subset X$.

So we have $\mu=\nu=\int_G \nu_x d\mu(x)$, the Ergodic Decomposition of $\mu$.

2. A foliation is a decomposition of a manifold into submanifolds (foliation box $\mathbb{R}^u\times\mathbb{R}^c$). A lamination is a partial foliation: $\mathbb{R}^u\times K$ for some compact subset $K\subset\mathbb{R}^c$

There are various versions of absolute continuity transversal abs cts (implied by bdd jacobian). leaf abs cts.

3. Let $\pi:G(M,k)\to M$ be the $k$-dimensional Grassmannian bundle over $M$. For each $k$-dimensional linear subspace $V\subset T_xM$, we denote $[V]$ the corresponding element in $G(x,k)\subset G(M,k)$. The topology of $G(M,k)$ is determined by the distance function $D$ such that for all $V\in G(x,k),W\in G(y,k)$
$D([V], [W]) = \inf\{L(\gamma) + \angle_{x}(V, P_{\gamma}W)|\gamma:y\to x\text{ is smooth}\}$,
where $P_\gamma$ is the parallel translation along $\gamma$. Under this topology the projection $\pi:G(M,k)\to M$ is a continuous map.

4. Let $f:M\to M$ be a diffeomorphism and $\Lambda$ be a compact invariant subset. Then $f$ is shadowing on $\Lambda$ if for each $\delta>0$ there exists $\epsilon>0$ such that every $\epsilon-$pseudo orbit can be $\delta-$shadowed by a genuine orbit of $f$.

Generalized to weakly shadowing property: for each $\delta>0$ there exists $\epsilon>0$ such that the $\epsilon-$chain is contained in the $\delta-$ neighborhood of a genuine orbit: $\{x_n\}\subset B(\mathcal{O}(x),\delta)$.

$\Lambda$ is stably weak shadowing at $f$ if there exist a nbhd $U\supset\Lambda$ and $\mathcal{U}\ni f$ such that $\bigcap_{n\in\mathbb{Z}}g^nU$ is weak shadowing under $g$ for each $g\in\mathcal{U}$. A special case is $M$ itself is weak shadowing.

$f$ is tame if there is a neighborhood $\mathcal{U}$ of $f$ such that each $g\in\mathcal{U}$ has only finitely many chain recurrent classes. (there are other different defintions with the same name.)

Conjecture:: $f$ is stably weak shadowing if and only if $f$ is tame.

Note: this is true if $\dim(M)=2$ and is conjectured for general case by GAN, Shaobo.

It is proved by YANG, Dawei that if a transitive set $\Lambda$ is stably weakly shadowing at $f$, then $\Lambda$ admits a dominated splitting.

Ergodic maps form a $G_\delta$ subset. Let $f\in\mathrm{Diff}^r_m(M)$ and $\phi\in C(M,\mathbb{R})$. Then Birkhoff ergodic theorem says that $A_n(\phi,f)(x)\to \phi^\ast(x)$ almost everywhere and in $L^2$. In particular the limit $\|A_n(\phi,f)-\int \phi dm\|\to \|\phi^\ast-\int \phi dm\|$ always exists, just may not be zero. Those maps $f$ with $\phi^\ast(x)\equiv \int \phi dm$ are characterized by
$\displaystyle \bigcap_{l\ge 1}\bigcap_{N\ge 1}\bigcup_{n\ge N}\{f\in\mathrm{Diff}^r_m(M):\|A_n(\phi,f)-\int \phi dm\|<1/l\}$, which is a $G_\delta$ subset. Therefore the ergodic ones form a $G_\delta$ subset of $\mathrm{Diff}^r_m(M)$.

Metrically transitive maps form a $G_\delta$ subset. A map $f\in\mathrm{Diff}^r_m(M)$ is said to be $m$-transitive, if $m$-a.e. points have dense orbits. Such a property also called weakly ergodic. These maps also form a $G_\delta$ subset of $\mathrm{Diff}^r_m(M)$. To this end, note that the open sets $U$ with $m(\partial U)=0$ form a basis of the topology on $M$, and the mapping $f\mapsto m(\bigcup_{\mathbb{Z}}f^nU)$ is lower semi-continuous. Therefore the set of maps with $m(\bigcup_{\mathbb{Z}}f^nU)> 1-1/l$ form an open subset.