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.

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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.