5. Let be a homeomorphism and be an invariant ergodic measure. Consider a bounded function with and the induced cocycle .

Then by Birkhoff ergodicity Theorem, we know , a.e.. A related statement is that, for a.e. , infinitely often.

Proof. Let and , . Clearly is invariant and hence has measure either zero or one. Suppose . Let be the set of points with for all large enough . Similarly we define . Clearly they are disjoint and both are invariant. Then by the choice , we see . By ergodic assumption, we can assume . So , which is absurd.

Moreover we have the following dichotomy:

– either is a coboundary: , -a.e.,

– or and -a.e..

Proof. Let (measurable). Let’s assume has positive measure. Clearly is invariant and hence full measure. In particular is well defined a.e.. Let . Clearly .

Note that for all . So , or equivalently, , or . So we need to show , -a.e.. A sufficient condition is , a.e..

Note that for -a.e. , there exists with and bounded, and hence stays bounded.

It seems that is dense in -a.e. in the second alternative.

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1. Let be a topological dynamical system. Birkhoff Pointwise Ergodic Theorem states that, there exists a full measure set ( for all -invariant measure ) such that the limit exists and is -invariant and ergodic for each . Then consider the measure (be careful) . It is an -invariant probability measure with that:

for each measurable subset .

So we have , the Ergodic Decomposition of .

2. A foliation is a decomposition of a manifold into submanifolds (foliation box ). A lamination is a partial foliation: for some compact subset

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

3. Let be the -dimensional Grassmannian bundle over . For each -dimensional linear subspace , we denote the corresponding element in . The topology of is determined by the distance function such that for all

,

where is the parallel translation along . Under this topology the projection is a continuous map.

4. Let be a diffeomorphism and be a compact invariant subset. Then is shadowing on if for each there exists such that every pseudo orbit can be shadowed by a genuine orbit of .

Generalized to weakly shadowing property: for each there exists such that the chain is contained in the neighborhood of a genuine orbit: .

is stably weak shadowing at if there exist a nbhd and such that is weak shadowing under for each . A special case is itself is weak shadowing.

is *tame* if there is a neighborhood of such that each has only finitely many chain recurrent classes. (there are other different defintions with the same name.)

**Conjecture:**: is stably weak shadowing if and only if is tame.

Note: this is true if and is conjectured for general case by GAN, Shaobo.

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

**Ergodic maps** form a subset. Let and . Then Birkhoff ergodic theorem says that almost everywhere and in . In particular the limit always exists, just may not be zero. Those maps with are characterized by

, which is a subset. Therefore the ergodic ones form a subset of .

**Metrically transitive maps** form a subset. A map is said to be -transitive, if -a.e. points have dense orbits. Such a property also called weakly ergodic. These maps also form a subset of . To this end, note that the open sets with form a basis of the topology on , and the mapping is lower semi-continuous. Therefore the set of maps with form an open subset.