5. Let be a stochastic process on , be the transition kernal (the probability for ). This induces an action on the space of Borel measures, . Suppose that

(1). there is a unique stationary measure for the discretized process ;

(2). is continuous.

Then is independent of , and is the unique stationary measure for the original process .

Proof. Note that is also stationary for , for all . Then by the uniqueness, we get for all and is closed and dense, hence coincides with .

4. There is a question about the mixing properties of the induced map. The answer is quite a surprise. Let be an ergodic measure-preserving isomorphism on the unit interval. Then Friedman and Ornstein proved (link) that the following two collections are dense in :

(4.1) such that is not ergodic for all .

(4.2) such that is mixing.

3. Abramov Entropy Formula. Let be a probability measure system, be a -preserving isomorphism on .

(3.1). Let such that , and be the conditional measure of on . For any point , let be the first return to (it is finite for -a.e. by Poincare recurrence theorem). Define the first-return map , , which preserves . Then

(3.2). Let be a measurable roof function, be the suspension space of wrt , be the suspension flow on , which preserves the (normalized) suspension measure . Then .

(3.1=>3.2). We assume for simplicity and then set . Then consider the set , and the induced map , which preserves . Note that , for which it is just a rotation. So (not that trivial). From (3.1), we see that , where . Combining terms, we get (3.2).

2. Some sharp contrast statements.

generic map (in particular, among the expanding ones) has no ACIP (by Avila and Bochi). Every expanding map admits a (unique) ACIP (due to Krzyzewski and Szlenk).

Consider an expanding map . Then every Holder potential has a unique equilibrium state . Consider the zero-temperature limit . Sometime is called an -maximizing measure. Let be the collection of -maximizing measures. Then for a general Lipschitz continuous potential, the following dichotomy holds:

(1) either is cohomologous to a constant (then contains all invariant measures);

(2) or it has a unique maximizing measure, which is supported on a periodic orbit.

Clearly the first case consists of a meager subset, and open and densely in the Lipschitz continuous potential, the ground state is supported on a periodic orbit.

An open question in ergodic optimization is: consider the doubling map , . Find such that (the Lebesgue measure).

Consider a hyperbolic basic set of . For generic (but with empty interior) potential , its has a unique ground state. Moreover, this state is fully supported.

A useful observation made by Jenkinson: let be continuous, be upper semi-continuous potential. Then the map is also upper semi-continuous.

Proof. Since is compact, is bounded and the map is well-defined. Let . We need to show that . First assume . Then pick a sequence of continuous functions pointwisely. Note that by the monotone convergence theorem.

1. Let be the set of area-preserving diffeomorphisms on a surface with topology.

Note that is open and dense (Pugh-Hayashi for ; for : Pixton for , Oliveira for and general surfaces with irreducible homology actions, Xia for Hamiltonian on general surface. still open for not that complicated action on general surface).

What about ?

In the case and , Bochi-Mane Theorem states that generically, and hence is of first category. So -generically, .

I don’t have a specific example with (even for ). See the following post here. Interesting cases: standard maps, convex billiards, geodesic flow on spheres with convex shape, perturbations of completely integrable ones. In particular, approximate ellipse with positive metric entropy.