6. Compact embedding of the spaces of Holder functions. Let be a compact metric space, be the space of -Holder continuous functions on . Note that we also denote , and . In the following we assume .

Let , whose -norm is given by , where , and .

**Proposition 1.** The embedding is compact.

A key step in the proof is to apply the following (then we pick a countable subset of and employ the Cantor’s Diagonal Method):

**Arzela–Ascoli Lemma.** A subset of is relatively compact in if and only if it is bounded and equicontinuous.

**Definition.** A subset is called equicontinuous at , if for every there exists such that for all , whenever .

Then is called equicontinuous if it is equicontinuous at every .

Proof of Proposiiton 1. Let be the unit ball. Clearly . Moreover, note that for all , whenever . So it suffices to pick for every . QED.

A stronger result is

**Proposition 2.** Let . Then the embedding is compact.

Proof. We need to show that the unit ball is relatively compact in . To this end let’s pick a sequence . By Proposition 1, there exists a subsequence . Without loss of generality we can assume , and hence a Cauchy sequence with respect to . We want to show that it is also a Cauchy sequence with respect to . Given , there exists such that whenever . A simple calculation shows that

for all with . So given , we just need to set , and . This completes the proof.

QED.

Given a continuous function on with as , we can define a -norm , where . Let be the set of functions with . One can prove

**Proposition 3.** Let be two functions as above with as . Then the embedding is compact.

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).

Assume (3.2) is true for some . Let . Note that , where . So

So we only need to check (3.2) for ergodic measures.

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.