Notes 3

6. Compact embedding of the spaces of Holder functions. Let X be a compact metric space, C^\alpha(X) be the space of \alpha-Holder continuous functions on X. Note that we also denote C(X)=C^0(X), and \text{Lip}(X)=C^1(X). In the following we assume \alpha\in(0,1).

Let f\in C^\alpha(X), whose \alpha-norm is given by \|f\|_\alpha=\|f\|+|f|_\alpha, where \|f\|=\max_X |f(x)|, and |f|_\alpha=\sup_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)^\alpha}.

Proposition 1. The embedding C^\alpha(X)\to C(X) is compact.

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

Arzela–Ascoli Lemma. A subset of C(X) is relatively compact in C(X) if and only if it is bounded and equicontinuous.

Definition. A subset E\subset C(X) is called equicontinuous at x_\ast\in X, if for every \epsilon there exists \delta such that for all f\in E, |f(x)-f(x_\ast)|\le \epsilon whenever d(x,x_\ast)\le\delta.

Then E\subset C(X) is called equicontinuous if it is equicontinuous at every x_\ast\in X.

Proof of Proposiiton 1. Let B_\alpha\subset C^\alpha(X) be the unit ball. Clearly \|f\|\le \|f\|_\alpha\le 1. Moreover, note that for all f\in B_\alpha, |f(x)-f(y)|\le |f|_\alpha\cdot d(x,y)^\alpha\le \delta^\alpha whenever d(x,x_\ast)\le\delta. So it suffices to pick \delta=\epsilon^{1/\alpha} for every \epsilon. QED.

A stronger result is

Proposition 2. Let \beta>\alpha. Then the embedding C^\beta(X)\to C^\alpha(X) is compact.

Proof. We need to show that the unit ball B_\beta\subset C^\beta(X) is relatively compact in C^\alpha(X). To this end let’s pick a sequence f_n\in B_\beta. By Proposition 1, there exists a subsequence f_{n_i}\to f\in C(X). Without loss of generality we can assume f_{n}\to f, and hence a Cauchy sequence with respect to \|\cdot\|. We want to show that it is also a Cauchy sequence with respect to |\cdot|_\alpha. Given \nu, there exists N(\nu) such that \|f_m-f_n\|\le \nu whenever m,n\ge N. A simple calculation shows that
|f_m-f_n|_\alpha\le \max\{\sup_{d(x,y)\le\delta},\sup_{d(x,y)\ge\delta}\}
\le \max\{ |f_m-f_n|_\beta\cdot \delta^{\beta-\alpha},2\|f_m-f_n\|\cdot \delta^{-\alpha}\}\le 4\delta^{\beta-\alpha}
for all m,n\ge N(\nu) with \nu=\delta^\beta. So given \epsilon, we just need to set \delta=\epsilon^{1/(\beta-\alpha)}, and \nu=\epsilon^{\beta/(\beta-\alpha)}. This completes the proof.

Given a continuous function \phi on \mathbb{R} with \phi(t)\searrow 0 as t\to 0+, we can define a \phi-norm \|f\|_\phi=\|f\|+|f|_\phi, where |f|_\phi=\sup_{x\neq y}\frac{|f(x)-f(y)|}{\phi(d(x,y))}. Let C^{\phi}(X) be the set of functions with \|f\|_\phi<+\infty. One can prove

Proposition 3. Let \phi,\psi be two functions as above with \frac{\psi(t)}{\phi(t)}\searrow 0 as t\to 0+. Then the embedding C^\psi(X)\to C^\phi(X) is compact.

5. Let \Phi_t be a stochastic process on X, P_t(x,A) be the transition kernal (the probability for \Phi_t(x)\in A). This induces an action on the space of Borel measures, P_t:\mu\mapsto\mu\circ P_t: A\mapsto \int_X P_t(x,A)\cdot\mu(dx). Suppose that

(1). there is a unique stationary measure \mu_t for the discretized process \{\Phi_{nt}\}_n;

(2). t\mapsto \mu_t is continuous.

Then \mu_t is independent of t, and \mu=\mu_1 is the unique stationary measure for the original process \{\Phi_t\}_t.

Proof. Note that \mu_t is also stationary for \{\Phi_{nkt}\}_n, for all k\ge1. Then by the uniqueness, we get \mu_{rt}=\mu_t for all r=p/q and \{t:\mu_t=\mu_1\} is closed and dense, hence coincides with \mathbb{R}.

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

(4.1) A\in \mathcal{B}_I such that T_A^k is not ergodic for all k\ge 2.

(4.2) A\in \mathcal{B}_I such that T_A is mixing.

3. Abramov Entropy Formula. Let (X,\mathcal{X},\mu) be a probability measure system, T:X\to X be a \mu-preserving isomorphism on X.

(3.1). Let A\in \mathcal{X} such that \mu(\bigcup_{n\ge 0}T^nA)=1, and \mu_A be the conditional measure of \mu on A. For any point x\in A, let n(x)=\inf\{n\ge1: T^nx\in A\} be the first return to A (it is finite for \mu-a.e. x\in A by Poincare recurrence theorem). Define the first-return map T_A:A\to A, x\mapsto T^{n(x)}x, which preserves \mu_A. Then h(T_A,\mu_A)\cdot \mu(A)=h(T,\mu).

(3.2). Let r:X\to (c,C) be a measurable roof function, X_r be the suspension space of X wrt r, \phi_t be the suspension flow on X_r, which preserves the (normalized) suspension measure \mu_r=\frac{1}{\mu(r)}\mu\times \ell. Then h(\phi_1,\mu_r)\cdot \mu(r)=h(T,\mu).

(3.1=>3.2). We assume c\ge 2 for simplicity and then set I=[0,1). Then consider the set A=X\times I\subset X_r, and the induced map \phi_A:=(\phi_1)_A, which preserves \mu_A:=(\mu_r)_A=\mu\times \ell_{I}. Note that \phi_A(\{x\}\times I)=\{Tx\}\times I, for which it is just a rotation. So h(\phi_A,\mu\times \ell_{I})=h(T,\mu) (not that trivial). From (3.1), we see that h(\phi_A,\mu_A)\cdot \mu_r(A)=h(\phi_1,\mu_r), where \mu_r(A)=\frac{1}{\mu(r)}. Combining terms, we get (3.2).

Assume (3.2) is true for some \mu_i. Let \mu=\sum_i p_i\cdot\mu_i. Note that \mu_r=\frac{1}{\mu(r)}\mu\times \ell=\sum_i q_i\mu_{i,r}, where q_i=\frac{p_i\cdot \mu_i(r)}{\mu(r)}. So

h(\phi_1,\mu_r)\cdot \mu(r)=(\sum_i q_i\cdot h(\phi_1,\mu_{i,r}))*\mu(r) =\sum_i p_i\cdot \mu_i(r)\cdot h(\phi_1,\mu_{i,r})=\sum_i p_i\cdot h(T,\mu_i)=h(T,\mu).

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

2. Some sharp contrast statements.

C^1 generic map (in particular, among the expanding ones) has no ACIP (by Avila and Bochi). Every C^{1+\alpha} expanding map admits a (unique) ACIP (due to Krzyzewski and Szlenk).

Consider an expanding map f:X\to X. Then every Holder potential \phi has a unique equilibrium state \mu_\phi. Consider the zero-temperature limit \mu^0_\phi=\lim_{\beta\to\infty}\mu_{\beta\phi}. Sometime \mu^0_\phi is called an \phi-maximizing measure. Let E(\phi) be the collection of \phi-maximizing measures. Then for a general Lipschitz continuous potential, the following dichotomy holds:

(1) either \phi is cohomologous to a constant (then E(\phi) 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 \tau:\mathbb{T}\mapsto\mathbb{T}, x\mapsto 2x. Find \phi such that E(\phi)=\{m\} (the Lebesgue measure).

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

A useful observation made by Jenkinson: let f:X\to X be continuous, \phi be upper semi-continuous potential. Then the map \Phi: \mu\in\mathcal{M}(f)\mapsto \mu(\phi) is also upper semi-continuous.

Proof. Since X is compact, \phi is bounded and the map \Phi is well-defined. Let \mu_n\in\mathcal{M}(f)\to \mu. We need to show that \limsup\mu_n(\phi)\le \mu(\phi). First assume \mu(\phi)\neq-\infty. Then pick a sequence of continuous functions \phi_i\ge\phi_{i+1}\to\phi pointwisely. Note that \mu(\phi-\phi_n)\to 0 by the monotone convergence theorem.

1. Let \text{Diff}_m^r(S) be the set of C^r area-preserving diffeomorphisms on a surface S with C^r topology.
Note that H^r=\{f\in \text{Diff}_m^r(S): h_{top}(f) > 0\} is open and dense (Pugh-Hayashi for r=1; for 2\le r\le \infty: Pixton for S^2, Oliveira for \mathbb{T}^2 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 H^r_m=\{f\in \text{Diff}_m^r(S): h_m(f) > 0\}?

In the case r=1 and S\neq \mathbb{T}^2, Bochi-Mane Theorem states that h_m(f)=0 generically, and hence H^r_m is of first category. So C^1-generically, h_m(f)=0 < h_{top}(f).

I don’t have a specific example with h_m(f)=0 < h_{top}(f) (even for r=1). 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.

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