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.
QED.

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.