## Symbolic extension and entropy

A diffeomorphism $f:M\to M$ is hyperbolic if the chain recurrent set $CR(f)$ is hyperbolic off $f$. It is showed to be equivalent to either one of the following:

1. $f$ satisfies Axiom A and no-cycle condition;
2. $f$ is $\Omega-$stable;
3. $f$ is chain stable;

Let $\alpha_k, k\ge1$ be an entropy structure of $(M,f)$, where $\alpha_k$ are partitions and get finer and $k$ increases such that the entropy map $h_k:\mathcal{M}(f)\to\mathbb{R},\mu\mapsto h(f,\mu,\alpha_k)$ converges pointwisely to $h(f,\mu)$.

$(M,f)$ has a principal symbolic extension iff $h_k$ converges uniformly to $h$.

$(M,f)$ has no symbolic extension at all iff there exist $c>0$ and a compact set $E\subset\mathcal{M}(f)$ such that for every $\mu\in E$ and every $k\ge1$,
$\limsup_{\nu\in E\to\mu}(h(f,\nu)-h(f,\nu,\alpha_k))>c.$

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Let $f:X\to X$ be a measurable map, $r:X\to\mathbb{R}$ be measurable function. For each $x\in X$, $n\ge1$ we define the scaled Bowen ball as
$B(x,r,n,f)=[y|d(f^ix,f^iy)\le r(f^ix),0\le ,
and the scaled local entropy for any Borel measure $m$ as
$h(x,f,r,m)=\limsup_{n\to\infty}\frac{-1}{n}\log m(B(x,r,n,f))$.

Proposition (R. Mane): For any $m\in\mathcal{M}(X)$, if $\mu\in\mathcal{M}(f)$ satisfies $\mu\ll m$, then for any $\mu-$integrable function $r$,
$h(f,\mu)\ge\int_X h(x,f,r,m) d\mu(x)$.

Lem 1. If $x_n\in(0,1)$ satisfies $\sum_{n\ge1}n\cdot x_n<\infty$, then $\sum_{n\ge1}x_n\log(x_n^{-1})<\infty$.

Lem 2. There exists a countable partition $\alpha$ with information $H_\mu(\alpha)<\infty$.
First partition the set $X_n=[x:e^{-n-1}\le r(x)< e^{-n}]$ and collect these together.

Then by Shannon–McMillan–Breiman theorem, we get
$h(f,\mu)\ge h(f,\mu,\alpha)=\int \lim_{n\to\infty}\frac{-1}{n} \log\mu(\alpha^n(x))d\mu(x)$
$\ \ \ \ =\int\lim_{n\to\infty}\frac{-1}{n}\log m(\alpha^n(x))d\mu(x)\ge\int h(x,f,r,m)d\mu(x)$.

Now let $f\in\mathrm{Diff}^{1+\epsilon}(M)$ and $\mu\ll m$ is an ACIP. Mane gave another proof of Pesin formula $h(f,\mu)=\int \chi^+(f,x)d\mu(x)$ for any ACIP $\mu$ of $f$.

Proof. Consider the zipped Oseledec splitting $E^u_x=\oplus_{\chi_i(f,,x)>0}E^i_x$ and $E^c==\oplus_{\chi_i(f,,x)\le0}E^i_x$. Let $X_j=[x:\dim E^u_x=j]$ and consider the conditional measure $\mu_j$ for each $j=0,\cdots,d$. It suffices to show that $h(f,\mu_j)=\int \chi^+(f,x)d\mu_j(x)$ for each $j$. We omit the subscript $j$ for a moment. For each $\delta>0$, there exists a compact set $K$ with $\mu(K)\ge 1-\delta$ in which the splitting $T_xM=E^u_x\oplus E^c_x$ is continuous and uniform:
There exist $\lambda>C>1$ and $N\ge1$, such that for $g=f^N$,
$\|D_xg^n(v)\|\ge\lambda^n\|v\|$
$\|D_xg^n(w)\|\le C$ for each $x\in K,n\ge0,v\in E^u_x,w\in E^c_x$.

Consider the first return map of $(f,K)$ as $n(x)=\inf[k\ge1:f^kx\in K]$ if $x \in K$ and $n(x)=0$ otherwisely. Note that $n:K\to\mathbb{N}$ is integrable with $\int n(x) d\mu(x)\le 1$. This induces a scaling funtion $r(x)\sim \xi^{n(x)}$ for some pretty small $\xi$. Note that $\log r$ is also $\mu-$integrable.

We are left to show that $h(x,r,g,m)\ge N\cdot (\chi^+(x,f)-C\varepsilon)$ for a tiny smaller set $K'\subset K$ ($\mu(K)\ge 1-2\varepsilon$…)