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.

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 <n],
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(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…)

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