A diffeomorphism is hyperbolic if the chain recurrent set is hyperbolic off . It is showed to be equivalent to either one of the following:

1. satisfies Axiom A and no-cycle condition;

2. is stable;

3. is chain stable;

Let be an entropy structure of , where are partitions and get finer and increases such that the entropy map converges pointwisely to .

has a principal symbolic extension iff converges uniformly to .

has no symbolic extension at all iff there exist and a compact set such that for every and every ,

—————————————————–

Let be a measurable map, be measurable function. For each , we define the scaled Bowen ball as

,

and the scaled local entropy for any Borel measure as

.

Proposition (R. Mane): For any , if satisfies , then for any integrable function ,

.

Lem 1. If satisfies , then .

Lem 2. There exists a countable partition with information .

First partition the set and collect these together.

Then by Shannon–McMillan–Breiman theorem, we get

.

Now let and is an ACIP. Mane gave another proof of Pesin formula for any ACIP of .

Proof. Consider the zipped Oseledec splitting and . Let and consider the conditional measure for each . It suffices to show that for each . We omit the subscript for a moment. For each , there exists a compact set with in which the splitting is continuous and uniform:

There exist and , such that for ,

for each .

Consider the first return map of as if and otherwisely. Note that $n:K\to\mathbb{N}$ is integrable with . This induces a scaling funtion for some pretty small . Note that is also integrable.

We are left to show that for a tiny smaller set (…)