Pesin set, closing lemma and shadowing lemma in $C^1$ non-uniformly hyperbolic systems with limit domination

Details Pesin set, closing lemma and shadowing lemma in $C^1$ non-uniformly hyperbolic systems with limit domination Pesin set, closing lemma and shadowing lemma in $C^1$ non-uniformly hyperbolic systems with limit domination

2010-04-04

arxiv.org/abs/1004.0486v1

Mathematics

Dynamical Systems

Scientific paper

For a $C^1$ diffeomorphism, we construct a new type of Pesin blocks and thus a Pesin set. All Pesin blocks have the same degree of mean hyperbolicity and all sufficiently long orbit segments with starting points and ending points at the same block are of the same type of quasi-hyperbolicity. We introduce a concept of limit domination, which is weaker than the usual domination. For a $C^1$ diffeomorphism preserving an hyperbolic ergodic measure $\mu$ with limit domination, we show the existence of our Pesin set with $\mu$ full measure, and we realize a closing lemma and shadowing lemma, comparable with Katok's closing lemma and shadowing lemma in the $C^{1+\alpha}\,(0<\alpha<1)$ setting. This enables us to get certain properties in $C^1$ setting with limit domination, which are similar to ones in the classical Pesin theory in the $C^{1+\alpha}$ setting. We present one of such properties, showing that all invariant measures supported on some $\mu$ full measure set can be approximated by periodic measures.

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