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arXiv · 1303.2375

Hadamard-Perron theorems and effective hyperbolicity

Vaughn Climenhaga·Yakov Pesin
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Abstract

We prove several new versions of the Hadamard-Perron Theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard-Perron Theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of "effective hyperbolicity" and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well-behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

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Vaughn Climenhaga, Yakov Pesin. 2013-03-10. Hadamard-Perron theorems and effective hyperbolicity. https://doi.org/10.1017/etds.2014.49

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