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Hadamard–Perron theorems and effective hyperbolicity

Published online by Cambridge University Press:  16 September 2014

VAUGHN CLIMENHAGA
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA email climenha@math.uh.edu
YAKOV PESIN
Affiliation:
Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, USA email pesin@math.psu.edu

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.

Type
Research Article
Copyright
© Cambridge University Press, 2014 

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