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Hadamard–Perron theorems and effective hyperbolicity

Published online by Cambridge University Press:  16 September 2014

VAUGHN CLIMENHAGA  and
YAKOV PESIN
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
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 1 , February 2016 , pp. 23 - 63
Copyright
© Cambridge University Press, 2014 

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References

Alves, J. F.. SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. Éc. Norm. Supér. (4) 33(1) (2000), 1–32.CrossRefGoogle Scholar
Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2) (2000), 351–398; MR 1757000 (2001j:37063b).CrossRefGoogle Scholar
Anosov, D. V.. Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, 90 (1967). American Mathematical Society, Providence, RI, 1969; translated from the Russian by S. Feder.Google Scholar
Barreira, L. and Pesin, Ya.. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
Climenhaga, V., Dolgopyat, D. and Pesin, Ya.. Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps. Preprint, arXiv:1405.6194.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Pliss, V.. On a conjecture of Smale. Differ. Uravn. 8 (1972), 268–282.Google Scholar