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$C^{1}$-openness of non-uniform hyperbolic diffeomorphisms with bounded $C^{2}$-norm

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  06 June 2019

CHAO LIANG ,
KARINA MARIN  and
JIAGANG YANG
CHAO LIANG
Affiliation:
School of Statistics and Mathematics, Central University of Finance and Economics, Beijing100081, China email chaol@cufe.edu.cn
KARINA MARIN
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas (ICEx), Universidade Federal de Minas Gerais, Brazil email kmarin@mat.ufmg.br
JIAGANG YANG
Affiliation:
Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil email yangjg@impa.br
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Abstract

We study the $C^{1}$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^{2}$ partially hyperbolic symplectic systems which have bounded $C^{2}$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.

Keywords

non-uniform hyperbolicitypartial hyperbolic diffeomorphismsLyapunov exponents

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 3078 - 3104
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Copyright
© Cambridge University Press, 2019

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