Some results on perturbations of Lyapunov exponents

Chao Liang; Wenxiang Sun; Jiagang Yang

doi:10.3934/dcds.2012.32.4287

Article Contents

2012, Volume 32, Issue 12: 4287-4305. Doi: 10.3934/dcds.2012.32.4287

This issue Previous Article Decay of solutions for a system of nonlinear Schrödinger equations in 2D Next Article Critical exponent for the semilinear wave equation with time-dependent damping

Some results on perturbations of Lyapunov exponents

1.
Applied Mathematical Department, Central University of Finance and Economics, Beijing, 100081
2.
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871
3.
Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, Niterói, 24020-140

Received: November 2010

Revised: May 2012

Published: December 2012

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Abstract

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.

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Mathematics Subject Classification: 37C40, 37D25, 37H15, 37A35.

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