Critical exponent for the semilinear wave equation with time-dependent damping
Decay of solutions for a system of nonlinear Schrödinger equations in 2D
December 2012, 32(12): 4287-4305. doi: 10.3934/dcds.2012.32.4287
Some results on perturbations of Lyapunov exponents
Chao Liang 1, , Wenxiang Sun 2, and Jiagang Yang 3,
Applied Mathematical Department, Central University of Finance and Economics, Beijing, 100081, China
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871
Departamento de Geometria, Instituto de Matemática, Universidade Federal Fluminense, Niterói, 24020-140, Brazil
Received November 2010 Revised May 2012 Published August 2012
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.
Keywords: dominated splitting, Integrated Lyapunov exponent, ergodic., far from tangencies, simple spectrum, Oseledec splitting.
Mathematics Subject Classification: 37C40, 37D25, 37H15, 37A3.
Citation: Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
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