Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

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Abstract

We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism f of a compact manifold X preserving a hyperbolic ergodic probability measure µ. The cocycle A over f is Hölder continuous and takes values in GL(d, ℝ) or, more generally, in the group of invertible bounded linear operators on a Banach space. For a GL(d, ℝ)-valued cocycle A we prove that the Lyapunov exponents of A with respect to µ can be approximated by the Lyapunov exponents of A with respect to measures on hyperbolic periodic orbits of f. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of A with respect to µ, but they cannot always be approximated by the exponents of A on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of A on hyperbolic periodic orbits of f.

Original languageEnglish (US)
Pages (from-to)5105-5118
Number of pages14
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume38
Issue number10
DOIs
StatePublished - Oct 2018

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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