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Forum Mathematicum

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Exponential closing property and approximation of Lyapunov exponents of linear cocycles

1Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China.

Citation Information: Forum Mathematicum. Volume 23, Issue 2, Pages 321–347, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/form.2011.011, April 2011

Publication History

Received:
2008-10-09
Revised:
2009-04-25
Published Online:
2011-04-02

Abstract

Let be a Hölder-continuous linear cocycle with a discrete-time, μ-measure-preserving driving flow ƒ: X × ℤ → X on a compact metric space X. We show that the Lyapunov characteristic spectrum of (, μ) can be approached arbitrarily by that of periodic points. Consequently, if all periodic points have only nonzero Lyapunov exponents and such exponents are uniformly bounded away from zero, then (, μ) also has only non-zero Lyapunov exponents. In our arguments, an exponential closing property of the driving flow is a basic condition. And we prove that every C 1-class diffeomorphism of a closed manifold obeys this closing property on its any hyperbolic invariant subsets.

Keywords.: Exponential closing; linear cocycle; Katok closing lemma; approximation of Lyapunov exponents and ergodic measures

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