Skip to main content

Ergodic Property of Lyapunov Exponents

  • 940 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1978)

Abstract

Generic property of SRB measures was first investigated by Bowen [10]. Theorem 4.12 in [10] says that if Ω is a hyperbolic attractor of a C 2 Axiom A diffeomorphism (M, f) and m is the volume measure on the compact Riemannian manifold M induced by the Riemannian metric, then for m-almost all x in the basin of attraction W s(Ω),

$$\mathop {{\rm lim}}\limits_{n \to+ \infty } \frac{1}{n}\mathop\sum \limits_{k = 0}^{n - 1} \delta _{^f } k_x= \mu+,$$
((VIII.1))

where µ + is the SRB measure for f on Ω. As µ + is an ergodic measure, the Lyapunov exponents of system f : M ← are µ +-almost everywhere constants. Recently, by exploiting a Ruelle’s perturbation theorem [79, Theorem 4.1] Jiang et al. [29] proved that m-almost all xW s(Ω) is positively regular and the Lyapunov spectrum of the system (i.e., the Lyapunov exponents associated with their multiplicities) at x are the constants

$${\rm \{ (}\lambda _{\rm 1} {\rm (}\mu _{\rm+ } {\rm , }f{\rm),}m_{\rm 1} {\rm (}\mu _{\rm+ } {\rm , }f{\rm)), \cdot \cdot\cdot , (}\lambda _{\rm r} {\rm (}\mu _{\rm+ } {\rm , }f{\rm ),}m_{\rm r} {\rm (}\mu _{\rm+ } {\rm , }f{\rm))\} }{\rm .}$$

This is called the ergodic property of Lyapunov exponents. Similar results have also been obtained in [29] for nonuniformly completely hyperbolic attractors of C 2 diffeomorphisms.

Keywords

  • Lyapunov Exponent
  • Absolute Continuity
  • Conditional Measure
  • Lyapunov Spectrum
  • Markov Partition

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

QUIAN, M., XIE, JS., ZHU, S. (2009). Ergodic Property of Lyapunov Exponents. In: Smooth Ergodic Theory for Endomorphisms. Lecture Notes in Mathematics(), vol 1978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01954-8_8

Download citation