Skip to main content

Axiom A Endomorphisms

  • 929 Accesses

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

Abstract

Axiom A system is an important class in smooth dynamical systems. In the ergodic theory of Anosov diffeomorphisms or of Axiom A attractors of diffeomorphisms, it was shown that there is an invariant Borel probability measure that is characterized by each of the following properties:

  1. (1)

    Pesin’s entropy formula (PEF) holds.

  2. (2)

    SRB property: its conditional measures on unstable manifolds are absolutely continuous with respect to the Lebesgue measures on the corresponding sub-manifolds.

  3. (3)

    Lebesgue a.e. point in an open set is generic with respect to this measure.

  4. (4)

    This measure is approximable by measures that are invariant under suitable stochastic perturbations.

Each one of these properties has been shown to be significant in its own right, but more striking is the fact that they are all equivalent to each one another. Many of these ideas are due to Sinai, Ruelle and Bowen. We refer the reader to [88], [10], [76] and [79].

In this chapter, we study the Axiom A endomorphisms along the line of [72].

Keywords

  • Unstable Manifold
  • Smooth Dynamical System
  • Unique Equilibrium State
  • Local Stable Manifold
  • Inverse Limit Space

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). Axiom A Endomorphisms. 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_4

Download citation