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Invariance principle in dynamical systems

Karina Marin, Mauricio Poletti

Abstract

In this survey we talk about what is known as Invariance Principle in dynamical systems. It states that the disintegration of measures with zero center Lyapunov exponents admits some extra invariance by holonomies. We focus on explaining the basic definitions and ideas behind a series of results about the Invariance Principle and give some basic applications on how this is used in dynamical systems.

Invariance principle in dynamical systems

Abstract

In this survey we talk about what is known as Invariance Principle in dynamical systems. It states that the disintegration of measures with zero center Lyapunov exponents admits some extra invariance by holonomies. We focus on explaining the basic definitions and ideas behind a series of results about the Invariance Principle and give some basic applications on how this is used in dynamical systems.

Paper Structure

This paper contains 12 sections, 23 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. disintegration along foliations
  3. holonomies
  4. the invariance principle
  5. Product structure and continuous extension
  6. Random dynamics and Furstenberg entropy
  7. Cocycles
  8. Linear cocycles
  9. Smooth Cocycles
  10. some applications
  11. Positivity of Lyapunov exponents
  12. Rigidity of measures

Key Result

Theorem (Invariance Principle)

Let $f:M\to M$ be a $C^1$ partially hyperbolic diffeomorphisms which is dynamically coherent and acts quasi-isometric along the center. If $\mu$ is an ergodic invariant measure such that $h^u_\mu(f)=h_\mu(f)$, then the center disintegration of $\mu$ is $u$-invariant.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 3.1
  • Theorem (Invariance Principle)
  • proof
  • Definition 4.1
  • Proposition 4.2: Theorem 6, BGV
  • proof
  • Corollary 4.3
  • Definition 4.4
  • Corollary 4.5
  • ...and 28 more