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Random 2D linear cocycles I: dichotomic behavior

Pedro Duarte, Marcelo Durães, Tomé Graxinha, Silvius Klein

Abstract

In this paper we establish a Bochi-Mañé type dichotomy in the space of two dimensional, nonnegative determinant matrix valued, locally constant linear cocycles over a Bernoulli or Markov shift. Moreover, we prove that Lebesgue almost every such cocycle has finite first Lyapunov exponent, which then implies a break in the regularity of the Lyapunov exponent, from analyticity to discontinuity.

Random 2D linear cocycles I: dichotomic behavior

Abstract

In this paper we establish a Bochi-Mañé type dichotomy in the space of two dimensional, nonnegative determinant matrix valued, locally constant linear cocycles over a Bernoulli or Markov shift. Moreover, we prove that Lebesgue almost every such cocycle has finite first Lyapunov exponent, which then implies a break in the regularity of the Lyapunov exponent, from analyticity to discontinuity.

Paper Structure

This paper contains 5 sections, 24 theorems, 67 equations.

Table of Contents

  1. Introduction and statements
  2. Avila-Bochi-Yoccoz Theory
  3. Projective uniform hyperbolicity
  4. Continuity dichotomy
  5. Examples

Key Result

Theorem 1.1

Given $\underline{A}\in {\rm Mat}^+_2(\mathbb{R})^k$ with at least one singular and one invertible component, the following dichotomy holds: either $\underline{A}$ is (projectively) uniformly hyperbolic or else it can be approximated by a sequence of cocycles $\underline{A}_n\in {\rm Mat}_2^+(\mathb

Theorems & Definitions (74)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 64 more