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Discontinuity of Lyapunov exponents for SL$(2,\mathbb{R})$ valued cocycles

Edhin Mamani, Raquel Saraiva

TL;DR

This paper studies the regularity of Lyapunov exponents for SL$(2,\mathbb{R})$-valued cocycles over a Bernoulli shift in the $\alpha$-Hölder topology. The authors construct explicit perturbations $B_k$ of a locally constant cocycle $A_{\sigma\eta}$ that exchange Oseledets subspaces, proving the existence of $\alpha$-Hölder cocycles arbitrarily close to $A_{\sigma\eta}$ with zero Lyapunov exponents, thereby establishing discontinuity points. The method leverages induced subsystems on cylinders, the Furstenberg-Kesten framework, and a careful cylinder-based perturbation to balance growth, extending Bocker-Viana, Butler, and Saraiva's work and generalizing to diagonal and higher-dimensional blocks. These results highlight robust discontinuities in Lyapunov exponents outside fiber-bunched regimes and demonstrate a dimensionally scalable mechanism for zero-exponent perturbations with potential implications for stability analyses in random matrix products.

Abstract

We exhibit an example of discontinuity point for the Lyapunov exponents as a function of the cocycle in the $α$-Hölder topology. The linear cocycle taking values in $SL(2, \mathbb{R})$ is locally constant and defined over a Bernoulli shift. Our example extends Bocker-Viana's and Butler's results. In particular, it gives a partial answer to a question raised by Clark Butler. Finally, we give an example of discontinuity in the setting of $SL(m,\mathbb{R})$-valued cocycles, which is constructed from $SL(2, \mathbb{R})$-valued cocycles.

Discontinuity of Lyapunov exponents for SL$(2,\mathbb{R})$ valued cocycles

TL;DR

This paper studies the regularity of Lyapunov exponents for SL-valued cocycles over a Bernoulli shift in the -Hölder topology. The authors construct explicit perturbations of a locally constant cocycle that exchange Oseledets subspaces, proving the existence of -Hölder cocycles arbitrarily close to with zero Lyapunov exponents, thereby establishing discontinuity points. The method leverages induced subsystems on cylinders, the Furstenberg-Kesten framework, and a careful cylinder-based perturbation to balance growth, extending Bocker-Viana, Butler, and Saraiva's work and generalizing to diagonal and higher-dimensional blocks. These results highlight robust discontinuities in Lyapunov exponents outside fiber-bunched regimes and demonstrate a dimensionally scalable mechanism for zero-exponent perturbations with potential implications for stability analyses in random matrix products.

Abstract

We exhibit an example of discontinuity point for the Lyapunov exponents as a function of the cocycle in the -Hölder topology. The linear cocycle taking values in is locally constant and defined over a Bernoulli shift. Our example extends Bocker-Viana's and Butler's results. In particular, it gives a partial answer to a question raised by Clark Butler. Finally, we give an example of discontinuity in the setting of -valued cocycles, which is constructed from -valued cocycles.
Paper Structure (5 sections, 6 theorems, 79 equations)

This paper contains 5 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Construction
  3. Lyapunov exponents of the perturbation
  4. Generalized Example
  5. Example in higher dimension

Key Result

Theorem 1.1

For any $\alpha>0$ and $\sigma > 1$ satisfying $\sigma^2 \geq 2^{3\alpha}$ there exist $\alpha$-Hölder continuous cocycles $B \colon \{0,1\}^{\mathbb{Z}} \to \mathrm{SL}(2,\mathbb{R})$ with zero Lyapunov exponents that are arbitrarily close to $A_\sigma$ in the $\alpha$-Hölder topology. In particula

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Example 4.1
  • ...and 3 more