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Asymptotic behavior of the spectral radius of locally constant strongly irreducible cocycles

Nicolas Martinez Ramos

Abstract

We establish some conditions under which $\text{GL}(d,\mathbb{R})$-valued cocycles over a subshift of finite type, equipped with an equilibrium state, exhibit exponential asymptotics for the spectral radius. Specifically, we show that the exponential growth rate of the spectral radius converges to the top Lyapunov exponent of the cocycle. This result provides a partial answer to a question posed by Aoun and Sert in their paper "Law of large numbers for the spectral radius of random matrix products" (2021). Our approach relies on large deviation estimates for linear cocycles, which may be of independent interest.

Asymptotic behavior of the spectral radius of locally constant strongly irreducible cocycles

Abstract

We establish some conditions under which -valued cocycles over a subshift of finite type, equipped with an equilibrium state, exhibit exponential asymptotics for the spectral radius. Specifically, we show that the exponential growth rate of the spectral radius converges to the top Lyapunov exponent of the cocycle. This result provides a partial answer to a question posed by Aoun and Sert in their paper "Law of large numbers for the spectral radius of random matrix products" (2021). Our approach relies on large deviation estimates for linear cocycles, which may be of independent interest.

Paper Structure

This paper contains 8 sections, 21 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Zero Weight on Hyperplanes
  4. Continuous local product structure
  5. Weightlessness of hyperplanes
  6. Large Deviation Estimates
  7. Distance between direction of maximum growth and hyperplane of least growth
  8. Proof of Main Theorem

Key Result

Theorem 1.1

Suppose $(\hat{\Sigma},T)$ is a topologically mixing two-sided subshift of finite type. Suppose that $\hat{\mu}$ is the equilibrium state of some Hölder potential. Let $A:\hat{\Sigma}\to GL(d,\mathbb{R})$ be a strongly irreducible locally constant linear cocycle such that $\lambda_{1}(\hat{\mu})>\la

Theorems & Definitions (48)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 3.1
  • proof
  • ...and 38 more