Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform quasi-multiplicativity of locally constant cocycles and applications

Reza Mohammadpour, Kiho Park

Abstract

In this paper, we show that a locally constant cocycle $\mathcal{A}$ is $k$-quasi multiplicative under the irreducibility assumption. More precisely, we show that if $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible for every $t \mid d$ and $1\leq m \leq d-1$, then $\mathcal{A}$ is $k$-uniformly spannable for some $k\in \mathbb{N}$, which implies that $\mathcal{A}$ is $k$-quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is $ψ$-mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets.

Uniform quasi-multiplicativity of locally constant cocycles and applications

Abstract

In this paper, we show that a locally constant cocycle is -quasi multiplicative under the irreducibility assumption. More precisely, we show that if and are irreducible for every and , then is -uniformly spannable for some , which implies that is -quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is -mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets.
Paper Structure (9 sections, 8 theorems, 37 equations)

This paper contains 9 sections, 8 theorems, 37 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Acknowledgements.
  3. Preliminary
  4. Set up
  5. Quasi-multiplicativity and Spannability
  6. Proof of Theorem \ref{['thm: main']}
  7. Applications
  8. Gibbs matrix equilibrium states have the Bernoulli property
  9. Shrinking target and recurrence sets

Key Result

Theorem 1.1

Let $\mathcal{A}\colon \Sigma_\ell \to \text{GL}_d(\mathbb{R})$ be a locally constant cocycle. Suppose $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible for every $t \mid d$ and $1\leq m \leq d-1$. Then $\mathcal{A}$ is $k$-uniformly spannable for some $k\in \mathbb{N}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • proof : Proof of Corollary \ref{['thm-k-qm']}
  • Remark 2.6
  • ...and 11 more