Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles

TL;DR

This paper investigates the properties of finite families of SL(2,R) matrices that exhibit uniform exponential growth, analyzing their structure, boundaries, and the effects of Markovian constraints on their products.

Contribution

It characterizes the open subsets of matrix families with uniform exponential growth and explores the impact of Markovian rules on these matrix products.

Findings

Identification of open subsets with uniform exponential growth

Analysis of boundary and complement of these subsets

Extension to Markovian product constraints

Abstract

We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We also consider the more general situation where the allowed products of matrices satisfy a Markovian rule.