Parameter spaces of locally constant cocycles

TL;DR

This paper studies the structure of hyperbolic cocycles in SL(2,R), introducing a new locus to analyze their properties and answering key open questions in the field.

Contribution

It introduces a novel locus in SL(2,R)^N using M"obius transformation semigroups to analyze hyperbolic cocycles and addresses open problems posed by previous researchers.

Findings

Characterization of the hyperbolic locus for locally constant cocycles

Introduction of a new locus for studying the complement of hyperbolic cocycles

Resolution of open questions by Avila, Bochi, Yoccoz, Jacques, and Short

Abstract

This article concerns the locus of all locally constant $\mathrm{SL}(2,\mathbb{R})$-valued cocycles that are uniformly hyperbolic, called the hyperbolic locus. Using the theory of semigroups of M\"obius transformations we introduce a new locus in $\mathrm{SL}(2,\mathbb{R})^N$ which allows us to study the complement of the hyperbolic locus. Our results answer a question of Avila, Bochi and Yoccoz, and Jacques and Short, while motivating a new line of investigation on the subject.