Continuity of Lyapunov exponents in all H\"older topologies for irreducible cocycles

TL;DR

This paper proves that Lyapunov exponents are continuous in all H"older topologies for irreducible $SL_2(\mathbb{R})$-valued cocycles, providing counterexamples to previous conjectures and extending results to certain $GL_2(\mathbb{R})$-valued cocycles.

Contribution

It establishes the continuity of Lyapunov exponents in all H"older topologies for irreducible cocycles, challenging prior conjectures and extending to cocycles with canonical holonomies.

Findings

Lyapunov exponents are continuous in all H"older topologies for irreducible cocycles.

Counterexamples to Viana's and the author's conjectures are provided.

Continuity results extend to $GL_2(\mathbb{R})$-valued cocycles with canonical holonomies.

Abstract

We prove that a locally constant $SL_{2}(\mathbb{R})$-valued cocycle over the shift generated by an irreducible collection of matrices is a continuity point for Lyapunov exponents in the $\alpha$-H\"older topology for every $\alpha > 0$. This gives negative answers to conjectures of Viana and the author; we pose a new conjecture to replace these conjectures. We show that an analogous continuity result also holds for $GL_{2}(\mathbb{R})$-valued cocycles that admit canonical holonomies.