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

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.