Monochromatic nonuniform hyperbolicity

TL;DR

This paper constructs examples of continuous GL(2,R)-cocycles over subshifts of finite type that are nonuniformly hyperbolic but not uniformly hyperbolic, challenging existing assumptions about hyperbolicity and regularity.

Contribution

It provides explicit constructions of nonuniformly hyperbolic cocycles with constant Lyapunov exponents, showing they cannot be Hölder continuous, thus advancing understanding of hyperbolic dynamics.

Findings

Existence of nonuniform hyperbolic cocycles with constant Lyapunov exponents

Such cocycles are not Hölder continuous

Construction uses Walters' nonuniformly hyperbolic cocycles

Abstract

We construct examples of continuous $\mathrm{GL}(2,\mathbb{R})$-cocycles which are not uniformly hyperbolic despite having the same non-zero Lyapunov exponents with respect to all invariant measures. The base dynamics can be any non-trivial subshift of finite type. According to a theorem of DeWitt--Gogolev and Guysinsky, such cocycles cannot be H\"older-continuous. Our construction uses the nonuniformly hyperbolic cocycles discovered by Walters in 1984.