Monochromatic nonuniform hyperbolicity

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 Hölder-continuous. Our construction uses the nonuniformly hyperbolic cocycles discovered by Walters in 1984.

v.3: Revised according to the referee reports. Final version, accepted for publication in Trans. Amer. Math. Soc