Discontinuity of Lyapunov Exponents Near Fiber Bunched Cocycles

Abstract

We give examples of locally constant $SL(2,\mathbb{R})$-cocycles over a Bernoulli shift which are discontinuity points for Lyapunov exponents in the Hölder topology and are arbitrarily close to satisfying the fiber bunching inequality. Backes, Brown, and the author have shown that the Lyapunov exponents vary continuously when restricted to the space of fiber bunched Hölder continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of Hölder cocycles.