Discontinuity of Lyapunov Exponents Near Fiber Bunched Cocycles

TL;DR

This paper provides examples of specific cocycles where Lyapunov exponents are discontinuous, demonstrating the limits of existing continuity results near fiber bunched cocycles in the H"older topology.

Contribution

It constructs explicit examples of cocycles with discontinuous Lyapunov exponents close to fiber bunched conditions, showing the optimality of previous continuity theorems.

Findings

Examples of discontinuous Lyapunov exponents near fiber bunched cocycles.

Discontinuity points are arbitrarily close to fiber bunched inequality.

Supports the optimality of existing continuity results.

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 H\"older 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 H\"older continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of H\"older cocycles.