Discontinuity of Lyapunov exponents for SL$(2,\mathbb{R})$ valued cocycles

TL;DR

This paper demonstrates that Lyapunov exponents can be discontinuous functions of SL(2,R)-valued cocycles, providing explicit examples and extending previous results, with implications for higher-dimensional cases.

Contribution

It constructs explicit examples of discontinuity points for Lyapunov exponents in SL(2,R) cocycles and extends these results to SL(m,R) cocycles, addressing open questions.

Findings

Discontinuity points for Lyapunov exponents are explicitly constructed.

Examples extend previous work by Bocker-Viana and Butler.

Discontinuity also shown in higher-dimensional SL(m,R) cocycles.

Abstract

We exhibit an example of discontinuity point for the Lyapunov exponents as a function of the cocycle in the $\alpha$-H\"older topology. The linear cocycle taking values in $SL(2, \mathbb{R})$ is locally constant and defined over a Bernoulli shift. Our example extends Bocker-Viana's and Butler's results. In particular, it gives a partial answer to a question raised by Clark Butler. Finally, we give an example of discontinuity in the setting of $SL(m,\mathbb{R})$-valued cocycles, which is constructed from $SL(2, \mathbb{R})$-valued cocycles.