Discontinuity example for the Lyapunov exponents on the boundary of the uniformly hyperbolic set

TL;DR

This paper provides an example demonstrating the discontinuity of Lyapunov exponents as a function of cocycles on the boundary of hyperbolic sets, highlighting subtle topological effects.

Contribution

It constructs a specific cocycle example showing Lyapunov exponent discontinuity on the boundary of hyperbolic sets in a refined topology.

Findings

Lyapunov exponents can be discontinuous at boundary points of hyperbolic sets.

Constructed cocycle is locally constant over Bernoulli shift.

Approximation by cocycles with zero Lyapunov exponents is possible.

Abstract

We present an example of a discontinuity point for the Lyapunov exponents when viewed as a function of the cocycle in a topology finer than the $C^0$-topology. The linear cocycle taking values in SL(2,R) is locally constant, defined over a Bernoulli shift, and lies on the boundary of the uniformly hyperbolic set. In particular, we show that it can be approximated, in the $C_{\delta-\log}$-topology, by cocycles whose Lyapunov exponents vanish.