Continuity of Lyapunov exponents in the C0 topology

TL;DR

This paper demonstrates that the Bochi-Ma ext{n}é theorem does not hold universally for linear cocycles over non-invertible maps, showing the existence of non-uniformly hyperbolic cocycles with bounded Lyapunov exponents.

Contribution

It provides a counterexample to the Bochi-Ma ext{n}é theorem in the context of non-invertible maps, expanding understanding of Lyapunov exponent behavior.

Findings

Existence of C0-open sets of linear cocycles not uniformly hyperbolic

Counterexamples where Lyapunov exponents are bounded away from zero

The Bochi-Ma ext{n}é theorem is false in this broader setting

Abstract

We prove that the Bochi-Ma\~{n}\'{e} theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.