Continuity of Lyapunov Exponents for Cocycles with Invariant Holonomies

TL;DR

This paper proves a conjecture by Viana showing that Lyapunov exponents vary continuously for certain cocycles over subshifts, provided they have invariant holonomies that depend continuously on the cocycle.

Contribution

It establishes the continuity of Lyapunov exponents for $GL(2,R)$-valued cocycles with invariant holonomies, confirming Viana's conjecture.

Findings

Lyapunov exponents are continuous under specified conditions.

Invariant holonomies depend continuously on the cocycle.

Supports conjecture by Viana in dynamical systems theory.

Abstract

We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to $GL(2,\mathbb{R})$-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle.