Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases

TL;DR

This paper extends Viana's theorem, showing that cocycles on noncompact classical semisimple Lie groups over hyperbolic bases almost always have positive Lyapunov exponents, indicating prevalent exponential divergence.

Contribution

It generalizes the positivity of the top Lyapunov exponent to cocycles on all noncompact classical semisimple Lie groups over hyperbolic systems.

Findings

Almost all cocycles on these groups have positive Lyapunov exponents.

Extension of Viana's theorem to a broader class of Lie groups.

Abstract

A theorem of Viana says that almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. In this note we extend this result to cocycles on any noncompact classical semisimple Lie group.