Extremal Lyapunov exponents in random dynamics

TL;DR

This paper derives a formula for extremal Lyapunov exponents in random dynamical systems generated by finitely many maps, using ergodic theory to characterize orbit instability.

Contribution

It provides a novel expression for extremal Lyapunov exponents in random dynamics with maps of degree greater than one, generalizing classical ergodic theorems.

Findings

Derived an explicit formula for extremal Lyapunov exponents.

Connected Lyapunov exponents to the limit of logarithmic operator norms.

Extended ergodic theory to subadditive sequences in this context.

Abstract

In this manuscript, we consider finitely many maps, all of which are defined on a smooth compact measure space, with at least one map in the collection having degree strictly bigger than 1. Working with random dynamics generated by this setting, we obtain an expression for the extremal Lyapunov exponents, that characterise the instability of typical orbits, as the limit of the averages of the logarithm of the operator norm of linear cocycles of generic orbits. We obtain this as a consequence to the Kingman's ergodic theorem for a subadditive sequence of measurable functions, which naturally generalises the Birkhoff's ergodic theorem.