A convex optimization approach to the Lyapunov exponents

TL;DR

This paper demonstrates that Lyapunov exponents of smooth measure-preserving systems can be characterized as solutions to a geodesically convex optimization problem on Riemannian metrics, providing new insights into their structure.

Contribution

It extends previous work by showing the convexity of the Lyapunov exponent optimization problem in the space of Riemannian metrics.

Findings

Lyapunov exponents are solutions to a convex optimization problem.

The optimization problem is geodesically convex with respect to the $L^2$-metric.

Consequences of convexity are derived for the structure of Lyapunov exponents.

Abstract

The aim of this paper is to shed more light on some recent ideas about Lyapunov exponents and clarify the formal structures behind these ideas. In particular, we show that the vector of (averaged) Lyapunov exponents of a smooth measure-preserving dynamical system can be regarded as the solution of a vector-valued optimization problem on a space of Riemannian metrics. This result was first formulated and proved by Jairo Bochi in the language of linear cocycles and their conjugacies. We go a step further and prove that the optimization problem is geodesically convex with respect to the $L^2$-metric. Moreover, we derive some consequences of this fact.