A convex optimization approach to the 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 to a vector-valued optimization problem on a space $\mathcal{M}$ of Riemannian metrics. Similar results were first proved by Jairo Bochi and Andrés Navas in the language of linear cocycles and their conjugacies. We go one step further and prove that the optimization problem is geodesically convex with respect to the $L^2$-metric on $\mathcal{M}$. Moreover, we derive some consequences of this fact.