Lyapunov exponents for uniformly hyperbolic random matrix products

TL;DR

This paper provides an explicit, rapidly convergent formula and a polynomial-time algorithm for approximating the top Lyapunov exponent of certain hyperbolic random matrix products driven by Markov shifts.

Contribution

It introduces a new explicit representation for the Lyapunov exponent under hyperbolicity conditions, enabling efficient approximation and analysis.

Findings

Explicit formula for the Lyapunov exponent in hyperbolic systems

Polynomial-time algorithm for approximating the exponent with error ε

Analytic dependence of the exponent on matrix entries and transition probabilities

Abstract

We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $\lambda$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the matrices are projectively uniformly hyperbolic with respect to the Markov shift, then $\lambda$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $\lambda$: only $O\big( (\log(1/\varepsilon))^3 \big)$ arithmetic operations are needed to achieve error $\varepsilon$. Furthermore, $\lambda$ depends real analytically on the matrix entries and the transition probabilities near a projectively uniformly hyperbolic system, and each Taylor coefficient can be approximated in polynomial time.