Lyapunov exponents for uniformly hyperbolic random matrix products

Abstract

We consider a finite family of invertible $2\times 2$ real matrices and a transitive Markov shift on the index set. Let $\hat{I}»$ 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 $\hat{I}»$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $\hat{I}»$: only $O((\log (1/\epsilon ){)}^3)$ arithmetic operations are needed to achieve error $\epsilon$. Furthermore, $\hat{I}»$ 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.

74 pages, no figures