Lyapunov exponents for random products of non-negative matrices

TL;DR

This paper investigates Lyapunov exponents for random products of non-negative 2x2 matrices, providing explicit formulas, conditions for conjugacy to positive matrices, and applications to fractal dimensions and recurrence growth rates.

Contribution

It introduces a new series formula for Lyapunov exponents, characterizes conjugacy conditions, and applies these results to fractal dimensions and recurrence analysis.

Findings

Explicit Neumann-series-type formula for Lyapunov exponents.

Characterization of conjugacy to positive matrices via heteroclinic connections.

Series formula for Hausdorff dimension of Cantor set intersections.

Abstract

We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite matrix. We further show that non-negative invertible $2 \times 2$ matrices are simultaneously conjugate to positive matrices if and only if ``generalized'' heteroclinic connections do not occur among products of length at most $2$. These results yield a series formula for the Hausdorff dimension of the intersection of the middle-$n$th Cantor set with a random translate of itself, for every natural number $n$ except $4$. Furthermore, our method applies to the intersection of thick Cantor sets under random translation. We also determine the almost sure growth rate of i.i.d. three-term recurrences with finitely many positive coefficients.