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arXiv · 2604.12244

Lyapunov exponents for uniformly hyperbolic random matrix products

Nima Alibabaei
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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.

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Nima Alibabaei. 2026-04-14. Lyapunov exponents for uniformly hyperbolic random matrix products. https://arxiv.org/abs/2604.12244

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