Analyticity of the Lyapunov exponents of random products of matrices

TL;DR

This work establishes that, for random products of matrices with compact symbol spaces, the top Lyapunov exponent $L_1$ depends real-analytically on the underlying data with respect to the total variation topology. The authors build an analytic framework in Banach spaces using holomorphy concepts and the Markov operator, proving that $L_1$ extends to a holomorphic function on an affine subspace of measures and remains analytic under Markov transitions when quasi-irreducibility or full support holds. Key steps include showing convergence of Markov-operator iterates to a rank-one limit, constructing a holomorphic extension through complex perturbations, and transferring these results to Markov cocycles with continuous kernels. The results generalize Peres’ finite-symbol case to infinite, compact-symbol settings and yield corollaries for absolutely continuous measures and locally constant Markov cocycles, with explicit dependence on total variation distance. The analysis highlights the essential role of compactness and irreducibility-like conditions in achieving high regularity for Lyapunov exponents in random matrix products.

Abstract

This paper is concerned with the study of random (Bernoulli and Markovian) product of matrices on a compact space of symbols. We establish the analyticity of the maximal Lyapunov exponent as a function of the transition probabilities, thus extending the results and methods of Y. Peres from a finite to an infinite (but compact) space of symbols. Our approach combines the spectral properties of the associated Markov operator with the theory of holomorphic functions in Banach spaces.

Theorems & Definitions (42)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.2
• Remark 1.5
• Definition 2.1
• Theorem 2.1
• Definition 2.2