Lyapunov exponents for products of matrices

TL;DR

This paper provides criteria to determine exponential growth rates of matrix products under certain conditions, with applications to measure regularity and fractal dimensions.

Contribution

It introduces checkable criteria for Lyapunov exponents of matrix products using symbolic dynamics and thermodynamic formalism, extending analysis to self-similar and self-affine measures.

Findings

Criteria for exponential growth rate of matrix products established

Applications to absolute continuity of self-similar measures

Applications to dimensional regularity of affine-invariant sets

Abstract

Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants $\lambda\in {\Bbb R}$ and $C>0$ such that for any $n\in {\Bbb N}$ and any $i_1, \ldots, i_n \in \{1,\ldots, k\}$, either $M_{i_1} \cdots M_{i_n}={\bf 0}$ or $C^{-1} e^{\lambda n} \leq \| M_{i_1} \cdots M_{i_n} \| \leq C e^{\lambda n}$, where $\|\cdot\|$ is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on ${\Bbb R}$, the absolute continuity of certain self-affine measures in ${\Bbb R}^d$ and the dimensional regularity of a class of sofic affine-invariant sets in the plane.