Lyapunov exponents for products of matrices

Abstract

Let $\mathbf{M}=(M_1,\dots ,M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether $\mathbf{M}$ possesses the following property: there exist two constants $\lambda \in \mathbb{R}$ and $C>0$ such that for any $n\in \mathbb{N}$ and any $i_1,\dots ,i_n\in \{1,\dots ,k\}$, either $M_{i_1}\cdots M_{i_n}=0$ or $C^{-1}{e}^{\lambda n}\le \Vert M_{i_1}\cdots M_{i_n}\Vert \le C{e}^{\lambda n}$, where $\Vert \cdot \Vert$ 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 $\mathbb{R}$, the absolute continuity of certain self-affine measures in ${\mathbb{R}}^d$ and the dimensional regularity of a class of sofic affine-invariant sets in the plane.

References in corpus (1)

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