Computation of Lyapunov exponents of matrix products

Abstract

For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(\omega_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(\omega):=\lim_{n\to \infty} \frac 1n \log \|A_{\omega_0} A_{\omega_2}\cdots A_{\omega_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{\omega_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(\omega)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $\omega$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $\omega$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.