Products of non-stationary random matrices and Multiperiodic equations of several scaling factors

Abstract

Let $\beta >1$ be a real number and $M: \mathbb{R}\to {\rm GL(\CC^d)}$ be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product $$P_n(x)=M({\beta }^{n-1}x)...M(\beta x)M(x).$$ Under some condition we prove a theorem of Furstenberg-Kesten type for such products of non-stationary random matrices. Theorems of Kingman and Oseledec type are also proved. The obtained results are applied to multiplicative functions defined by commensurable scaling factors. We get a positive answer to a Strichartz conjecture on the asymptotic behavior of such multiperiodic functions. The case where $\beta$ is a Pisot--Vijayaraghavan number is well studied.