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

Let $\beta >1$ be a real number and $M:\mathbb{R}\to \mathrm{G}\mathrm{L}({\text{CC}}^{\mathrm{d}})$ be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product $$P_n(x)=M({\beta }^{n-1}x)\mathrm{.}\mathrm{.}\mathrm{.}M(\beta x)M(x)\mathrm{.}$$ 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.