  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.
