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

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

2002-10-22

arxiv.org/abs/math/0210347v1

Mathematics

Dynamical Systems

Scientific paper

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.

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