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

Ai-Hua Fan, Benoît Saussol and Jörg Schmeling

Vol. 214 (2004), No. 1, 31–54

DOI: 10.2140/pjm.2004.214.31

Abstract

Let β > 1 be a real number and M : ℝ → GL(ℂ) be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product

Under some conditions 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 β is a Pisot–Vijayaraghavan number is well studied.

Milestones

Received: 8 October 2002

Revised: 1 August 2003

Published: 1 March 2004