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

2002/10/22 by Ai-Hua Fan, Fan, Ai-Hua, Benoit Saussol +3
Mathematics · #28A80 #42A38 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CA #math.DS #msc:28A80 #msc:42A38

paper · pdf · doi:10.48550/arxiv.math/0210347

arxiv created 2002/10/22 · arxiv updated 2009/11/30

Abstract

Let β>1 be a real number and M: ℝ→ \rm GL(\CC^d) be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product P_n(x) =M(β^n-1x)... M(βx) 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 β is a Pisot--Vijayaraghavan number is well studied.

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