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Random matrix products and applications to cellular automata

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Abstract

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X i) 0/∞i over ℂ. Assume that theX i's are chosen from a finite set {D 0,D 1...,D t-1(ℂ), withP(X i=Dj)>0, and that the monoid generated byD 0, D1,…, Dq−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22].

Our results on λ enable us to provide an approximation for the numberN ≠0(F(x)n,r) of nonzero coefficients inF(x) n.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN ≠0(F(x)n,r) ≈n α for “almost” everyn.

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  1. Hte'ena 39, P.O.BOX 5549, 81541, Yavne, Israel

    Yossi Moshe

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  1. Yossi Moshe
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Correspondence to Yossi Moshe.

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Supported in part by FWF Project P16004-N05

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Moshe, Y. Random matrix products and applications to cellular automata. J. Anal. Math. 99, 267–294 (2006). https://doi.org/10.1007/BF02789448

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  • DOI: https://doi.org/10.1007/BF02789448

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