HOW DO RANDOM FIBONACCI SEQUENCES GROW

HOW DO RANDOM FIBONACCI SEQUENCES GROW? ELISE JANVRESSE, BENOIT RITTAUD, THIERRY DE LA RUE Abstract. We study the random Fibonacci sequences defined by F1 = F2 = eF1 = eF2 = 1 and for n ≥ 1, Fn+2 = Fn+1 ± Fn (linear case) and eFn+2 = | eFn+1 ± eFn| (non-linear case), where each ± sign is independent and either + with probability p or ? with probability 1 ? p (0 < p ≤ 1). Our main result is that the exponential growth of Fn for 0 < p ≤ 1, and of eFn for 1/3 ≤ p ≤ 1 is almost surely given by Z ∞ 0 log x d??(x), where ? is an explicit function of p depending on the case we consider, and ?? is an explicit probability distribution on R+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of p, since we prove that it is equal to zero for 0 < p ≤ 1/3. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative. 1. Introduction In this article, we wish to investigate the exponential growth of random Fibonacci sequences (Fn)n≥1 and (F˜n)n≥1, defined inductively by F1 = F2 = F˜1 = F˜2 = 1, and for all n ≥ 1, (1) Fn+2 = Fn+1 ± Fn (linear

• stern-brocot intervals


• successive reduced sequence


• pattern rll


• reduced sequences


• random fibonacci


• lyapunov exponent


• ?1? ?