How Do Random Fibonacci Sequences

We study the random Fibonacci sequences defined by F 1 = F 2 = e F 1 = e F 2 = 1 and for n ≥ 1, F n+2 = F n+1 ± Fn (linear case) and e F n+2 = | e F n+1 ± e Fn| (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 e Fn for 1/3 ≤ p ≤ 1 is almost surely given by Z ∞ 0 log x dνα(x), where α is an explicit function of p depending on the case we consider, and να is an explicit probability distribution on Ê + 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.