How do random Fibonacci sequences grow?

Abstract

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by $$\int_0^\infty \log x d\nu_\alpha (x), $$ where $\alpha$ is an explicit function of $p$ depending on the case we consider, and $\nu_\alpha$ is an explicit probability distribution on $\RR_+$ 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 $0<p\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative

Dynamics of $\lambda$-continued fractions and $\beta$-shifts

“…Moreover, φ λ is increasing, and since it is onto, φ λ is continuous. We observed in [5] some connection, when λ = 1, with Minkowski's question mark function. Translated in the context of the present paper, this connection can be written as follows:…”

“…The motivations for the present article stem from several works by the same authors [5,6], where the exponential growth of random Fibonacci sequences with parameter λ is studied. In [6], the case λ = λ k for some integer k ≥ 3 is solved and involves a probability distribution on R + invariant under some dynamics.…”

Random fibonacci sequences and capacity/power scaling in cooperative multihop networks

“…β > 0, and studied further in [14], where it was shown that there was a critical threshold β such that β > β would cause exponential growth in the sequence, while β < β would cause exponential decay. Further work was performed in [15], where the exponential growth rate of…”

Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

“…The method involves using Stern-Brocot sequences, Furstenberg's Theorem (see Theorem 2.3) and the invariant measure to compute λ. We also point to the work of [14,13,12] where the authors generalized the results of Viswanath by letting 0 < p ≤ 1 and treating λ as a function of p which bears some similarity to the model we study in Section 3. They also considered the non-linear case.…”