随机斐波那契序列如何增长？
How do random Fibonacci sequences grow?
摘要：
本文研究了由$F_1=F_2=1$定义的两类随机Fibonacci序列，对于$n\ge 1$，分别为$F_{n+2} = F_{n+1} \pm F_{n}$（线性情况）和$F_{n+2} = |F_{n+1} \pm F_{n}|$（非线性情况），其中每个符号独立，且概率为+$p$或概率为-$1-p$（$0<p\le 1$）.我们的主要结果是，$F_n$的$0<p\le 1$（线性情况）或$1/3\le p\le 1$（非线性情况）的指数增长几乎肯定是由$$\int_0^\infty \log x dν_α(x), $$，其中$α$是$p$的显式函数取决于我们考虑的情况下，和$ν_α$是一个显式的概率分布上的$\RR_+$归纳定义的Stern-Brocot间隔。在非线性的情况下，最大的李雅普诺夫指数不是$p$的解析函数，因为我们证明了它等于零的$0<p\le1/3$。给出了最大李雅普诺夫指数变化的一些结果，并给出了最大李雅普诺夫指数的导数公式。
英文摘要：
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 $$\int_0^\infty \log x dν_α(x), $$ where $α$ is an explicit function of $p$ depending on the case we consider, and $ν_α$ 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.
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