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تسجيل مستخدم جديدHow do random Fibonacci sequences grow?
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تأليف
Elise Janvresse
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We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $nge 1$, $F_{n+2} = F_{n+1} pm F_{n}$ (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 $1-p$ ($0<ple 1$). Our main result is that the exponential growth of $F_n$ for $0<ple 1$ (linear case) or for $1/3le ple 1$ (non-linear case) is almost surely given by $$int_0^infty log x d u_alpha (x), $$ where $alpha$ is an explicit function of $p$ depending on the case we consider, and $ u_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 $0<ple1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.
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We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $nge 1$, $F_{n+2} = lambda F_{n+1} pm F_{n}$ (linear case) and $widetilde F_{n+2} = |lambda widetilde F_{n+1} pm widetilde F_{n}|$ (non-linear case),
where each $pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<ple 1$). Our main result is that, when $lambda$ is of the form $lambda_k = 2cos (pi/k)$ for some integer $kge 3$, the exponential growth of $F_n$ for $0<ple 1$, and of $widetilde F_{n}$ for $1/k < ple 1$, is almost surely positive and given by $$ int_0^infty log x d u_{k, rho} (x), $$ where $rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ u_{k, rho}$ is an explicit probability distribution on $RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<ple 1$ in the easier case $lambdage 2$. Finally, we study the variations of the exponent as a function of $p$.
An infinite sequence of real random variables $(xi_1, xi_2, dots)$ is said to be rotatable if every finite subsequence $(xi_1, dots, xi_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(xi_1, xi_2, dots)$ is
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Recent experiments have focused attention on the properties of chains of atoms in which the atoms are either in their ground states or in highly excited Rydberg states which block similar excitations in their immediate neighbors. As the low energy Hi
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