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Random Fibonacci Sequences

Clément Sire·Paul L. Krapivsky
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Abstract

Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent lambda(B) in powers of B and B^{-1}, respectively. For the classical case of $β=1$ we obtain exact non-perturbative results. In particular, an invariant measure associated with Ricatti variable r_n=x_{n+1}/x_{n} is shown to exhibit plateaux around all rational.

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BibTeXRIS

Clément Sire, Paul L. Krapivsky. 2001-06-22. Random Fibonacci Sequences. https://doi.org/10.1088/0305-4470%2F34%2F42%2F322

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