Random Fibonacci sequences

Solutions to the random Fibonacci recurrence x(n+1) = x(n) +/- betax(n-1) decrease (increase) exponentially, x(n) similar to exp(lambdan), for sufficiently small (large) beta. In the limits beta --> 0 and beta --> infinity, we expand the Lyapunov exponent lambda(beta) in powers of beta and beta (-1), respectively. For the classical case of beta = 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 r.