Random Fibonacci sequences

Solutions to the random Fibonacci recurrence x(n+1) = x(n) +/- betax(n-1) d ecrease (increase) exponentially, x(n) similar to exp(lambdan), for suffici ently small (large) beta. In the limits beta --> 0 and beta --> infinity, w e 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-pert urbative results. In particular, an invariant measure associated with Ricat ti variable r(n) = x(n+1)/x(n) is shown to exhibit plateaux around all rati onal r.