Statistics of finite-time Lyapunov exponents in the Ulam map.

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length n. The variance anomalously decays as 1/n(2). The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting 2(n-1) spikes that narrow and accumulate close to the mean value with increasing n. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small n, where the finiteness of bin size trims the sharp peaks.