Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

One-dimensional intermittent maps with stretched exponential delta x(t) similar to delta x(0)e(lambda alpha t alpha) separation of nearby trajectories are considered. When t --> infinity the standard Lyapunov exponent lambda = Sigma(t-1)(i=0)ln|M'(x(i))|/t is zero (M' is a Jacobian of the map). We investigate the distribution of lambda(alpha) = Sigma(t-1)(i=0)ln|M'(x(i))|/t(alpha), where alpha is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of lambda(alpha) is determined by the infinite invariant density. Using semianalytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it we obtain excellent agreement between numerical simulation and theory. We show that alpha <lambda(alpha)> is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that <lambda(alpha)> and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.