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

One-dimensional intermittent maps with stretched exponential δ xt ∼δ x0 e λα tα separation of nearby trajectories are considered. When t→∞the standard Lyapunov exponent λ= Σ i=0 t-1 ln ΙM^′ (xi) Ι/t is zero (M^′ is a Jacobian of the map). We investigate the distribution of λα = Σ i=0 t-1 ln ΙM^′ (xi) Ι/ tα, where α is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of λα 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 α λα is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that λα and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.