Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy hKS and the Lyapunov exponent λ. It is well known that many systems exhibit subexponential separation of nearby trajectories and then λ=0. In many cases such systems are nonergodic and do not obey usual statistical mechanics. Here we investigate the nonergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows δxₜ=δx₀e^λ_αt^α with 0<α<1. The limit distribution of λ_α is the inverse L\'evy function. The average ⟨λ_α⟩ is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.