  Pesin's identity provides a profound connection between entropy $h_{KS}$
(statistical mechanics) and the Lyapunov exponent $\lambda$ (chaos theory). It
is well known that many systems exhibit sub-exponential separation of nearby
trajectories and then $\lambda=0$. In many cases such systems are non-ergodic
and do not obey usual statistical mechanics. Here we investigate the
non-ergodic phase of the Pomeau-Manneville map where separation of nearby
trajectories follows $\delta x_t= \delta x_0 e^{\lambda_{\alpha} t^{\alpha}}$
with $0<\alpha<1$. The limit distribution of $\lambda_{\alpha}$ is the inverse
L{\'e}vy function. The average $< \lambda_{\alpha} >$ 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.
