Pesin-Type Identity for Weak Chaos

Abstract

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{é}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.

5 pages, 3 figures

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