Pesin-Type Identity for Weak Chaos

TL;DR

This paper extends Pesin's identity to non-ergodic systems with infinite invariant densities, specifically analyzing the Pomeau-Manneville map where trajectory separation follows a sub-exponential pattern.

Contribution

It introduces a generalized Pesin's identity applicable to systems exhibiting weak chaos and infinite invariant densities, expanding the theoretical framework beyond ergodic systems.

Findings

Derived the inverse Lévy distribution as the limit for Lyapunov exponents.

Linked the average Lyapunov exponent to the infinite invariant density and entropy.

Validated the generalized Pesin's identity for the Pomeau-Manneville map.

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{\'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.