Beyond Lyapunov

TL;DR

This paper introduces new ergodic parameters that provide a more comprehensive understanding of dynamical systems, extending classical concepts like Lyapunov exponents and relating them to entropy and self-organization.

Contribution

It proposes a new family of ergodic parameters, generalizes the Pesin formula, and explores their relation to Rényi entropies and measures of self-organization.

Findings

Derived a generalized Pesin formula under weak correlation conditions.

Linked ergodic parameters to Rényi entropies and self-organization measures.

Enhanced the characterization of invariant measures in dynamical systems.

Abstract

Ergodic parameters like the Lyapunov and the conditional exponents are global functions of the invariant measure, but the invariant measure itself contains more information. A more complete characterization of the dynamics by new families of ergodic parameters is discussed, as well as their relation to the dynamical R\'{e}nyi entropies and measures of self-organization. A generalization of the Pesin formula is derived which holds under some weak correlation conditions.