Infinite ergodic theory and Non-extensive entropies

TL;DR

The paper investigates how infinite ergodic theory informs non-extensive entropies by focusing on systems with infinite invariant measure and zero Lyapunov exponent. It leverages the dual operator and Darling-Kac theorems to show that time averages are intrinsically random and that sub-exponential instability lacks a single universal descriptor; the return sequence $a_n$ is regularly varying and governs the limits of ergodic sums. For classical models such as the $Pomeau{-}Manneville$ maps, Boole maps, parabolic Julia maps, and Horocycle flows, the theory yields explicit $a_n$ and Mittag-Leffler limit laws for normalized sums, highlighting rich asymptotics beyond polynomial growth. These findings bridge non-extensive entropy formalisms with infinite-measure dynamics, informing modeling of weakly chaotic systems and clarifying the scope of Pesin-like relations in infinite-measure contexts.

Abstract

We bring into account a series of result in the infinite ergodic theory that we believe that they are relevant to the theory of non-extensive entropies

Theorems & Definitions (6)

• Theorem 1
• Remark 1
• Theorem 2
• Definition 1
• Theorem 3
• Theorem 4