q-deformed statistical-mechanical structure in the dynamics of the Feigenbaum attractor

TL;DR

This paper reveals a q-deformed statistical-mechanical framework underlying the dynamics of the Feigenbaum attractor, linking universal constants to a novel entropy measure that tracks ensemble trajectories over time.

Contribution

It introduces a new q-deformed statistical-mechanical structure for Feigenbaum attractor dynamics, connecting universal constants with a time-dependent entropy.

Findings

Partition function sums distances between neighboring attractor points.

Derived a q-entropy measuring trajectories away from the attractor.

Identified q-indexes linked to universal constants.

Abstract

We show that the two complementary parts of the dynamics associated to the Feigenbaum attractor, inside and towards the attractor, form together a q -deformed statistical-mechanical structure. A time-dependent partition function produced by summing distances between neighboring positions of the attractor leads to a q-entropy that measures the fraction of ensemble trajectories still away at a given time from the attractor (and the repellor). The values of the q-indexes are given by the attractor's universal constants, while the thermodynamic framework is closely related to that first developed for multifractals.