Tsallis entropy composition and the Heisenberg group

TL;DR

This paper embeds Tsallis entropy into the Heisenberg group to explore its geometric and fractal properties, revealing its implications for the structure of phase spaces with polynomial growth.

Contribution

It introduces a geometric framework linking Tsallis entropy to the Heisenberg group and sub-Riemannian geometry, providing new insights into its composition and underlying space properties.

Findings

Tsallis entropy induces fractal properties on Euclidean space.

Configuration space of Tsallis systems exhibits polynomial growth.

A geometric interpretation of Abe's formula via Pansu derivative.

Abstract

We present an embedding of the Tsallis entropy into the 3-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.