Entropy of geometric structures

TL;DR

This paper introduces a generalized entropy concept for geometric structures, extending existing notions to include singular objects like foliations and Poisson structures, and explores its fundamental properties and invariants.

Contribution

It defines a new entropy for geometric structures that encompasses singular cases and establishes its basic properties, including additivity and invariance for Poisson structures.

Findings

Entropy generalizes topological and geometric entropy concepts.

Additivity property analogous to thermodynamic entropy.

Entropy serves as a new invariant for Poisson structures.

Abstract

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic properties for this entropy, including the \emph{additivity property}, analogous to the additivity of Clausius--Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the characteristic foliation by symplectic leaves.