On finsler entropy of smooth distributions and Stefan-Sussman foliations

TL;DR

This paper introduces a Finsler entropy concept for smooth distributions and Stefan-Sussmann foliations, generalizing classical topological entropy, and shows it vanishes for controllable distributions and singular Riemannian foliations.

Contribution

It defines Finsler entropy for these structures, extending classical entropy notions, and proves its nullity in specific controllable and singular cases.

Findings

Finsler entropy generalizes classical topological entropy.

Finsler entropy is zero for controllable distributions.

Finsler entropy vanishes for singular Riemannian foliations.

Abstract

Using the definition of entropy of a family of increasing distances on a compact metric set given in [10] we introduce a notion of Finsler entropy for smooth distributions and Stefan-Sussmann foliations. This concept generalizes most of classical topological entropy on a compact Riemannian manifold : the entropy of a flow ([9]), of a regular foliation ([11]), of a regular distribution ([5]) and of a geometrical structure ([22]). The essential results of this paper is the nullity of the Finsler entropy for a controllable distribution and for a singular Riemannian foliation.