Hyperbolic foliated entropy of suspensions

TL;DR

This paper investigates the hyperbolic entropy of foliations created by suspensions, linking it to existing entropy concepts, and provides new results including exact estimates and variational principles for specific cases.

Contribution

It establishes a connection between hyperbolic foliation entropy and pseudo-group entropy, showing non-invariance under diffeomorphisms and deriving new estimates and principles for certain representations.

Findings

Hyperbolic entropy is not invariant under diffeomorphisms.

Minimal entropy suspensions admit an invariant measure.

Exact estimates and variational principles are obtained for representations isomorphic to Z.

Abstract

We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguy\^en and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted version of an entropy defined by Ghys, Langevin and Walczak for pseudo-groups of homeomorphisms. Such a link has various consequences. Among them, it implies that the hyperbolic entropy of foliations is not invariant by diffeomorphisms, and that a minimal entropy suspension admits an invariant measure. Finally, this allows us to study thoroughly the simple case in which the image of the representation is isomorphic to~$\mathbb{Z}$. In that case, we give the first exact estimate of the hyperbolic entropy, and prove a Brin--Katok type theorem and a variational principle, relying strongly on the standard ones for the entropy of maps.