Pointwise partial hyperbolicity in 3-dimensional nilmanifolds

TL;DR

This paper investigates partially hyperbolic systems on 3-dimensional nilmanifolds, establishing their dynamical coherence and classifying these systems up to leaf conjugacy, with implications for entropy and attractors.

Contribution

It proves dynamical coherence for all partially hyperbolic systems on certain nilmanifolds and classifies these systems up to leaf conjugacy, extending understanding of their dynamics.

Findings

All partially hyperbolic systems on these nilmanifolds are dynamically coherent.

Classified partially hyperbolic systems on the 3-torus without certain periodic tori.

Established existence and uniqueness of measures of maximal entropy and quasi-attractors.

Abstract

We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3-manifolds with nilpotent, non-abelian fundamental group. We further classify the partially hyperbolic systems on these manifolds up to leaf conjugacy. We also classify those systems on the 3-torus which do not have an attracting or repelling periodic 2-torus. These classification results allow us to prove some dynamical consequences, including existence and uniqueness results for measures of maximal entropy and quasi-attractors.