Robust dynamics, invariant structures and topological classification

TL;DR

This paper discusses the classification of partially hyperbolic dynamical systems in three dimensions, focusing on geometric structures, topological invariants, recent challenges, and future research directions.

Contribution

It advances the understanding of 3D partially hyperbolic systems through foliation theory and topological methods, highlighting recent examples and proposing future steps.

Findings

Classification of 3D partially hyperbolic systems using foliation theory

Identification of recent examples challenging existing classification

Suggestions for future research after classification is achieved

Abstract

This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classification program and we propose some steps to continue this classification. Finally, we give some suggestions on what to do after classification is achieved.