The dynamics of generic Kuperberg flows

TL;DR

This paper investigates the complex dynamical behavior of Kuperberg flows on 3-manifolds, introducing the concept of zippered laminations and analyzing the properties of their minimal sets, including chaos and topological features.

Contribution

It introduces the notion of zippered lamination and characterizes the minimal sets of Kuperberg flows under generic conditions, revealing their topology and chaotic dynamics.

Findings

Minimal sets are invariant zippered laminations.

Presence of non-zero entropy-type invariants indicating chaos.

Minimal sets lack stable shape but satisfy Mittag-Leffler condition.

Abstract

In this work, we study the dynamical properties of Krystyna Kuperberg's aperiodic flows on $3$-manifolds. We introduce the notion of a ``zippered lamination'', and with suitable generic hypotheses, show that the unique minimal set for such a flow is an invariant zippered lamination. We obtain a precise description of the topology and dynamical properties of the minimal set, including the presence of non-zero entropy-type invariants and chaotic behavior. Moreover, we show that the minimal set does not have stable shape, yet satisfies the Mittag-Leffler condition for homology groups.