The dynamics of generic Kuperberg flows

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.

This is the final version of the manuscript. Section 23 has been extended with many more details of the proof that the unique minimal set does not have stable shape, but does satisfy the Mittag-Leffler condition on homology groups