Existence of periodic orbits for geodesible vector fields on closed 3-manifolds

In this paper we deal with the existence of periodic orbits of geodesible vector fields on closed 3-manifolds. A vector field is geodesible if there exists a Riemannian metric on the ambient manifold making its orbits geodesics. In particular, Reeb vector fields and vector fields that admit a global section are geodesible. We will classify the closed 3-manifolds that admit aperiodic volume preserving real analytic geodesible vector fields, and prove the existence of periodic orbits for real analytic geodesible vector fields (not volume preserving), when the 3-manifold is not a torus bundle over the circle. We will also prove the existence of periodic orbits of C2 geodesible vector fields in some closed 3-manifolds

The dynamics of generic Kuperberg flows

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.Comment: 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 group

Aperiodicity at the boundary of chaos

We consider the dynamical properties of $C^{\infty}$-variations of the flow on an aperiodic Kuperberg plug ${\mathbb K}$. Our main result is that there exists a smooth 1-parameter family of plugs ${\mathbb K}_{\epsilon}$ for $\epsilon \in (-a,a)$ and $a<1$, such that: (1) The plug ${\mathbb K}_0 = {\mathbb K}$ is a generic Kuperberg plug; (2) For $\epsilon <0$, the flow in the plug ${\mathbb K}_{\epsilon}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) For $\epsilon > 0$, the flow in the plug ${\mathbb K}_{\epsilon}$ has positive topological entropy, and an abundance of periodic orbits.Comment: Minor edits and text revisions from version 1. arXiv admin note: text overlap with arXiv:1306.502