Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

Abstract

In this paper we use broken book decompositions to study Reeb flows on closed 3-manifolds. We show that if the Liouville measure of a nondegenerate contact form can be approximated by periodic orbits, then there is a Birkhoff section for the associated Reeb flow. In view of Irie's equidistribution theorem, this is shown to imply that the set of contact forms whose Reeb flows have a Birkhoff section contains an open and dense set in the C ∞ -topology. We also show that the set of contact forms whose Reeb flows have positive topological entropy is open and dense in the C ∞ -topology.

Periodic orbits and Birkhoff sections of stable Hamiltonian structures

“…The aim of this section is to generalize the main results of [12] and [13,14], from Reeb vector fields of a contact form to Reeb vector fields of a SHS. Broken book decompositions, introduced in [12] (consult Definition 5.7), provide a strong tool for studying the dynamics of a vector field.…”

“…Colin, Dehornoy, Hryniewicz, and the second author in [13] and Contreras and Mazzuchelli in [14], established independently that the set of Reeb vector field of contact structures admitting a Birkhoff section contains an open and dense set in C ∞ -topology. In analogy, our main result is Theorem 1.1.…”

“…If dλ is a constant non-zero multiple of ω, then the SHS provides the same information as λ, which is a contact form. Since we will extend results from [12,13], we are concerned with the case in which dλ is a non-constant multiple of ω (hence the set {dλ = 0} can be a non-empty proper subset of the ambient manifold). In this case, Cieliebak and Volkov's Structure Theorem [10] (Theorem 2.2), establishes a way to cut the manifold along invariant tori.…”

“…If we assume that f is non-constant, in the contact parts of M (confer Section 2.1), the results in [12] and [14,13] imply the existence of a broken book decomposition or a Birkhoff section (both adapted to have boundary) respectively. Theorems 1.1 and 1.2 are proved by pasting either enough leaves of a broken book decomposition or a Birkhoff section of the contact parts of M with the sections that the flow has in the suspension parts.…”

Surfaces of section for geodesic flows of closed surfaces

“…It is not known whether this latter condition holds for a C ∞ generic Riemannian metric on a closed surface; indeed, even the fact that, for such a C ∞ generic Riemannian metric, the lifts of the closed geodesics form a dense subset of the unit tangent bundle is an open problem. Nevertheless, Irie's equidistribution theorem [Iri21] implies that the contact volume form can be approximated by periodic orbits for a C ∞ generic contact form on any closed 3-manifold, and together with [CDHR22] this provides an alternative proof of the C ∞ generic existence of Birkhoff sections for closed contact 3-manifolds.…”