Topological entropy for Reeb vector fields in dimension three via open book decompositions

Marcelo R.R Alves,Vincent Colin,Ko Honda

doi:10.5802/jep.89

Marcelo R.R Alves, Vincent Colin + Show 1 more

Open Access

https://doi.org/10.5802/jep.89

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Abstract

Given an open book decomposition of a contact three man-ifold (M, ξ) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n , we construct a Legendrian knot Λ close to the stable foliation of a page, together with a small Legendrian pushoff Λ. When k ≥ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of Λ → Λ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for ξ have positive topological entropy.

Highlights

In this paper we combine the techniques of [CH13] and [Alv19] to obtain results on the topological entropy of a large class of contact 3-manifolds.Theorem 1.1. — Let (M, ξ) be a closed cooriented contact 3-manifold which admits a supporting open book decomposition whose binding is connected and whose monodromy is isotopic to a pseudo-Anosov homeomorphism with fractional Dehn twist coefficient k/n
A cooriented contact structure on a 3-manifold M is a plane field ξ given by ξ = ker α for a 1-form α with α ∧ dα > 0. Such an α is called a contact form on (M, ξ), and it determines a Reeb vector field Rα defined by ιRα dα = 0, α(Rα) = 1
The Legendrian contact homology of a pair Λ → Λ. — The homology we present in this subsection is the one described in [BEE12, Rem. 7.4]; a similar construction appears in [Ekh08]

Summary

Introduction

In this paper we combine the techniques of [CH13] and [Alv19] to obtain results on the topological entropy of a large class of contact 3-manifolds. — Let (M, ξ) be a closed cooriented contact 3-manifold which admits a supporting open book decomposition whose binding is connected and whose monodromy is isotopic to a pseudo-Anosov homeomorphism with fractional Dehn twist coefficient k/n. One uses properties of diffeomorphisms isotopic to pseudo-Anosov homeomorphisms: in every Nielsen class of periodic points, the total Lefschetz index is −1 and the number of Nielsen classes grows exponentially with the period This is enough to conclude that the dimension of cylindrical contact homology generated by periodic orbits of action less than T grows. — Every closed contact 3-manifold (M, ξ) admits a (tight) degree five branched cover along a transverse knot on which every Reeb flow has positive topological entropy. Acknowledgements. — Our special thanks to Frédéric Bourgeois for explaining us the construction of the linearized Legendrian contact homology of a directed pair of Legendrian submanifolds, which is part of his joint work with Ekholm and Eliashberg [BEE12]

Preliminaries

Review of contact homology

Cobordisms and almost complex structures

Constructions

Control of once-and twice-punctured holomorphic disks

Growth of the number of Reeb chords and positivity of topological entropy