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

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 $\ge$ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of $Λ$ $\rightarrow$ $Λ$ 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.