$C^0$-Stability of Topological Entropy for Contactomorphisms

Positive topological entropy of positive contactomorphisms

A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $Λ$, and that $(M,Λ)$ is fillable by a Liouville domain $(W,ω)$ with exact Lagrangian $L$ such that $ω|_{π_2(W,L)}=0$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive cont…

Lower complexity bounds for positive contactomorphisms

Let $S^*Q$ be the spherization of a closed connected manifold of dimension at least two. Consider a contactomorphism $φ$ that can be reached by a contact isotopy that is everywhere positively transverse to the contact structure. In other words, $φ$ is the time-1-map of a time-dependent Reeb flow. We show that the volume growth of $φ$ is bounded from below by the topological complexity of the loop space of $Q$. Denote by $ΩQ_0(q)$ the component of the based loop space that contains the constant loop. We show that if the fundamental group or the homology of $ΩQ_0(q)$ grows exponentially, then th…

Chern-Weil theory and the group of strict contactomorphisms

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group U(n+1) is homotopically essential in the group of contactomorphisms of the standard contact sphere S^(2n+1) and prove that in the case of the 3-sphere the contactomorphism group is in fact homotopy equivalent to U(2). In the second part of the paper we focus on the group of strict contactomorphisms -- using the framework of Chern-Weil theory we introduce an…

Reeb orbits that force topological entropy

We develop a forcing theory of topological entropy for Reeb flows in dimension $3$. A transverse link $L$ in a closed contact $3$-manifold $(Y,ξ)$ is said to force topological entropy if $(Y,ξ)$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,ξ)$ realizing $L$ as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link $L$ which imply that $L$ forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on …

哈密顿微分同胚的拓扑熵：持久同调与Floer理论视角

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

本文从持久同调与Floer理论视角引入条码熵，证明其受拓扑熵上下界约束，并得出曲面上哈密顿微分同胚的条码熵等于拓扑熵。

Cylindrical contact homology and topological entropy

We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold $(M,ξ)$ admits a hypertight contact form $λ_0$ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on $(M,ξ)$ has positive topological entropy. Using this result, we provide numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy.