Pseudorotations of the 2-disc and Reeb flows on the 3-sphere

TL;DR

This paper establishes conditions under which Hamiltonian diffeomorphisms of compact surfaces can be embedded as Poincaré return maps in Reeb flows on 3-manifolds, specifically demonstrating the embedding of certain irrational pseudorotations into the 3-sphere.

Contribution

It introduces a contact cut construction to embed Hamiltonian diffeomorphisms into Reeb flows, including irrational pseudorotations of the 2-disc on the 3-sphere.

Findings

Irrational pseudorotations of the 2-disc embed into Reeb flows on the 3-sphere.

A sufficient condition for Hamiltonian diffeomorphisms to embed as Poincaré return maps.

Construction of a dynamically convex contact form on the 3-sphere for these embeddings.

Abstract

We use Lerman's contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed 3-manifold as Poincar\'e return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the 2-disc constructed by Fayad-Katok embed into the Reeb flow of a dynamically convex contact form on the 3-sphere.