Reeb orbits trapped by Denjoy minimal sets

TL;DR

This paper demonstrates how any compact invariant set of a flow on a torus, derived from a diffeomorphism, can be realized as an invariant set of a Reeb flow in standard contact space, extending previous constructions.

Contribution

It generalizes the construction of Reeb flows with prescribed invariant sets to higher dimensions and broader classes of flows on tori.

Findings

Realizes invariant sets of flows as Reeb flow invariant sets

Extends previous constructions to higher dimensions

Uses contact forms equal to the standard outside a compact set

Abstract

Let $\varphi$ be any flow on $T^n$ obtained as the suspension of a diffeomorphism of $T^{n-1}$ and let $\mathcal A$ be any compact invariant set of $\varphi$. We realize $(\mathcal A, \varphi|_{\mathcal A})$ up to reparametrization as an invariant set of the Reeb flow of a contact form on $\mathbb R^{2n+1}$ equal to the standard contact form outside a compact set and defining the standard contact structure on all of $\mathbb R^{2n+1}$. This generalizes the construction of Geiges, R\"ottgen and Zehmisch.