Beltrami fields with hyperbolic periodic orbits enclosed by knotted invariant tori

TL;DR

This paper constructs Beltrami fields in Euclidean space and on the 3-torus with complex invariant structures, including knotted tori and hyperbolic periodic orbits, demonstrating rich dynamical behavior.

Contribution

It establishes the existence of Beltrami fields with prescribed knotted invariant tori and hyperbolic orbits, extending understanding of their topological and dynamical complexity.

Findings

Existence of Beltrami fields with knotted invariant tori and hyperbolic orbits.

Construction of Beltrami fields with prescribed invariant structures.

High-frequency Beltrami fields on the 3-torus exhibit similar properties.

Abstract

We prove that there exist Beltrami fields in Euclidean space, with sharp decay at infinity, which have a prescribed set of invariant tori (possibly knotted or linked) that enclose an arbitrarily large number of hyperbolic periodic orbits. These hyperbolic orbits are cablings over the core curve of each torus. Moreover, the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori. We show that an analogous result holds for high-frequency Beltrami fields on the flat 3-torus.