Beltrami fields exhibit knots and chaos almost surely

TL;DR

This paper proves that a typical random Beltrami field in three dimensions almost surely contains chaotic regions, invariant tori with complex topologies, and knotted trajectories, confirming a long-standing conjecture about their complexity.

Contribution

It establishes the almost sure presence of chaos and complex topologies in random Beltrami fields, extending Nazarov--Sodin theory and applying advanced dynamical systems tools.

Findings

Chaotic regions coexist with invariant tori in Beltrami fields.

Asymptotic bounds for horseshoes, zeros, and knotted structures are derived.

Results apply to both $ extbf{R}^3$ and high-frequency Beltrami fields on the 3-torus.

Abstract

In this paper we show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold's 1965 conjecture that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros, and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov--Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including KAM theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on $\mathbf{R}^3$ and of high-frequency Beltrami fields on the 3-torus.