Beltrami fields exhibit knots and chaos almost surely.

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 speculation 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 Kolmogorov-Arnold-Moser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on R3 and of high-frequency Beltrami fields on the 3-torus. [ABSTRACT FROM AUTHOR]

Copyright of Forum of Mathematics, Sigma is the property of Cambridge University Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)