Beltrami fields exhibit knots and chaos almost surely
Forum of Mathematics, Sigma, Tome 11 (2023)
Voir la notice de l'article provenant de la source Cambridge University Press
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 ${\mathbb {R}}^3$ and of high-frequency Beltrami fields on the 3-torus.
@article{10_1017_fms_2023_52,
author = {Alberto Enciso and Daniel Peralta-Salas and \'Alvaro Romaniega},
title = {Beltrami fields exhibit knots and chaos almost surely},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.52},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.52/}
}
TY - JOUR AU - Alberto Enciso AU - Daniel Peralta-Salas AU - Álvaro Romaniega TI - Beltrami fields exhibit knots and chaos almost surely JO - Forum of Mathematics, Sigma PY - 2023 VL - 11 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.52/ DO - 10.1017/fms.2023.52 LA - en ID - 10_1017_fms_2023_52 ER -
%0 Journal Article %A Alberto Enciso %A Daniel Peralta-Salas %A Álvaro Romaniega %T Beltrami fields exhibit knots and chaos almost surely %J Forum of Mathematics, Sigma %D 2023 %V 11 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.52/ %R 10.1017/fms.2023.52 %G en %F 10_1017_fms_2023_52
Alberto Enciso; Daniel Peralta-Salas; Álvaro Romaniega. Beltrami fields exhibit knots and chaos almost surely. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.52
Cité par Sources :