论文标题
Beltrami田地几乎肯定会表现出结和混乱
Beltrami fields exhibit knots and chaos almost surely
论文作者
论文摘要
在本文中，我们表明，有了概率1，一个随机的Beltrami场表现出与复杂拓扑结构不变的托里（Tori）并存的混乱区域。考虑到固定欧拉在维度3中的研究中引起的这个问题的动机是V.I.阿诺德（Arnold）1965年的猜想是，典型的贝尔特拉米（Beltrami）田地表现出与限制具有两个自由度的通用哈密顿系统的限制相同的复杂性。 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双曲线。我们的结果在$ \ mathbf {r}^3 $上的Beltrami字段中都持有我们的结果，以及3个腹部上的高频Beltrami字段。
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.
