贝尔特拉米场几乎肯定会呈现出纽结和混沌
Beltrami fields exhibit knots and chaos almost surely
摘要：
在本文中，我们表明，概率为1，一个随机贝尔特拉米场表现出混乱的地区共存的复杂拓扑结构不变的环面。考虑这个问题的动机是在三维定常欧拉流的研究中产生的。Arnold在1965年猜想，一个典型的Beltrami场表现出与一般的具有两个自由度的哈密顿系统的能量超曲面的限制相同的复杂性。证明铰链上获得的马蹄铁，零，打结不变环面和周期轨迹的高斯随机Beltrami场的展览，我们通过一个非平凡的扩展Nazarov-Sodin理论高斯随机单色波和应用不同的工具，从动力系统的理论，包括KAM理论，Melnikov分析和双曲的数量的渐近界。我们的结果在$\mathbf{R}^3$上的Beltrami场和3-环面上的高频Beltrami场的情况下都成立。
英文摘要：
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.
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