标题： 螺旋度到拓扑流的普遍扩展 显示英文标题

标题： A universal extension of helicity to topological flows

摘要： 螺旋度是在由向量场支配的物理系统中的一种基本守恒量，其演化由三维流形上的体积保持变换描述。 显著的例子包括无粘性、不可压缩流体流动，由三维欧拉方程建模，以及导电等离子体，由磁流体动力学（MHD）方程描述。 螺旋度的一个关键性质是它在体积保持微分同胚下的不变性。 在1973年一篇有影响力的论文中，阿诺德将螺旋度解释为“渐近霍普不变量”，并提出了这个问题：这种不变性是否在体积保持同胚下仍然成立。 更一般地，他问螺旋度是否可以扩展到拓扑体积保持流。 我们对没有静止点的流给出了这两个问题的肯定回答。 我们的方法在我们称之为$C^0$哈密顿结构的框架中重新表述了阿诺德的问题。 这一观点使我们能够利用$C^0$辛几何中的最新进展，特别是关于面积保持同胚群的代数结构的结果。 显示英文摘要

摘要： Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows, modeled by the three-dimensional Euler equations, and conducting plasmas, described by the magnetohydrodynamics (MHD) equations. A key property of helicity is its invariance under volume-preserving diffeomorphisms. In an influential article from 1973, Arnold, having provided an ergodic interpretation of helicity as the "asymptotic Hopf invariant", posed the question of whether this invariance persists under volume-preserving homeomorphisms. More generally, he asked whether helicity can be extended to topological volume-preserving flows. We answer both questions affirmatively for flows without rest points. Our approach reformulates Arnold's question in the framework of what we call $C^0$ Hamiltonian structures. This perspective enables us to leverage recent developments in $C^0$ symplectic geometry, particularly results concerning the algebraic structure of the group of area-preserving homeomorphisms.