标题： 湍流闭合关系 显示英文标题

标题： Turbulent Closed Relations

摘要： 本文将经典的动力系统中湍流的概念从由连续函数生成的动力系统推广到紧致度量空间上的闭关系所定义的动力系统。 利用Mahavier乘积和相关的移位映射，我们引入并探讨了CR-湍流和反向CR-湍流，分析它们与拓扑熵之间的关系。 关键在于理解湍流何时蕴含熵以及反之亦然，结果表明对于有限的闭关系，这些性质是等价的。 然而，提供了例子来证明对于更一般的的关系，这种等价性可能失效。 我们还构造了一类在单位区间上具有动态丰富但结构简单的显式湍流闭关系。 此外，由于同胚不能容纳湍流，我们研究了一个弱化的湍流概念：同胚的分离的、连通集意义上的半湍流，然后证明光滑扇形甚至不能支持这种较弱形式的湍流动力学。 本文包含新的例子、反例和开放问题，以加深对非经典设置下湍流的理解。 显示英文摘要

摘要： This paper generalizes the classical notion of turbulence from dynamical systems generated by continuous functions to those defined by closed relations on compact metric spaces. Using the Mahavier product and the associated shift map, we introduce and explore CR-turbulence and reverse CR-turbulence, analyzing their relationship to topological entropy. A key focus is understanding when turbulence implies entropy and vice versa, with results showing that for finite closed relations, these properties are equivalent. However, examples are provided to demonstrate that this equivalence can fail for more general relations. We also construct a large class of explicit turbulent closed relations on the unit interval that are dynamically rich yet structurally simple. Additionally, since homeomorphisms cannot admit turbulence, we investigate a weakened notion of turbulence: separated, continuum-wise semi-turbulence for homeomorphisms, and then prove that smooth fans cannot support even this weaker form of turbulent dynamics. The paper includes new examples, counterexamples, and open questions that deepen the understanding of turbulence in non-classical settings.