标题： 通过闭关系的Mahavier系统规范 显示英文标题

标题： Specification in Mahavier Systems via Closed Relations

摘要： 我们研究了拓扑动力系统两个基本性质，规范性与初始规范性，并探讨它们在更广泛的CR-动力系统中的推广，其中动力由闭关系而非连续函数来支配。 虽然在许多经典系统中这两个性质是等价的，但我们证明它们在CR-动力系统中的推广通常会导致不同的行为。 将它们应用于Mahavier动力系统，我们引入了新的规范类型性质。 这些广义概念扩展了经典理论，并揭示了动力行为中的丰富结构差异。 此外，每个新性质在限制到连续函数时都会退化为标准的规范性。 显示英文摘要

摘要： We study two fundamental properties of topological dynamical systems, the specification property and the initial specification property, and explore their generalizations to the broader setting of CR-dynamical systems, where the dynamics are governed by closed relations rather than continuous functions. While these two properties are equivalent for many classical systems, we demonstrate that their generalizations to CR-dynamical systems often lead to distinct behaviors. Applying them to Mahavier dynamical systems, we introduce new specification-type properties. These generalized notions extend the classical theory and reveal rich structural differences in dynamical behavior. Moreover, each of the new properties reduces to the standard specification property when restricted to continuous functions.