标题： 抽象动作空间及其拓扑和动态性质 显示英文标题

标题： Abstract Action Spaces and their topological and dynamic properties

摘要： 我们引入了动作空间的概念，即一个集合$\boldsymbol{X}$，其上具有满足适当公理的动作成本$\mathsf{a}:(0,+\infty)\times \boldsymbol{X}\times \boldsymbol{X}\to [0,+\infty)$，这些公理实际上提供了经典度量空间概念的“动态”推广。 动作成本自然地作为最小化运动方案中出现的耗散项出现，这可以在一般动作空间中得到解决。 如同度量空间的情况一样，我们将证明动作成本在$\boldsymbol{X}$上诱导了一个内在的拓扑和度量结构。 此外，我们在$\boldsymbol{X}$中的路径上引入了相关的动作泛函，研究了有限动作曲线的性质，并讨论了它们的绝对连续性。 最后，在类似于度量空间的近似中点性质的条件下，我们提供了动作成本的动态解释。 显示英文摘要

摘要： We introduce the concept of action space, a set $\boldsymbol{X}$ endowed with an action cost $\mathsf{a}:(0,+\infty)\times \boldsymbol{X}\times \boldsymbol{X}\to [0,+\infty)$ satisfying suitable axioms, which turn out to provide a `dynamic' generalization of the classical notion of metric space. Action costs naturally arise as dissipation terms featuring in the Minimizing Movement scheme for gradient flows, which can then be settled in general action spaces. As in the case of metric spaces, we will show that action costs induce an intrinsic topological and metric structure on $\boldsymbol{X}$. Moreover, we introduce the related action functional on paths in $\boldsymbol{X}$, investigate the properties of curves of finite action, and discuss their absolute continuity. Finally, under a condition akin to the approximate mid-point property for metric spaces, we provide a dynamic interpretation of action costs.