标题： 关于流的扩张测度的不变性 显示英文标题

标题： On the Invariance of Expansive Measures for Flows

摘要： 我们研究在紧致度量空间上无不动点的连续流的扩张测度，如[6]中所引入的。 我们通过动力球对扩张测度提供了新的刻画，与[6]中考虑的动力球不同，这些动力球实际上是博雷尔集。 这使得该理论更易于进行测度论分析。 然后，我们利用这些广义的动力球建立了流的布林-卡托局部熵公式的一个版本。 作为应用，我们证明了每个正熵的遍历不变测度都是正向扩张的，从而将[1]的结果扩展到常规流的设置中。 这意味着具有正拓扑熵的流拥有扩张的不变测度。 此外，我们证明了这些测度的稳定类具有零测度。 最后，我们证明了扩张测度的集合在弱*-拓扑下形成一个$G_{\delta\sigma}$子集，并且每个扩张测度（无论是否为不变测度）都可以由支撑在不变集上的扩张测度逼近。 显示英文摘要

摘要： We study expansive measures for continuous flows without fixed points on compact metric spaces, as introduced in [6]. We provide a new characterization of expansive measures through dynamical balls that, in contrast to the dynamical balls considered in [6], are actually Borel sets. This makes the theory more amenable to measure-theoretic analysis. We then establish a version of the Brin-Katok local entropy formula for flows using these generalized dynamical balls. As an application, we prove that every ergodic invariant measure with positive entropy is positively expansive, thus extending the results of [1] to the setting of regular flows. This implies that flows with positive topological entropy admit expansive invariant measures. Furthermore, we show that the stable classes of such measures have zero measure. Lastly, we prove that the set of expansive measures forms a $G_{\delta\sigma}$ subset in the weak*-topology and that every expansive measure (invariant or not) can be approximated by expansive measures supported on invariant sets.