论文标题
关于度量熵和相关数量的通用行为，在波兰度量空间上均匀连续地图
On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces
论文作者
论文摘要
在这项工作中，我们表明，如果$ f $是在波兰度量空间上定义的均匀连续地图，那么零公制熵的$ f $ invariant措施是$g_Δ$集合（在弱拓扑中）。特别是，如果在$ f $ invariant的措施中，$ f $ periodic措施的集合密集，则该集合是通用的。这解决了Sigmund（Sigmund，K。）对具有规范属性的动力系统提出的猜想。Trans。Amer。Math。Soc。190（1974），285-299），该系统指出，满足周期性规格属性的拓扑动态系统的不变度度量的度量熵是典型的。 我们还表明，如果$ x $是紧凑的，并且如果$ f $是具有密集的定期措施的扩张或Lipschitz地图，则通常是$ q \ in（0,1）$的较低相关熵等于零。此外，我们表明，如果$ x $是一个紧凑的公制空间，并且如果$ f $是具有密集的定期措施的扩展地图，则具有包装尺寸的一组不变措施，复发的上限和相当于零等于零的上限等待时间指示器是残留的。 最后，我们提供了一个替代证明，证明了一组扩展措施是一组$ g_ {δσ} $在一组概率度量中设置的$ \ m（x）$，如果$ x $是波兰度量空间，并且如果$ f $均匀连续；该结果最初是由Lee，Morales and Shin（Lee，K.，Morales，C。A.和Shin，B。的B。
In this work, we show that if $f$ is a uniformly continuous map defined over a Polish metric space, then the set of $f$-invariant measures with zero metric entropy is a $G_δ$ set (in the weak topology). In particular, this set is generic if the set of $f$-periodic measures is dense in the set of $f$-invariant measures. This settles a conjecture posed by Sigmund (Sigmund, K. On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285-299) which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if $X$ is compact and if $f$ is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for $q\in(0,1)$ is equal to zero. Moreover, we show that if $X$ is a compact metric space and if $f$ is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equal to zero is residual. Finally, we present an alternative proof of the fact that the set of expansive measures is a $G_{δσ}$ set in the set of probability measures $\M(X)$, if $X$ is a Polish metric space and if $f$ is uniformly continuous; this result was originally proved by Lee, Morales and Shin (Lee, K., Morales, C. A., and Shin, B. On the set of expansive measures. Commun. Contemp. Math. 20, 7 (2018), 1750086, 10) for compact metric spaces.
