标题： 即使模糊的规格性质也意味着遍历测度的稠密性 显示英文标题

标题： Even the vague specification property implies density of ergodic measures

摘要： 我们证明，如果一个拓扑动力系统$(X,T)$是满射的并且具有模糊指定性质，那么其遍历测度在所有不变测度的空间中是稠密的。 模糊指定性质推广了鲍文的经典指定性质，并涵盖了迄今为止引入的指定性质的大多数扩展。 证明首先考虑系统$(X,T)$的自然扩张$X_T$作为空间$X^\mathbb{Z}$上的移位作用的一个子系统，该空间是$X$值的双无限序列的空间。 We then construct a sequence of subsystems of $X^\mathbb{Z}$ that approximate $X_T$ in the Hausdorff metric induced by a metric compatible with the product topology on $X^\mathbb{Z}$. The approximating subsystems consist of $\delta$-chains for $\delta$ decreasing to $0$. We show that chain mixing implies that each approximating system possesses the classical periodic specification property. 此外，我们使用模糊规范证明，我们的近似子系统$X^\mathbb{Z}$在由 Besicovitch 伪度量诱导的 Hausdorff 度量下收敛到$X_T$。 由此可知，这些子系统的$\delta$链的不变测度单纯形收敛到$X_T$的不变测度单纯形，相对于 Ornstein 的$\bar{d}$度量的一个推广版本。 更重要的是，遍历测度的密度在极限中被保留。 证明通过观察到$X_T$和$(X,T)$的不变测度单纯形重合而完成。 本文开发的近似技术似乎具有独立的兴趣。 显示英文摘要

摘要： We prove that if a topological dynamical system $(X,T)$ is surjective and has the vague specification property, then its ergodic measures are dense in the space of all invariant measures. The vague specification property generalises Bowen's classical specification property and encompasses the majority of the extensions of the specification property introduced so far. The proof proceeds by first considering the natural extension $X_T$ of $(X,T)$ as a subsystem of the shift action on the space $X^\mathbb{Z}$ of $X$-valued biinfinite sequences. We then construct a sequence of subsystems of $X^\mathbb{Z}$ that approximate $X_T$ in the Hausdorff metric induced by a metric compatible with the product topology on $X^\mathbb{Z}$. The approximating subsystems consist of $\delta$-chains for $\delta$ decreasing to $0$. We show that chain mixing implies that each approximating system possesses the classical periodic specification property. Furthermore, we use vague specification to prove that our approximating subsystems of $X^\mathbb{Z}$ converge to $X_T$ in the Hausdorff metric induced the Besicovitch pseudometric. It follows that the simplices of invariant measures of these subsystems of $\delta$-chains converge to the simplex of invariant measures of $X_T$ with respect to a generalised version of Ornstein's $\bar{d}$ metric. What is more, the density of ergodic measures is preserved in the limit. The proof concludes by observing that the simplices of invariant measures for $X_T$ and $(X,T)$ coincide. The approximation technique developed in this paper appears to be of independent interest.