Even the vague specification property implies density of ergodic measures

Abstract

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.