Universality of G-subshifts with specification

Abstract

Let $G$ be an infinite countable amenable group and let $(X,G)$ be a $G$-subshift with specification, containing a free element. We prove that $(X,G)$ is universal, i.e., has positive topological entropy and for any free ergodic $G$-action on a standard probability space, $(Y,ν,G)$, with $h(ν)<h_{top}(X)$, there exists a shift-invariant measure $μ$ on $X$ such that the systems $(Y,ν,G)$ and $(X,μ,G)$ are isomorphic. In particular, any $K$-shift (consisting of the indicator functions of all maximal $K$-separated sets) containing a free element is universal.