Modeling ergodic, measure preserving actions on \(\mathbb{Z}^d\) shifts of finite type (Q5945904)

A topological \(\mathbb{Z}^d\)-action \((Y,S)\) is a universal model if for any aperiodic, ergodic, measure-preserving \(\mathbb{Z}^d\)-action \((X,\mu,T)\) with \(h(X,\mu,T) \leq h_{\text{top}}(Y,S)\), there is an \(S\)-invariant Borel probability measure \(\nu\) on \(Y\) such that \((X,\mu,T)\) and \((Y,\nu,S)\) are measurably isomorphic. For \(d=1\), \textit{W. Krieger} [Proc. int. Congr. Math., Vancouver 1974, Vol. 2, 303-308 (1975; Zbl 0334.28019)] showed that a topologically mixing shift of finite type is universal. For \(d>1\) topological mixing does not even guarantee positive topological entropy, so it is not clear what the analogue should be. Here the authors consider \(\mathbb{Z}^d\) shifts of finite type with the uniform filling property (a strong kind of topological mixing), and under the additional assumption of dense periodic points, they show that such a \(\mathbb{Z}^d\) shift of finite type is a universal model.