Rescaling of Markov shifts

Given a Zd topological Markov shift S and a d×d integer matrix M with det(M) ¹ 0, we introduce the M-rescaling of S, denoted S(M). We show that some (internal) power of the Zd-action on S(M) is isomorphic to some (Cartesian, or external) power of S, and deduce that the two Markov shifts have the same topological entropy. Several examples from the theory of group automorphisms are discussed. Full shifts in any dimension are shown to be invariant under rescaling, and the problem of whether the reverse is true is interpreted as a higher-dimensional analogue of William's problem.