On coincidence of entropies for two classes of dynamical systems

Abstract

Let $ \Gamma $ be a periodic graph with the vertex set ${\Bbb Z}^d $. A subgraph of $ \Gamma $ is called an essential spanning forest if it contains all vertices of $ \Gamma $, has no cycles, and if all its connected components are infinite. The set of all essential spanning forests in $ \Gamma $ is compact in a suitable topology, and ${\Bbb Z}^d $ acts on it by translations. Burton and Pemantle computed the topological entropy of such an action. Their formula turned out to be the same as the formula for the topological entropy of ${\Bbb Z}^d $-actions on certain subgroups of $({\Bbb R}/{\Bbb Z})^{{\Bbb Z}^d}$ obtained previously by Lind, Schmidt and Ward. The question was to explain the coincidence. Here we prove directly that the entropies for two systems must be equal.