ON COINCIDENCE OF ENTROPIES FOR 2 CLASSES OF DYNAMICAL-SYSTEMS

Let Gamma be a periodic graph with the vertex set Z(d). A subgraph of Gamma is called an essential spanning forest if it contains all vertic es of Gamma, has no cycles, and if all its connected components are in finite. The set of all essential spanning forests in Gamma is compact in a suitable topology, and Z(d) acts on it by translations. Burton an d Pemantle computed the topological entropy of such an action. Their f ormula turned out to be the same as the formula for the topological en tropy of Z(d)-actions on certain subgroups of (R/Z)(Zd) obtained previ ously by Lind, Schmidt and Ward. The question was to explain the coinc idence. Here we prove directly that the entropies for two systems must be equal.