ORBIT EQUIVALENCE FOR CANTOR MINIMAL z2-SYSTEMS

摘要

In this paper, we consider dynamical systems on the Cantor set. By a Cantor set,X, we mean a metrizable topological space which is compact, totally disconnected (the closed and open sets form a base for the topology) and has no isolated points.Any two such spaces are homeomorphic. Moreover, dynamical systems on such spaces, which include many symbolic systems, have a fundamental role in the theory. Our results will be concerned with actions of the group Z2 on a Cantor set X.For the moment, it will be convenient to consider the case that G is a countable abelian group with an action, on X by homeomorphisms. For each g in G,we have a homeomorphism 9 : X→ X such that cpf(oxr al)l x =in X axnd,P~ h = p~F(0=gg.urtheхr~m orxe, hwe) w,il asume the action is fre, meaning for some x in X, only if g = 0, and minimal, meaning that, for every x in X, the set 9 (x) g E G} is dense in X, or equivalently, there are no non-trivial closed -invariant subsets of X.