Some Transformations Having a Unique Measure with Maximal Entropy

1. Introduction Let X be a compact metric space and T: X-> X a homeomorphism of X onto X. Let M(T) denote the collection of all T-invariant Borel probability measures on X. By Krylov and Bogolioubov's work we know M(T) is non-empty (see [10]). M{T) is a convex set and closed in the weak topology. For /x e M(T), h(T, p) will denote the measure-theoretic entropy of T with respect to p. Ifh top (T) denotes the topological entropy of T ([1]) then h t0V (T) = sup h(T,p) ([5]). Gurevic has given