The Mean and Transitive Points of Homeomorphisms

Abstract

A point p of a compact metric space Q is called a mean point of a homeomorphism T of Q onto itself if the sequence of arithmetic means F.(p) = 1/n J:= f(T'p) converges for every real valued continuous function f(x) on a. This notion of a mean point was introduced by Kryloff and Bougoliouboff who called such a point a quasi-regular point [cf. 5, p. 93].' The mean points derive their importance from the fact that, as was shown by Kryloff and Bougoliouboff, they determine completely the totality of all normalized invariant Borel measures in Q. Indeed, to each mean point there corresponds a unique normalized invariant Borel measure and every normalized invariant Borel measure is a limit of convex linear combinations of measures corresponding to mean points. In fact a more restricted class of points called transitive points is already sufficient to determine all normalized invariant Borel measures in Q. These points are the mean points to which correspond ergodic normalized invariant Borel measures. The question of whether there always exist mean or transitive points in Q2 was answered by Kryloff and Bougoliouboff who showed that the set of transitive (and hence also mean) points of Q is not empty. Moreover they have also shown that this set has maximal invariant measure. I.e., it has measure one for every normalized invariant measure. In this paper it is shown that there are always infinitely many transitive (and hence mean) points in Q provided I has infinitely many points (cf. Theorems 1 and 1' below). Moreover under the added restriction that Q is locally connected it is shown that there are c transitive (and hence mean) points in Q (cf. Theorems 2 and 2'). It is also shown that while the mean points are the general case as far as measure is concerned, they are by no means the general case as far as category is concerned. Indeed, under conditions which occur in many examples of dynamical systems, the set of mean points is a set of first category (cf. Theorem 3). That an added restriction on Q is needed in order to insure the existence of c mean points in Q is shown in the last paragraph by an example of a non locally connected compact metric space Q with a homeomorphism of it onto itself, for which there are only a countable infinity of mean points although the space Q has c points. The proofs of our main theorems are based partly on the theory of invariant measures in dynamical systems and partly on modifications of theorems due