Strict Ergodicity and Transformation of the Torus

Abstract

Introduction. If T is a measure preserving transformation ofl a probability space Q with measure Iu, the ergodic theorem assures the existence N-1 almost everywhere with respect to /i of the average limN-'Ef(T i ), where Noo n=O f is an integrable function. Can this statement be improved in case Q is a compact topological space, T a suitable homeomorphism of ?2 with itself, and f a continuous function on Q? In particular, can convergence almost everywhere be replaced by convergence everywhere? In what follows we shall examine this question for the case that Q is an r-dimensional torus. When r = 1, i. e. Q is a circle, the answer is in the affirmative and the averages in question always exist (a fact implicit in the results of Denjoy [1] and van Kampen [5]). For r > 1, however, further restrictions must be imposed on the transformation T and part of our objective will be to exhibit a class of T for which this sharpened form of the ergodic theorem holds. The question we are considering is closely tied up with that of the strict ergodicity of a transformation. A transformation T of a compact Hausdorff space Q is strictly ergodic if it leaves invariant a unique probability measure on the borel field of ?. This notion was first introduced in conlnection with the theory of dynamical systems by Kryloff and Bogoliuboff ([6]; cf. also [8], [9]). When T is a strictly ergodic transformation, then (Theorem N-1 1. 1) the limits of N-1 E f(TAw) necessarily exist for f continuous and all n=O X C EQ, and moreover, this limit is independent of w. In the case of a strictly ergodic transformation, these conclusions are in fact a good deal more elementary than the usual ergodic theorem. Thus it is quite natural to inquire when a transformation of a given space will be strictly ergodic. As we will see, an important condition for the validity of some of our conclusions is that the transformation T not be homotopic to the identity transformation. This implies that the transformation T cannot be embedded in a continuous transformation group T(t) and so, in particular, could :not arise from the consideration of dynamical systems on the torus. The homotopy