Powers of Homeomorphisms with Almost Periodic Properties

Let X be a topological space (an "accessible space," a "1-space," or a " TV-space " in the terminology of Fréchet, Kuratowski, or Alexandroff-Hopf, respectively) and let f(X) = X be a homeomorphism. We use the following terminology, which was suggested by G. A. Hedlund and which is to be carefully distinguished from those terminologies used by Birkhoff, Ayres, Whyburn, and others. A point x of X is said to be recurrent under ƒ provided that to each neighborhood U of x there corresponds a positive integer n such that f(x) £ U. The mapping ƒ is said to be pointwise recurrent provided that each point of X is recurrent under ƒ. A point x of X is said to be almost periodic under ƒ provided that to each neighborhood U of x there corresponds a monotone increasing sequence nif n2, • • • of positive integers with the properties that the numbers ni+\—ni (i = l, 2, • • • ) are uniformly bounded andƒ**'(#)£ U (i = l, 2, • • • ). The mapping/ is said to be pointwise almost periodic provided each point of X is almost periodic under/. Following Birkhoff [l, p. 198], a subset F of X is said to be minimal under ƒ provided that F is nonvacuous, closed and invariant under ƒ, that is, ƒ( F) = F, and furthermore F does not contain a proper subset with these properties. For xÇzX, the set n£±Zf(x) is called the otbit of x under ƒ and the set ]Cn-°o/(#) is called the semi-orbit of x under/. A decomposition of X is a collection of nonvacuous pairwise disjoint closed subsets of X which fill up X.