Orbit-closure Decompositions and Almost Periodic Properties

Let X be a metric space with metric p, let f(X)QX be a continuous mapping, and let h(X) ^X be a homeomorphism. For x&X, the s e t 23^-°o/(^) i called the semi-orbit of # under ƒ and the set n£XZh{x) is called the orbit of # under h. For # £ X , the closure of the semi-orbit of x under ƒ is called the semi-orbit-closure of x under ƒ and the closure of the orbit of x under h is called the orbit-closure of x under h. A nonvacuous subset Y of X is said to be semi-minimal (minimal) under f(h) provided that the semi-orbit-closure (orbit-closure) of each point of Y is F. Clearly, any two semi-minimal (minimal) sets are either coincident or disjoint. It is easily proved that a subset Y of X is semi-minimal (minimal) under ƒ (h) if and only if Fis nonvacuous, closed, ƒ( F) C Y(h(Y) = F), and furthermore F contains no proper subset with these properties. We follow Birkhoff [2, p. 198] in the terminology of "minimal set." A decomposition of X is defined to be a collection of nonvacuous closed pairwise disjoint subsets of X which fill up X. We say that the mapping ƒ gives a semi-orbit-closure (a semi-minimal set) decomposition provided that the collection of semi-orbit-closures (semi-minimal sets) is a decomposition of X. Also, it is said that the homeomorphism h gives an orbit-closure (a minimal-set) decomposition provided that the collection of orbit-closures (minimal sets) is a decomposition of X. A point x of X is said to be almost periodic under ƒ provided that to each € > 0 there corresponds a positive integer N with the property that in every set of N consecutive positive integers appears an integer n such that p(x,f(x)) < e. The mapping ƒ is said to be pointwise almost periodic provided that each point of X is almost periodic under ƒ. It is to be noted that various writers use the above terms in different senses and employ other terminologies for these notions.