On the Iteration of Transformations in Noncompact Minimal Dynamical Systems

Let A be a Hausdorff space, jfc-o Tkg\\ is uniformly bounded for all positive /. In a reflexive Banach space, this characterization of the range of il—T) is valid for any linear transformation T for which ||rn|| is bounded for all n. A sufficient condition in a general Banach space would seem to require an assumption that the elements { X)i=o Tki} lie f°r aH j m a fixed weakly compact subset K of X. It would be interesting to obtain a proof of Theorem 1 along these lines. We shall content ourselves with showing by these methods that if m is a totally-finite measure on a c-algebra on A, Lmim) the space of ?ra-essentially bounded measurable functions, a measure preserving mapping of A into A, then in order that for an element g in L°°im), there should exist an/ in L°°im) such that fika) | should be uniformly bounded for all positive/2 Such a result raises another sort of question. For a topological space A if g is continuous and / is a solution of the equation /(a)