Some topological properties of weakly almost periodic mappings

Abstract

A map φ: X→ X induces a linear operator T: C( X)→ C( X) by composition: Tf( x)= f° φ( x). T and φ are termed weakly almost periodic if the sequence { T n } is precompact in the weak operator topology. Using general structure theorems for weakly almost periodic operators, the properties of these point maps are studied from the viewpoint of dynamical systems. The structure of individual minimal sets and of the union M of all minimal sets of φ are investigated. One key result is that, if X is compact, then φ is a strongly almost periodic (i.e., has uniformly equicontinuous iterates) homeomorphisms of M and M is a retract of X. These and other general results are applied to the case where X is a manifold. Several results in which weak implies strong almost periodicity are obtained.