Weaker forms of specification for maps on uniform spaces

Abstract

We introduce and study here some weaker forms of specification property for uniformly continuous surjective self-maps on uniform spaces, namely topological quasi-weak specification property, topological semi-weak specification property and topological periodic quasi-weak specification property. We also introduce and study the notion of topological specification point for uniformly continuous surjective self-maps on uniform spaces. It is proved that for uniformly continuous surjective self-maps on uniform spaces the pointwise topological periodic specification property implies Devaney chaos. Moreover, the relation between pointwise notions of mixing, topological shadowing and topological specification is explored. It is shown that for uniformly continuous self-maps on uniform spaces the existence of two distinct topological specification points implies that the map has positive uniform entropy. Also, the limiting behaviour of a topological specification point under orbital convergence of maps is studied.