On invariant distributionally scrambled sets in non-compact systems

Abstract

We study here the invariance of topological distributionally scrambled sets for maps on uniform spaces (not necessarily compact or metrizable). For a uniformly continuous surjective self-map defined on a uniformly locally compact Hausdorff uniform space having topological weak specification, a fixed point, and countably many periodic points with distinct periods, we prove that the map admits an invariant topological distributionally scrambled set of type 1. Further, if the uniform space is second countable, the map admits a dense Mycielski invariant topological distributionally scrambled set of type 1.