Minimal Sets in Almost Equicontinuous Systems

Supplying necessary and sufficient conditions such that a transitive system (as a subsystem of the Bebutov system) is uniformly rigid and using the fact that each transitive uniformly rigid system has an almost equicontinuous extension, we construct almost equicontinuous systems containing $n$ ($n\in\mathbb N$), countably many, and uncountably many minimal sets, which serve as new examples of almost equicontinuous systems. Our method is quite general as each transitive uniformly rigid system has a factor that is a subsystem of the Bebutov system. Moreover, we explore how the number of connected components in a transitive pointwise recurrent system is related to the connectedness of the minimal sets contained in the system.