Dynamical Properties of Proximal Group Spaces

In this paper, we introduce a new class of dynamical systems, Proximal Dispersive Group Spaces (PDG - spaces), to study the structure of topological group actions. In these spaces, we observe a mixture of two different behaviors: points come close to each other (proximal convergence) under some group elements, while remaining separated (dispersive separation) under others. This framework generalizes the classical theory to allow for selective convergence without global asymptotic collapse. We give a formal definition of PDG - spaces and study some of their basic properties, like invariance under group actions and incompatibility with equicontinuous systems. Our results show the existence of non-trivial minimal subsets that are both proximal and dispersive. In this paper, we prove that PDG - spaces constitute a special intermediate regime between purely proximal and distal systems.