quantum topological dynamical systems: foundations

This paper introduces a framework for quantum topological dynamical systems (qtds), reformulating classical topological dynamics in a quantum-inspired probabilistic context. the fundamental objects are maps or flows on spaces of probability densities over a manifold, rather than on the manifold itself. we provide a general construction for such dynamics from classical systems and analyze their properties. the central contribution is the introduction and characterization of microscopic fixed and periodic points and invariant sets, with necessary and sufficient conditions linking these probabilistic concepts to classical deterministic counterparts. this work establishes a dictionary for translating classical topological dynamics into the language of evolving probability densities.