Recurrence and Entropy for Discrete-Time Deterministic Dynamical Systems

We investigate discrete-time deterministic systems on finite state spaces equipped with symmetry groups, extending the analysis to actions of arbitrary countable linearly ordered groups. Under the assumption of strong recurrence, characterized by the absence of weakly wandering sets of positive measure, we establish the structural constraints governing dynamical invariants. For systems associated with amenable groups, we employ Følner sequences to rigorously define asymptotic frequencies and demonstrate that maximal Shannon entropy emerges naturally from the system's architecture rather than stochastic assumptions. We show that the interplay of strong recurrence and symmetry enforces specific distribution patterns; while transitive symmetry leads to a uniform stationary distribution and maximal entropy, we provide a generalized formula for non-transitive cases based on orbit decomposition. These results bridge classical recurrence theory and ergodic decomposition with modern measure-theoretic entropy, illustrated through concrete examples for both finite and infinite countable settings.