Recurrence and Entropy for Discrete-Time Deterministic Dynamical Systems

We investigate discrete-time deterministic systems whose trajectories are indexed by the positive cone of a countable linearly ordered group G and evolve on a σ-finite measure space (Ω, B, m). The paper operates at two levels. At the measure-theoretic level, the ambient space (Ω, m) may be non-atomic and infinite, where strong recurrence, characterized by the absence of weakly wandering sets of positive measure, governs structural properties such as syndeticity of return-time sets, invariance under measure equivalence, and inheritance to subgroups. At the combinatorial level, the dynamics is compressed onto the finite effective state space Ωeff, the set of states actually visited by a trajectory, on which a canonical atomic probability measure μ is constructed via a deliberate support-switch from m. At this level, the entropy results additionally require G to be amenable, where Følner sequences are employed to show that μ is the limiting empirical distribution along almost every trajectory. When G acts transitively on Ωeff, the measure μ is necessarily uniform and the stationary Shannon entropy H (X) =−∑xμ (x) logμ (x) achieves its maximum log|Ωeff| from structural constraints alone. A generalized orbit-decomposition formula covers the non-transitive case. Results at both levels are illustrated through cyclic shifts, rational rotations, Sturmian shifts, and Z2-actions.