Entropy and its variational principle for noncompact metric spaces

In the present paper, we introduce a natural extension of AKM-topological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the classical formula for the entropy of an automorphism of a noncompact Lie group is just an upper bound for its topological entropy.

💡 Research Summary

The paper develops a comprehensive theory of entropy for continuous self‑maps on non‑compact metric spaces, extending the classical Adler‑Konheim‑McAndrew (AKM) topological entropy and establishing a full variational principle that unifies three notions of entropy: the extended topological entropy, the supremum of measure‑theoretic entropies over all invariant probability measures, and the infimum of metric (Bowen‑type) entropies taken over a suitable class of metrics.

The authors begin by redefining topological entropy for a proper map (T:X\to X) (i.e., the pre‑image of any compact set is compact). They introduce the concept of an admissible covering: a finite open cover (\alpha) such that for each element (A\in\alpha) the closure or the complement of (A) is compact. For each (n) they form the refinement (\alpha_n={A_0\cap T^{-1}A_1\cap\cdots\cap T^{-n}A_n: A_i\in\alpha}) and denote by (N(\alpha_n)) the minimal cardinality of a sub‑cover of (\alpha_n). The limit \