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.