  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.
