  We introduce a notion of entropy for automorphisms of discrete groups which
admit amenable actions on a compact space. This entropy is dual to classical
topological entropy in the sense that if G is discrete and abelian then our
notion of entropy agrees with the topological entropy of the induced
automorphism on the (compact) dual group of G.
  We prove a number of basic properties of this dual entropy and give a few
calculations. In particular, we are able to give precise calculations for
arbitrary automorphisms of crystallographic groups.
