Dual entropy in discrete groups with amenable actions

Let G be a discrete group which admits an amenable action on a compact space and γ ∈ Aut(G) be an automorphism. We define a notion of entropy for y and denote the invariant by ha(γ). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(γ) = h_Top(γ̂) where h_Top(γ̂) denotes the topological entropy of the induced automorphism γ̂ of the (compact, abelian) dual group Ĝ. ha(·) enjoys a number of basic properties which are useful for calculations. For example, it decreases in invariant subgroups and certain quotients. These basic properties are used to compute the dual entropy of an arbitrary automorphism of a crystallographic group.