In this paper we will consider
entropy of automorphisms on locally compact abelian groups. Bowen’s definition of
entropy of a uniformly continuous mapping applies in particular to topological
automorphisms of l.c.a. groups. If hB(α,G) denotes the Bowen entropy of
α ∈Aut(G), we investigate the appropriate dual notion h∞(α,Ĝ) of the adjoint
automorphism α on the dual group Ĝ, and show hB(α,G) = h∞(α,Ĝ). We define
the total entropy h(α,G) of α on G to be the sum hB(α,G) + h∞(α,G) and show
that with this definition, h(α,G) coincides with Kolmogorov-Sinai entropy if G is
compact and furthermore the invariance properties present in the compact case are
retained for an arbitrary l.c.a. group G. We also obtain the addition theorem for
entropy and a formula for the entropy on projective limits. In conclusion we mention
some questions which arise.
