Entropy on discrete abelian groups

Abstract

Let G be a discrete group and • an automorphism of G. Then there is an automorphism t% the adjoint of a, on the dual group F defined by (~ar)(x) = z(~-lx), x ~ G, r E / ' , and the mapping cz ~ %~ is an isomorphism of Aut(G) onto Aut(F). (See [2; 26.9].) Denote by h(*a) the Kolmogorov-Sinai entropy of ~o~ on F (with respect to Haar measure on / ' ) , or equivalently, the topological entropy of *oz on F. Since entropy is viewed as a measure of expansiveness, it is plausible that the entropy h(t~) of to should be reflected in some way in the behavior of the adjoint, ~. I f for example we take G = Z • so that ~ ~ T ~, the n-dimensional torus, and a ~ Ant(G), then o~ is identified with an n X n matrix with integer coefficients whose determinant has absolute value one, and tc~ is just the inverse transpose of the matrix of ~. Now by the well-known formula of Kolmogorov.