GROUP AUTOMORPHISMS WITH FINITE ENTROPY

A compact metrie abelian group X with the normalized Haar measure is a Lebesgue probability space. A group automorphism σ of X is an invertible measure preserving transformation of the probability space. This paper is to show that if the entropy of σ is finite, then there exist totally disconnected subgroups H and N, a finite-dimensional subgroup S and a subgroup T satisfying the conditions: (i)H, N, S and T are strictly σ-invariant, (ii)N=H∩S∩T, (iii)h(σ|N)=0, (iv) if S/N is non-trivial then it is a finite-dimensional solenoidal group with condition (**) (see the definition in §1), (v) if T/N is non-trivial then it is connected and locally connected, such that X/N splits into a direct sum X/N=H/N⊕S/N⊕T/N. This result characterizes the structure of finite entropy automorphisms. © 1979 Springer-Verlag.