Institute for Mathematical Physics Entropy of Automorphisms of Ii 1 {factors Arising from the Dynamical Systems Theory Entropy of Automorphisms of Ii 1 -factors Arising from the Dynamical Systems Theory

Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X;), preserving the measure. If T 2 Aut (X;) is an automorphism of the equivalence relation deened by G then T can be extended to an automorphism T of the II 1-factor M = L 1 (X;)oG. We prove that if T commutes with the action of G then H(T) = h(T), where H(T) is the Connes-Sttrmer entropy of T , and h(T) is the Kolmogorov{Sinai en-tropy of T. We prove also that for given s and t, 0 s t 1, there exists a T such that h(T) = s and H(T) = t.