The Addition Theorem for the Entropy of Transformations of G-spaces(1)

For a measure-preserving transformation T which is a skew-product of a measure-preserving transformation S and a topological group endomorphism a, it is shown that the entropy h satisfies the following "addition theorem": h(T) = h(S) + h(o). Introduction. In a previous paper [4], conditions were given for a certain type of transformation of a G-space (G being a compact separable group) to have completely positive entropy. It is useful to be able to calculate the actual numerical value of the entropy ; the purpose of the present paper is to extend previously known formulae to cover this type of transformation. As in [4], the notation of Rohlin's survey article [3] is used : the entropy of a measure-preserving transformation / of a Lebesgue space (M, SS, p.) is denoted by h(T) ; //(£) denotes the entropy of the (measurable) partition f of M and H(£/r]) denotes the mean conditional entropy of | with respect to r¡. Throughout this paper, the basic measure space (M, 38, p.) will be a direct product of a Lebesgue space (X, (é>, v) and a compact separable group G with Borel sets and Haar measure m (this also being a Lebesgue space) ; all the measures are normalized, i.e. p.(M) = v(X) = m(G) = l. The measure-preserving transformation Z will act as follows : Z(x, g) = (Sx, cr(g)G is some measurable map; throughout this paper, such a transformation will be described as a skew-product of 5 and a (the map <p not being specified). It will be proved that (1) h(T) = h(S)+h(a). For the case where M is itself a compact separable group, Z is a group endomorphism and G is a Z-invariant (ZG<=G) closed normal subgroup (a=the restriction of Zto G), this result was proved by Juzvinskiï in [2] as an essential step Received by the editors March 18, 1970 and, in revised form, October 27, 1970. AMS 1970 subject classifications. Primary 28A65, 22D40; Secondary 22D05, 22EI5.