TOPOLOGICAL PROPERTIES OF COMPLETE AUTOMORPHISMS OF A MEASURE SPACE S. I. Bezuglyi and V. Ya. Golodets GROUPS OF UDC 513.88+519.46 Considerable progress has recently been achieved in the study and classification of dynamical systems with a precision up to a weak equivalence [1-3]. The obtained results have applications to the group represen- tation theory and to the theory of operator algebras, which in turn, stimulate to a considerable extent develop- ments in this branch of ergodie theory. The study of the weak equivalence (or trajectory equivalence) of auto- morphism groups of the Lebesgue spaee naturally leads to the investigation of complete automorphism groups. Such groups were first introdueed by Dye [4], and further studied in [i, 2, 5-9]. In the present paper we study the topologieal properties of complete automorphism groups preserving the quasiinvariance of measure, viz., their connectedness, simple connectedness, and the presence of closed normal divisors. Moreover, it has been shown that any two aperiodic automorphisms of the complete group are isomorphic modulo a small measure. It should be noted that some topologieM properties of automorphism groups had been discussed earlier as well. For instanee, the linear eonneetedness of the group of all automorphisms in a weak topology had been proved [i0]. For eomplete groups preserving the measure of automorphisms, Dye described the c~osed normal divisors [8], andBelinskaya studied the set of aperiodic automorphisms of a complete group generated by one automorphism [7]. We shall discuss an arbitrary complete group leavh~g the measure quasiinvariant. So far, no one has discussed the simple conneetedness of complete groups. The present paper is eomposed of four sections. In the first, we diseuss the group ~ of all the auto-. morphisms of the Lebesgue spaee leaving the measure quasiinvariant, as well as two metrics, d and 6, on the group ~i~ It is well known that in the ease of measure-preserving automorphisms these metrics are equivalent [11]. In the general case it is not so, but nevertheless the topologies induced by these metrics on the group are equivalent. This allows us to prove that ~I and any complete group as well, is a topological group. In the seeond seetion, the linear eonnectech~ess of complete automorphism groups is established, and then their simple eonnectedness is proved. Basically, the proof depends on the theorem about the approximation of an arbitrary autoraorphism by a periodie one, and also on the possibility of approximating any line in a complete group by lines consisting only of periodie automorphisms. In the third section we prove a theorem about the structure of the elosed normal divisors of a complete group. In the last section we introduee a classification of ergodie complete groups of automorphisms, whieh agrees with von Neumann's classification of factors, We bring out the group types: I n (n - I, 2, .... ~), Ill, II~, and III. Taking advantage of this classifieation~ we prove that in a conlplete group the set of automorphisms which are conjugate with a given aperiodic autemor- phism is everywhere dense in the set of all aperiodic automorphisms of this group. Essentially, we use the definitions and concepts introduced in [Ii]. Automorphisms are considered modulo changes on sets of measure 0. The fact that all the constructions, and in particular the definition of a complete group, are correet when we pass to sueh etasses of automorphisms is obvious as a rule, and will not be discussed separately. 1. An automorphism g of the Lebesgme spaee (M, F, t~),(~M = I), [eaves the measure ~ quasiinvariant if the measures # and pg (#g(A) = p(gA),.A E F)are equivalent. The group of aH classes of automorphisms coinciding modulo 0 in the Lebesgue space (M, F, I~) which leave the measure # quasiinvariant will be denoted by g and we shah neglect the differenees between classes and their representatives. For g~I and e > 0 we shall denote by ~(g, s) such a positive number that #A < u(g, e) implies ~(gA) < (A ~ F). Physicoteehnica[ Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Kharkov. TransLated from Sibirskii Matematicheskii Zhurnal, Vol. 21, No. 2, pp. 3-13, March-April, 1980. Origtnal article submitted Mareh I0, 1978. 0037-4466/80/2102-0147507.50 9 1981 Plenum Publishing Corporation 1.47