ILLINOIS JOURNAL OF MATHEMATICS Volume 37, Number 4, Winter 1993 UNIFORMLY SWEEPING OUT DOES NOT IMPLY MIXING TERRENCE ADAMS 1. Introduction Let T be an invertible measure preserving transformation on a measure space that is isomorphic to the unit interval with Lebesgue measure. It was shown in [F1] that if T is mixing, then T is uniformly sweeping out (see 2 for definitions). A sequential counterexample to the converse was given in [F2] where a transformation was constructed that is not mixing on a sequence but is uniformly sweeping out on the sequence. In [C], Chacon constructed another example of a rank one transformation that is weakly mixing but not mixing that is different from Chacon’s transfor- mation [F3, 86-89]. In [FK] the example in [C] was shown to be lightly mixing, not partially mixing, and not lightly 2-mixing which implies not sweeping out of order 2. Our purpose is to show the transformation T in [C] is uniformly sweeping out. Thus T is rank one, not partially mixing, uniformly sweeping out, but not sweeping out of order 2. This is in contrast to Kalikow’s theorem which states that rank one mixing implies 2-mixing [KA]. We also note that it is not difficult to construct a partially mixing transfor- mation that is not uniformly sweeping out. It was shown in [FT] that (2k 1)-mixing implies uniformly sweeping out of order k, k > 1. Thus mixing of all orders implies uniformly sweeping out of all orders. Concerning the converse, we do not know if uniform sweeping out of all orders implies mixing. Acknowledgement. The present work is part of my doctoral dissertation at the State University of New York at Albany written under the guidance of Nathaniel A. Friedman. I wish to thank him for his support and many helpful discussions. 2. Preliminaries Let (X, ,/x) be a measure space isomorphic to the unit interval with Lebesgue measure and let T be an invertible measure preserving map of X Received June 19, 1990. 1991 Mathematics Subject Classification. Primary 28D05. (C) 1993 by the Board of Trustees of the University of Illinois Manufactured in the United States of America 525