Weakly mixing transformations which are not strongly mixing

Abstract

2. Preliminaries. The examples will be of an invertible measure preserving transformation of the unit interval (equipped with the usual Lebesgue sets and Lebesgue measure). The transformation will be constructed in a sequence of steps, each step enlarging the domain of definition of the transformation. The construction utilizes a geometric approach described in [1] which consists of mapping subintervals of the same length linearly onto each other, and of representing this geometrically by a figure in which each interval is placed below its image. Thus, if II, . . . , In are pairwise disjoint subintervals of the unit interval, if they have the same length, and if we define r on Un, Ik by mapping Ik linearly onto Ik+', k =1, * *, n -1 (and leave r undefined elsewhere), the geometric figure that corresponds to this map consists of the intervals I', . . . ,In arranged in a stack with I' on the bottom and In on the top, and with the remaining intervals arranged in order between them, so that each point is located below its image. The action of the transformation can thus be regarded as an upward flow through the stack which ends when the top layer is reached, since the transformation is undefined there. Note that we have not assumed that the union of I', * . *, In is the unit interval. The main advantage of the geometric approach is that many properties of transformations become much clearer when viewed in this way. Of course, the geometric figure associated with each stage of the definition is not necessary for the definition of the transformation nor for the proof of its properties. It serves simply as an aid in understanding the construction.