A canonical structure theorem for finite joining-rank maps

A numerical isomorphism invariant,joining-rank, was introduced in [1] as a quantitative generalization of Rudolph’s property of minimal selfjoinings. Therein, a structure theory was developed for those transformationsT whose joining-rank, jr (T), is finite. Here, we sharpen the theorem and show it to be canonical: If jr (T)<∞ then there is a unique triple 〈e, p, S〉 wheree andp are natural numbers andS is a map with minimal self-joinings, such thatT is ane-point extension ofS ^P. Furthermore, the producte·p equals the joining-rank ofT. This theorem applies to any finite-rank mixing map, since for such maps the rank dominates the joining-rank. Another corollary is that any rank-1 transformation which is partial-mixing has minimal self-joinings. This partially answers a question of [3]. Partially supported by a National Science Foundation Postdoctoral Research Fellowship.