Finite Rank Z^d Actions and the Loosely Bernoulli Property

. We define finite rank for Z d actions and show that those finite rank actions with a certain tower shape are loosely Bernoulli for d 1. Contents 1. Introduction 125 2. Background 126 3. Finite Rank 127 4. The Matching Lemma 127 5. The Theorem for Finite Rank 131 References 134 1. Introduction In [4], the authors show that rank 1 Z d transformations are loosely Bernoulli, extending the Z result from [7]. D. Ornstein, D. Rudolph, and B. Weiss also show in [7] that all finite rank transformations are loosely Bernoulli. In this paper we give the Z d generalization of this result. Intuitively, a zero entropy loosely Bernoulli (LB) action has one name up to the f metric on processes (cf [6, 2]). The proof in [4] rests on the fact that for a rank one action most large enough names are well covered by towers of various sizes. Given two large names we use this fact to identify towers in one name who have a same size tower close by in the second name. We use these pairs of neighbours..