Robinson construct a class of Z actions and study the joinings within this class. These actions are the natural 2-dimensional analogs to the Chacon transformation and are called Chacon Z 2 Actions. They are produced by a 2-dimensional rank 1 cutting and stacking construction which is reviewed in Subsection 2.1. Some of the w x results in PR follow from the general theory of joinings but others depend on the specific algebraic and geometric properties of the group Z 2 . In particular, the group Z 2 has Z for a nontrivial subgroup and the purpose of this paper is to study this particular subgroup action. Let the 2 Γ i j Ε½ .
2 4 Chacon Z action be denoted by T S : i, j g Z . Then T and S each generate a Z-subaction. Ergodicity of the Chacon Z 2 action is clear from the construction but it is the ergodicity of the Z action that will be needed for this paper. This result is a consequence of the careful choice of pattern 2 w x used in the Chacon Z actions and is proven in PR . In this paper we will write statements in terms of T and leave the analogous statements and proofs for S to the reader. We conjecture that in fact they can be extended i o j o Ε½ . to T S for any fixed i , j . We divide this paper into two parts, o o *The second author would like to acknowledge grant KOSEF-6ARC and KOSEF 95-0701-02-01-3.