A Map with Topological Minimal Self-joinings in the Sense of Del Junco

Andr es del Junco has proposed a deenition of topological minimal self-joinings intended to parallel Dan Rudolph's measure-theoretic concept. By means of a rank-two \cutting and stacking", this article constructs the rst example of a system (a subshift) satisfying his proposed deenition of 2-fold topological minimal self-joinings. The second part of the article shows that 2-fold topological minimal self-joinings does not imply 3-fold and that no map has 4-fold topological minimal self-joinings. This latter result follows from a generalization of a theorem of Schwartzman. x0 Introduction In 1979 Dan Rudolph showed that Don Ornstein's rank-1 mixing map, T, could be built to possess a measure-theoretic property which has come to be called minimal self-joinings of all orders. For order 2 (our tacit assumption, if no adjective is present) this says that whenever two copies of T sit simultaneously as factors of a third, ergodic, transformation then the two copies are either independent or are identiied by some power of T. Equivalently, letting (X;) denote the space on which T acts, any T T-invariant ergodic measure on X X projecting to on each coordinate is either product measure or else is supported on the graph of some T n. One can similarly deene K-fold minimal self-joinings by considering the possible ergodic measures living on X K , the cartesian K-th power of X. If T has minimal self-joinings of all orders then any automorphism of T N is simply a cartesian product of powers of T composed with a permutation of the coordinates. Rudolph constructed such a map in R] and used the automorphism property to fabricate a menagerie of counterexample transformations. Later investigations showed, K] and K,T], that the maps with minimal self-joinings play the role of elementary building blocks for a class of maps containing the nite-rank mixing maps. What analogue does this notion have in the topological category of a homeomorphism T of a compact metric space X? Using minimal sets as the analogue of ergodic measures, Nelson Markley proposed in M] a deenition paralleling Rudolph's measure-theoretic deenition. He and Joe Auslander proved in A,M] that this property for order 2 implies the property for all orders and powers. They then went on to establish a structure theorem roughly analogous to the measure-theoretic one of K]. However, an exact analogue of Rudolph's theory is not obtained because, for such transformations, the automorphism group of T N seems diicult …