Tail field representations and the zero-two law

The zero-two law was proved for a positive L-1-contraction T by Ornstein an d Sucheston, and gives a condition which implies T-n f - Tn+1 f --> 0 for a ll f. Extensions of this result to the case of a positive L-p-contraction, 1 less than or equal to p < infinity, have been obtained by several authors . In the present paper we prove a theorem which is related to work of Wittm ann. We will say that a positive contraction T contains a circle of length m if there is a nonzero function f such that the iterated values f, Tf,..., Tm-1 f have disjoint support, while T-m f = f. Similarly, a contraction T conta ins a line if for every m there is a nonzero function f (which may depend o n m) such that f, T f,...,Tm-1 f have disjoint support. Approximate forms o f these conditions are defined, which are referred to as asymptotic circles and fines, respectively. We show (Theorem 3) that if the conclusion T-n f - Tn+1 f --> 0 of the zero-two law does not hold for all f in Lpr then eith er T contains an asymptotic circle or T contains an asymptotic line. The po int of this result is that any condition on T which excludes circles and li nes must then imply the conclusion of the zero-two law. Theorem 3 is proved by means of the representation of a positive L-p-contra ction in terms of an L-p-isometry. Asymptotic circles and lines for T corre spond to exact circles and lines for the isometry on tail-measurable functi ons, and exact circles and lines for the isometry are obtained using the Ro hlin tower construction for point transformations.