Tail field representations and the zero-two law

The zero-two law was proved for a positiveL ₁-contractionT by Ornstein and Sucheston, and gives a condition which impliesT ⁿ f−T ⁿ⁺¹ f → 0 for allf. Extensions of this result to the case of a positiveL _p -contraction, 1≤p<∞, have been obtained by several authors. In the present paper we prove a theorem which is related to work of Wittmann. We will say that a positive contractionT contains a circle of lengthm if there is a nonzero functionf such that the iterated valuesf, T f,…,T ^m-1 f have disjoint support, whileT ^m f=f. Similarly, a contractionT contains a line if for everym there is a nonzero functionf (which may depend onm) such thatf,…,T ^m-1 f have disjoint support. Approximate forms of these conditions are defined, which are referred to as asymptotic circles and lines, respectively. We show (Theorem 3) that if the conclusionT ⁿ f−T ⁿ⁺¹ f→0 of the zero-two law does not hold for allf inL _p , then eitherT contains an asymptotic circle orT contains an asymptotic line. The point of this result is that any condition onT which excludes circles and lines must then imply the conclusion of the zero-two law. Theorem 3 is proved by means of the representation of a positiveL _p -contraction in terms of anL _p -isometry. Asymptotic circles and lines forT correspond to exact circles and lines for the isometry on tail-measurable functions, and exact circles and lines for the isometry are obtained using the Rohlin tower construction for point transformations. Research supported in part by NSERC.