Tail field representations and the zero-two law

The zero-two law was proved for a positive L₁-contraction T by Ornstein and Sucheston, and gives a condition which implies Tⁿ f - Tⁿ⁺¹ f → 0 for all f. Extensions of this result to the case of a positive L_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 contraction T contains a circle of length m if there is a nonzero function f such that the iterated values f,T f,..., T^m-1 f have disjoint support, while T^m f = f. Similarly, a contraction T contains a line if for every m there is a nonzero function f (which may depend on m) such that f, T f,...,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 conclusion Tⁿ f - Tⁿ⁺¹ f → 0 of the zero-two law does not hold for all f in L_p, then either T contains an asymptotic circle or T contains an asymptotic line. The point of this result is that any condition on T which excludes circles and lines must then imply the conclusion of the zero-two law.