TY - JOUR

T1 - Tail field representations and the zero-two law

AU - Akcoglu, Mustafa

AU - Baxter, John

PY - 2001/1/1

Y1 - 2001/1/1

N2 - 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 Lp-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,..., Tm-1 f have disjoint support, while Tm 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,...,Tm-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 Lp, 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.

AB - 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 Lp-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,..., Tm-1 f have disjoint support, while Tm 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,...,Tm-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 Lp, 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.

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U2 - 10.1007/BF02784130

DO - 10.1007/BF02784130

M3 - Article

AN - SCOPUS:0039436497

VL - 123

SP - 253

EP - 272

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -