Fully generic sequences and a multiple-term return-times theorem

Abstract

The return-times theorem of Bourgain, Furstenberg, Katznelson, and Ornstein states: Theorem [BFKO]. Given any measure-preserving transformation of a probability space and , there is a subset of full measure so that for any and second dynamical system and , there is a subset of full measure so that for any , We will prove here a multiple term version of this result first proposed by I. Assani. This will be accomplished by recasting the result completely as a fact about infinite names in where is a compact metrizable space. We define what it means for such an infinite name to be fully generic for some shift invariant measure. The core result here is Theorem 1 which verifies that if is fully generic in some then the collection of names for which the pair is fully generic in will have measure one with respect to any shift-invariant measure. The multiple term return-times theorem is an obvious induction on this statement. By replacing the role played by -orthogonality in the proofs in [BFKO] and [R2] by a disjointness result on joinings the argument avoids the need to identify distinguished factor algebras for the higher term averages.