Ergodicity and conservativity of products of infinite transformations and their inverses

Abstract

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T of the transformation with itself is ergodic, but the product T ×T −1 of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

Infinite symmetric ergodic index and related examples in infinite measure

“…For every odd n, choose R n and H n = R n ⊔ S n such that x, y, z, z ′ ∈ H n with x ∈ R n , (x, y) = (z, z ′ ) =⇒ |x − z − y + z ′ | ≫ 2h n ≥ 2 max D(I 0 , n). This is a stronger version of condition (2-1) in [7], and similar to the restriction discussed in Remark 1, [4]. Add max{H n } + h n spacers on the rightmost subcolumn for every n, and choose δ n such that n odd δ k n → ∞ but n odd δ k+1 n < ∞.…”

“…holds, and it is impossible to have a ℓ,q − d ℓ,q = a ℓ ′ ,q − d ℓ ′ ,q , as was the case of Lemma 3.2 in [4]. Since ε is a fixed constant, Proposition 2.4 implies that T (p+1) is not conservative.…”

“…Soon after [2], Bergelson asked if there existed a transformation T that has infinite ergodic index but such that T × T −1 is not ergodic. In [4], Clancy et al construct rank-one transformations T such that T × T is ergodic but T × T −1 is not ergodic; it is also shown in [4] that for a rank-one T , the product T × T −1 must always be conservative. The example with T × T −1 not ergodic was later generalized to more general group actions in Danilenko [7], but the full Bergelson question remains open.…”

“…In Section 5 we show that power weak mixing holds in a class of transformations containing the primary example of [8] (and also the examples in [13]), and the bounded-recurrence example of [9]. For terms not defined here the reader may refer for [14] or [4].…”

Strict doubly ergodic infinite transformations

“…Such z and j do not satisfy (11), so k + D ip+z (B, n + 1) has empty intersection with all D(B, n)-copies D j+z (B, n + 1) with j ≤ j p . In addition, when z ≥ − 9rn jp−ip and j ≥ j p + 1, then we should have…”