# On infinite transformations with maximal control of ergodic two-fold product powers

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## Abstract

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure-preserving transformations. A class of transformations is constructed such that for any subset *R* ⊂ ℚ ∩ (0, 1) there exists *T* in this class such that *T* ^{ p } × *T* ^{ q } is ergodic if and only if \(\frac{p}{q}\) ∈ *R*. This contrasts with the finite measure-preserving case where *T* ^{ p } × *T* ^{ q } is ergodic for all nonzero *p* and *q* if and only if *T* × *T* is ergodic. We also show that our class is rich in the behavior of conservative products.

For each positive integer *k*, a family of rank-one infinite measure-preserving transformations is constructed which have ergodic index *k*, but infinite conservative index.

## Keywords

Positive Integer Positive Measure Markov Shift Measure Preserve Transformation Conservative Index## Preview

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