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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.

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Authors and Affiliations

  1. Department of Defense, 9161 Sterling Dr., Laurel, MD, 20723, USA

    Terrence M. Adams

  2. Department of Mathematics and Statistics, Williams College, Williamstown, MA, 01267, USA

    Cesar E. Silva

Authors
  1. Terrence M. Adams
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  2. Cesar E. Silva
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Correspondence to Terrence M. Adams.

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Adams, T.M., Silva, C.E. On infinite transformations with maximal control of ergodic two-fold product powers. Isr. J. Math. 209, 929–948 (2015). https://doi.org/10.1007/s11856-015-1241-1

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  • DOI: https://doi.org/10.1007/s11856-015-1241-1

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