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A simple map with no prime factors

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Abstract

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμ S , i.e.,μ S (A×B)=μ(A∩S ;−n;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕ ∞ i=1 ℤ2, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada

    Andrés del Junco

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  1. Andrés del Junco
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Correspondence to Andrés del Junco.

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del Junco, A. A simple map with no prime factors. Isr. J. Math. 104, 301–320 (1998). https://doi.org/10.1007/BF02897068

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  • DOI: https://doi.org/10.1007/BF02897068

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