A simple map with no prime factors |
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Authors: | Andrés del Junco |
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Institution: | (1) Department of Mathematics, University of Toronto, M5S 1A1 Toronto, Ontario, Canada |
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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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Keywords: | |
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