2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension |
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Authors: | Thierry de la Rue |
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Institution: | 1. Laboratoire de Mathématiques Rapha?l Salem, UMR 6085 CNRS – Université de Rouen, Avenue de l'Université, 12, F76801, Saint-étienne-du-Rouvray Cedex, France
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Abstract: | V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process (ξi )i2ℤ, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold
mixing problem, the socalled 3-dot system, but in the context of stationary random fields indexed by ℤ2.
In this work, we first present an attempt to adapt Ledrappier's construction to the onedimensional case, which finally leads
to a stationary process which is 2-fold but not 3-fold mixing conditionally to the σ-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample
can not be fully adapted to one-dimensional stationary processes. |
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Keywords: | multifold mixing self-joinings 3-dot system |
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