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Weak limits of powers of Chacon’s automorphism

Published online by Cambridge University Press:  27 September 2013

É. JANVRESSE ,
A. A. PRIKHOD’KO ,
T. DE LA RUE  and
V. V. RYZHIKOV
É. JANVRESSE
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Normandie Université, Université de Rouen, CNRS, Avenue de l’Université, 76801 Saint Étienne du Rouvray, France email elise.janvresse@univ-rouen.frThierry.de-la-Rue@univ-rouen.fr
A. A. PRIKHOD’KO
Affiliation:
Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, 119991 Russia email sasha.prihodko@gmail.comvryzh@mail.ru
T. DE LA RUE
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Normandie Université, Université de Rouen, CNRS, Avenue de l’Université, 76801 Saint Étienne du Rouvray, France email elise.janvresse@univ-rouen.frThierry.de-la-Rue@univ-rouen.fr
V. V. RYZHIKOV
Affiliation:
Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, 119991 Russia email sasha.prihodko@gmail.comvryzh@mail.ru
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Abstract

We completely describe the weak closure of the powers of the Koopman operator associated with Chacon’s classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of $\alpha $-weak mixing for Chacon’s automorphism.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 1 , February 2015 , pp. 128 - 141
Copyright
© Cambridge University Press, 2013 

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References

Ageev, O. N.. On asymmetry of the future and the past for limit self-joinings. Proc. Amer. Math. Soc. 131 (7) (2003), 2053–2062 (electronic).Google Scholar
Chacon, R. V.. Weakly mixing transformations which are not strongly mixing. Proc. Amer. Math. Soc. 22 (1969), 559–562.Google Scholar
del Junco, A., Rahe, M. and Swanson, L.. Chacon’s automorphism has minimal self-joinings. J. Anal. Math. 37 (1980), 276–284.Google Scholar
del Junco, A.. A simple measure-preserving transformation with trivial centralizer. Pacific J. Math. 79 (2) (1978), 357–362.Google Scholar
del Junco, A. and Lemańczyk, M.. Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc. 115 (3) (1992), 725–736.Google Scholar
El Abdalaoui, E. H., Lemańczyk, M. and de la Rue, T.. On spectral disjointness of powers for rank-one transformations and Möbius orthogonality. Preprint, arXiv:1301.0134, 2013.Google Scholar
Friedman, N. A.. Introduction to Ergodic Theory (Van Nostrand Reinhold Mathematical Studies, 29). Van Nostrand Reinhold Co., New York, 1970.Google Scholar
Katok, A.. Combinatorial Constructions in Ergodic Theory and Dynamics (University Lecture Series, 30). American Mathematical Society, Providence, RI, 2003.Google Scholar
Prikhod’ko, A. A. and Ryzhikov, V. V.. Disjointness of the convolutions for Chacon’s automorphism. Colloq. Math. 84/85 (part 1) (2000), 67–74 (dedicated to the memory of Anzelm Iwanik).Google Scholar
Prikhod’ko, A. A. and Ryzhikov, V. V.. Several questions and hypotheses concerning the limit polynomials for Chacon transformation. Preprint, arXiv:1207.0614, 2012.Google Scholar
Purkayastha, S.. Simple proofs of two results on convolutions of unimodal distributions. Statist. Probab. Lett. 39 (2) (1998), 97–100.Google Scholar
Ryzhikov, V.. Minimal self-joinings, bounded constructions, and weak closure of ergodic actions. Preprint, arXiv:1212.2602, 2012.Google Scholar
Stepin, A. M.. Spectral properties of generic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 50 (4) (1986), 801–834; 879.Google Scholar