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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

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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.

Information

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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