Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-24T06:44:42.907Z Has data issue: false hasContentIssue false
  • English
  • Français

IP-rigidity and eigenvalue groups

Published online by Cambridge University Press:  08 March 2013

JON AARONSON ,
MARYAM HOSSEINI  and
MARIUSZ LEMAŃCZYK
JON AARONSON
Affiliation:
School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel email aaro@tau.ac.il
MARYAM HOSSEINI
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON, K1N 6N5, Canada email maryhoseini@uottawa.ca
MARIUSZ LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland email mlem@mat.umk.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the class of increasing sequences of natural numbers which are IP-rigidity sequences for some weakly mixing probability-preserving transformation. This property is closely related to the uncountability of the eigenvalue group of a corresponding non-singular transformation. We give examples, including a super-lacunary sequence which is not IP-rigid.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1057 - 1076
Copyright
Copyright ©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
Aaronson, J.. Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, a collection of invited papers on ergodic theory. Israel J. Math. 33 (3–4) (1979), 181–197.Google Scholar
Aaronson, J.. The eigenvalues of non-singular transformations. Israel J. Math. 45 (1983), 297–312.Google Scholar
Aaronson, J. and Nadkarni, M. G.. ${L}_{\infty } $ eigenvalues and ${L}_{2} $ spectra of non-singular transformations. Proc. Lond. Math. Soc. (3) 55 (1987), 538–570.CrossRefGoogle Scholar
Bergelson, V., del Junco, A., Lemańczyk, M. and Rosenblatt, J.. Rigidity and non-recurrence along sequences. Preprint, submitted, arxiv.org/pdf/1103.0905.Google Scholar
Cornfeld, I. P., Fomin, S. V. and Sinai, Y. G.. Ergodic Theory. Springer, New York, 1982.Google Scholar
Eggleston, H. G.. Sets of fractional dimensions which occur in some problems of number theory. Proc. Lond. Math. Soc. (2) 54 (1952), 42–93.Google Scholar
Eisner, T. and Grivaux, S.. Hilbertian Jamison sequences and rigid dynamical systems. J. Funct. Anal. 261 (2011), 2013–2052.CrossRefGoogle Scholar
Erdős, P. and Taylor, S. J.. On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences. Proc. Lond. Math. Soc. (3) 7 (1957), 598–615.CrossRefGoogle Scholar
Frostman, O.. Potentiel d’equilibre et capacité des ensembles avec quelques applications la théorie des fonctions. Meddelanden Mat. Sem. Univ. Lund. 3 115s (1935).Google Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.Google Scholar
Furstenberg, H. and Weiss, B.. The finite multipliers of infinite ergodic transformations. The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State University, Fargo, ND, 1977) (Lecture Notes in Mathematics, 668). Springer, Berlin, 1978, pp. 127–132.Google Scholar
Grivaux, S.. IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products. Preprint, arXiv:1209.2884.Google Scholar
Hindman, N. and Strauss, D.. Algebra in the Stone–Čech compactification. De Gruyter, Berlin, 1998.Google Scholar
Host, B., Mela, J.-F. and Parreau, F.. Analyse harmonique des mesures (Asterisque, 135–136). Société Mathématique de France, Paris, 1986, 261 pp.Google Scholar
Host, B., Mela, J.-F. and Parreau, F.. Nonsingular transformations and spectral analysis of measures. Bull. Soc. Math. France 119 (1) (1991), 33–90.Google Scholar
Ito, Sh. and Nakada, H.. Approximations of real numbers by the sequence $\{ n\alpha \} $ and their metrical theory. Acta Math. Hungar. 52 (1988), 91–100.Google Scholar
Kahane, J. P. and Salem, R.. Ensembles parfaites et series trigonometriques. Hermann, Paris, 1963.Google Scholar
Katok, A. and Thouvenot, J.-P.. Spectral properties and combinatorial constructions in ergodic theory. Handbook of Dynamical Systems. Vol. 1B. Elsevier, Amsterdam, 2006, pp. 649–743.Google Scholar
Nadkarni, M. G.. Basic Ergodic Theory. Hindustan Book Agency, New Delhi, 1998.Google Scholar
Nadkarni, M. G.. Spectral Theory of Dynamical Systems. Hindustan Book Agency, New Delhi, 1998.Google Scholar
Parreau, F.. Ergodicité et pureté des produits de Riesz. Ann. Inst. Fourier 40 (1990), 391–405.Google Scholar
Zimmer, R. J.. Ergodic Theory and Semisimple Groups (Monographs in Mathematics, 81). Birkhäuser, Basel, 1984.Google Scholar