Some new examples of universal hypercyclic operators in the sense of Glasner and Weiss
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- by Sophie Grivaux PDF
- Trans. Amer. Math. Soc. 369 (2017), 7589-7629 Request permission
Abstract:
A bounded operator $A$ on a real or complex separable infinite-dimensional Banach space $Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $T$ of a standard Lebesgue probability space $(X,\mathcal {B},\mu )$, there exists an $A$-invariant probability measure $\nu$ on $Z$ with full support such that the two dynamical systems $(X,\mathcal {B},\mu ;T)$ and $(Z,\mathcal {B}_{Z},\nu ;A)$ are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or bilateral weighted shifts on $\ell _{p}$ or $c_{0}$, to show the existence of universal operators on a large class of Banach spaces and to give a criterion for universality in terms of unimodular eigenvectors. We also obtain similar results for operators which are universal for all ergodic systems (not only for invertible ones) and study necessary conditions for an operator on a Hilbert space to be universal.References
- Viviane Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1793194, DOI 10.1142/9789812813633
- Frédéric Bayart and Sophie Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083–5117. MR 2231886, DOI 10.1090/S0002-9947-06-04019-0
- Frédéric Bayart and Sophie Grivaux, Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 181–210. MR 2294994, DOI 10.1112/plms/pdl013
- Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
- Frédéric Bayart and Étienne Matheron, Mixing operators and small subsets of the circle, J. Reine Angew. Math. 715 (2016), 75–123. MR 3507920, DOI 10.1515/crelle-2014-0002
- Frédéric Bayart and Imre Z. Ruzsa, Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory Dynam. Systems 35 (2015), no. 3, 691–709. MR 3334899, DOI 10.1017/etds.2013.77
- A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 383–404. MR 2308137, DOI 10.1017/S014338570600085X
- N. Feldman, Linear chaos?, preprint, 2001, available at http://home.wlu. edu/~feldmann/ research.html.
- Sophie Grivaux and Étienne Matheron, Invariant measures for frequently hypercyclic operators, Adv. Math. 265 (2014), 371–427. MR 3255465, DOI 10.1016/j.aim.2014.08.002
- Eli Glasner and Benjamin Weiss, A universal hypercyclic representation, J. Funct. Anal. 268 (2015), no. 11, 3478–3491. MR 3336730, DOI 10.1016/j.jfa.2015.02.002
- Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812, DOI 10.1007/978-1-4471-2170-1
- Roger Jones, Joseph Rosenblatt, and Arkady Tempelman, Ergodic theorems for convolutions of a measure on a group, Illinois J. Math. 38 (1994), no. 4, 521–553. MR 1283007
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, Mathematics and its Applications (Soviet Series), vol. 14, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian and with a preface by Wojbor A. Woyczynski. MR 1435288, DOI 10.1007/978-94-009-3873-1
Additional Information
- Sophie Grivaux
- Affiliation: CNRS, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France
- MR Author ID: 705957
- Email: sophie.grivaux@u-picardie.fr
- Received by editor(s): October 12, 2014
- Received by editor(s) in revised form: October 3, 2015, and October 13, 2015
- Published electronically: March 6, 2017
- Additional Notes: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 7589-7629
- MSC (2010): Primary 47A16, 37A35, 47A35, 47B35, 47B37
- DOI: https://doi.org/10.1090/tran/6855
- MathSciNet review: 3695839