Two problems on weighted shifts in linear dynamics
HTML articles powered by AMS MathViewer
- by Frédéric Bayart
- Proc. Amer. Math. Soc. 149 (2021), 5255-5266
- DOI: https://doi.org/10.1090/proc/15597
- Published electronically: September 9, 2021
Abstract:
We show that an invertible bilateral weighted shift is strongly structurally stable if and only if it has the shadowing property. We also exhibit a Köthe sequence space supporting a frequently hypercyclic weighted shift, but no chaotic weighted shifts.References
- 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 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
- Nilson C. Bernardes Jr., Patricia R. Cirilo, Udayan B. Darji, Ali Messaoudi, and Enrique R. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018), no. 1, 796–816. MR 3759568, DOI 10.1016/j.jmaa.2017.11.059
- Nilson C. Bernardes Jr. and Ali Messaoudi, A generalized Grobman-Hartman theorem, Proc. Amer. Math. Soc. 148 (2020), no. 10, 4351–4360. MR 4135302, DOI 10.1090/proc/15077
- Nilson C. Bernardes Jr. and Ali Messaoudi, Shadowing and structural stability for operators, Ergodic Theory Dynam. Systems 41 (2021), no. 4, 961–980. MR 4223416, DOI 10.1017/etds.2019.107
- Antonio Bonilla and Karl-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions, Rev. Mat. Complut. 31 (2018), no. 3, 673–711. MR 3847081, DOI 10.1007/s13163-018-0260-y
- S. Charpentier, K. G. Grosse-Erdmann, and Q. Menet, Chaos and frequent hypercyclicity for weighted shifts, Ergodic Theory Dynam. Systems, to appear, DOI 10.1017/etds.2020.122.
- Robert M. Gethner and Joel H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 2, 281–288. MR 884467, DOI 10.1090/S0002-9939-1987-0884467-4
- K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000), no. 1, 47–68. MR 1763044, DOI 10.4064/sm-139-1-47-68
- Philip Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620. MR 121542, DOI 10.1090/S0002-9939-1960-0121542-7
- Carol Kitai, INVARIANT CLOSED SETS FOR LINEAR OPERATORS, ProQuest LLC, Ann Arbor, MI, 1982. Thesis (Ph.D.)–University of Toronto (Canada). MR 2632793
- Quentin Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc. 369 (2017), no. 7, 4977–4994. MR 3632557, DOI 10.1090/tran/6808
- J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Ci. 40 (1968), 263–266. MR 246331
- Charles C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969), 363–367. MR 257533, DOI 10.2307/2373513
- Héctor N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993–1004. MR 1249890, DOI 10.1090/S0002-9947-1995-1249890-6
Bibliographic Information
- Frédéric Bayart
- Affiliation: Laboratoire de Mathématiques Blaise Pascal UMR 6620 CNRS, Université Clermont Auvergne, Campus universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière Cedex, France
- MR Author ID: 683115
- Email: frederic.bayart@uca.fr
- Received by editor(s): January 8, 2021
- Received by editor(s) in revised form: January 21, 2021, and March 23, 2021
- Published electronically: September 9, 2021
- Additional Notes: The author was partially supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front)
- Communicated by: Javad Mashreghi
- © Copyright 2021 by Frédéric Bayart
- Journal: Proc. Amer. Math. Soc. 149 (2021), 5255-5266
- MSC (2020): Primary 47A16
- DOI: https://doi.org/10.1090/proc/15597
- MathSciNet review: 4327429