Topologically transitive skew-products of backward shift operators and hypercyclicity
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Abstract:
In this article we look at skew-products of multiples of the backward shift and examine conditions under which the skew-product is topologically transitive or hypercyclic in the second coordinate. We also give an application of the theory to iterated function systems of multiples of backward shift operators.References
- Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
- Frédéric Bayart and Sophie Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal. 226 (2005), no. 2, 281–300. MR 2159459, DOI 10.1016/j.jfa.2005.06.001
- 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
- L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math. 157 (2003), no. 1, 17–32. MR 1980114, DOI 10.4064/sm157-1-2
- Juan Bès and Alfredo Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94–112. MR 1710637, DOI 10.1006/jfan.1999.3437
- J. Bonet, F. Martínez-Giménez, and A. Peris, Linear chaos on Fréchet spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 7, 1649–1655. Dynamical systems and functional equations (Murcia, 2000). MR 2015614, DOI 10.1142/S0218127403007497
- 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
- Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
- Sophie Grivaux, Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory 54 (2005), no. 1, 147–168. MR 2168865
- Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
- K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 2, 273–286 (English, with English and Spanish summaries). MR 2068180
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto (1982).
- Fernando León-Saavedra, Notes about the hypercyclicity criterion, Math. Slovaca 53 (2003), no. 3, 313–319. MR 2025025
- Alfonso Montes-Rodríguez and Héctor N. Salas, Supercyclic subspaces, Bull. London Math. Soc. 35 (2003), no. 6, 721–737. MR 2000019, DOI 10.1112/S002460930300242X
- S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22. MR 241956, DOI 10.4064/sm-32-1-17-22
- J. H. Shapiro, Notes on the Dynamics of Linear Operators, Unpublished Lecture Notes, (available at www.math.msu.edu/ ˜shapiro).
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
Additional Information
- George Costakis
- Affiliation: Department of Mathematics, University of Crete, Knossos Avenue, GR-714 09, Heraklion, Crete, Greece
- Email: costakis@math.uoc.gr
- Demetris Hadjiloucas
- Affiliation: The School of Computer Science and Engineering, Cyprus College, 6 Diogenes Street, Engomi, P. O. Box 22006, 1516 Nicosia, Cyprus
- Email: dhadjiloucas@cycollege.ac.cy
- Received by editor(s): August 22, 2006
- Received by editor(s) in revised form: November 7, 2006
- Published electronically: November 30, 2007
- Communicated by: Joseph A. Ball
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 937-946
- MSC (2000): Primary 47A16, 28D99
- DOI: https://doi.org/10.1090/S0002-9939-07-09001-6
- MathSciNet review: 2361867