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Topologically transitive skew-products of backward shift operators and hypercyclicity
Authors:
George Costakis and Demetris Hadjiloucas
Journal:
Proc. Amer. Math. Soc. 136 (2008), 937-946
MSC (2000):
Primary 47A16, 28D99
Posted:
November 30, 2007
MathSciNet review:
2361867
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Additional Information
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.
- 1.
Ludwig
Arnold, Random dynamical systems, Springer Monographs in
Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992
(2000m:37087)
- 2.
Frédéric
Bayart and Sophie
Grivaux, Hypercyclicity and unimodular point spectrum, J.
Funct. Anal. 226 (2005), no. 2, 281–300. MR 2159459
(2006i:47014), http://dx.doi.org/10.1016/j.jfa.2005.06.001
- 3.
Frédéric
Bayart and Sophie
Grivaux, Frequently hypercyclic
operators, Trans. Amer. Math. Soc.
358 (2006), no. 11, 5083–5117 (electronic). MR 2231886
(2007e:47013), http://dx.doi.org/10.1090/S0002-9947-06-04019-0
- 4.
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 (2003m:47013), http://dx.doi.org/10.4064/sm157-1-2
- 5.
Juan
Bès and Alfredo
Peris, Hereditarily hypercyclic operators, J. Funct. Anal.
167 (1999), no. 1, 94–112. MR 1710637
(2000f:47012), http://dx.doi.org/10.1006/jfan.1999.3437
- 6.
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 (2004i:47016), http://dx.doi.org/10.1142/S0218127403007497
- 7.
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
(88g:47060), http://dx.doi.org/10.1090/S0002-9939-1987-0884467-4
- 8.
Gilles
Godefroy and Joel
H. Shapiro, Operators with dense, invariant, cyclic vector
manifolds, J. Funct. Anal. 98 (1991), no. 2,
229–269. MR 1111569
(92d:47029), http://dx.doi.org/10.1016/0022-1236(91)90078-J
- 9.
Sophie
Grivaux, Hypercyclic operators, mixing operators, and the bounded
steps problem, J. Operator Theory 54 (2005),
no. 1, 147–168. MR 2168865
(2006k:47021)
- 10.
Karl-Goswin
Grosse-Erdmann, Universal families and hypercyclic
operators, Bull. Amer. Math. Soc. (N.S.)
36 (1999), no. 3,
345–381. MR 1685272
(2000c:47001), http://dx.doi.org/10.1090/S0273-0979-99-00788-0
- 11.
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
(2005c:47010)
- 12.
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
(96c:58055)
- 13.
C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto (1982).
- 14.
Fernando
León-Saavedra, Notes about the hypercyclicity
criterion, Math. Slovaca 53 (2003), no. 3,
313–319. MR 2025025
(2004k:47010)
- 15.
Alfonso
Montes-Rodríguez and Héctor
N. Salas, Supercyclic subspaces, Bull. London Math. Soc.
35 (2003), no. 6, 721–737. MR 2000019
(2004d:47022), http://dx.doi.org/10.1112/S002460930300242X
- 16.
S.
Rolewicz, On orbits of elements, Studia Math.
32 (1969), 17–22. MR 0241956
(39 #3292)
- 17.
J. H. Shapiro, Notes on the Dynamics of Linear Operators, Unpublished Lecture Notes, (available at www.math.msu.edu/ ˜shapiro).
- 18.
Peter
Walters, An introduction to ergodic theory, Graduate Texts in
Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR 648108
(84e:28017)
- 1.
- L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992 (2000m:37087)
- 2.
- F. Bayart, S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal. 226 (2005) 281-300. MR 2159459 (2006i:47014)
- 3.
- F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006) 5083-5117. MR 2231886 (2007e:47013)
- 4.
- L. Bernal-Gonzalez and K. -G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math. 157 (2003) 17-32. MR 1980114 (2003m:47013)
- 5.
- J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999) 94-112. MR 1710637 (2000f:47012)
- 6.
- J. Bonet, F. Martinez-Gimenez and A. Peris, Linear chaos on Frechet spaces, Dynamical systems and functional equations (Murcia, 2000). Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003) 1649-1655. MR 2015614 (2004i:47016)
- 7.
- R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987) 281-288. MR 884467 (88g:47060)
- 8.
- G. Godefroy and J. H. Shapiro, Operators with dense invariant cyclic manifolds, J. Funct. Anal. 98 (1991) 229-269. MR 1111569 (92d:47029)
- 9.
- S. Grivaux, Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory 54 (2005) 147-168. MR 2168865 (2006k:47021)
- 10.
- K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999) no. 3, 345-381. MR 1685272 (2000c:47001)
- 11.
- K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. (2003) 273-286. MR 2068180 (2005c:47010)
- 12.
- A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995. MR 1326374 (96c:58055)
- 13.
- C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto (1982).
- 14.
- F. León-Saavedra, Notes about the hypercyclicity criterion, Math. Slovaca, 53 (2003) 313-319. MR 2025025 (2004k:47010)
- 15.
- A. Montes-Rodriguez and H. N. Salas, Supercyclic subspaces, Bull. London Math. Soc. 35 (2003) 721-737. MR 2000019 (2004d:47022)
- 16.
- S. Rolewicz, On orbits of elements, Studia Math., 32 (1969) 17-22. MR 0241956 (39:3292)
- 17.
- J. H. Shapiro, Notes on the Dynamics of Linear Operators, Unpublished Lecture Notes, (available at www.math.msu.edu/ ˜shapiro).
- 18.
- P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. MR 648108 (84e:28017)
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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
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-09001-6
PII:
S 0002-9939(07)09001-6
Keywords:
Hypercyclic operators,
skew-product
Received by editor(s):
August 22, 2006
Received by editor(s) in revised form:
November 7, 2006
Posted:
November 30, 2007
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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