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Li–Yorke chaos in linear dynamics

Published online by Cambridge University Press:  10 July 2014

N. C. BERNARDES JR ,
A. BONILLA ,
V. MÜLLER  and
A. PERIS
N. C. BERNARDES JR
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil email bernardes@im.ufrj.br
A. BONILLA
Affiliation:
Departamento de Análisis Matemático, Universidad de la Laguna, 38271, La Laguna (Tenerife), Spain email abonilla@ull.es
V. MÜLLER
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Zitná 25, 115 67 Prague 1, Czech Republic email muller@math.cas.cz
A. PERIS
Affiliation:
IUMPA, Universitat Polit\`ecnica de Val\`encia, Departament de Matemàtica Aplicada, Edifici 7A, 46022 Val\`encia, Spain email aperis@mat.upv.es
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Abstract

We obtain new characterizations of Li–Yorke chaos for linear operators on Banach and Fréchet spaces. We also offer conditions under which an operator admits a dense set or linear manifold of irregular vectors. Some of our general results are applied to composition operators and adjoint multipliers on spaces of holomorphic functions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 6 , September 2015 , pp. 1723 - 1745
Copyright
© Cambridge University Press, 2014 

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References

Ansari, S. I.. Existence of hypercyclic operators on topological vector spaces. J. Funct. Anal. 148(2) (1997), 384–390.CrossRefGoogle Scholar
Bartoll, S., Martínez-Giménez, F. and Peris, A.. The specification property for backward shifts. J. Differ. Eq. Appl. 18(4) (2012), 599–605.CrossRefGoogle Scholar
Bayart, F. and Matheron, É.. Dynamics of Linear Operators (Cambridge Tracts in Mathematics, 179). Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Beauzamy, B.. Introduction to Operator Theory and Invariant Subspaces. North-Holland, Amsterdam, 1988.Google Scholar
Bermúdez, T., Bonilla, A., Martínez-Giménez, F. and Peris, A.. Li–Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1) (2011), 83–93.CrossRefGoogle Scholar
Bernal-González, L.. On hypercyclic operators on Banach spaces. Proc. Amer. Math. Soc. 127(4) (1999), 1003–1010.CrossRefGoogle Scholar
Bernal-González, L. and Montes-Rodríguez, A.. Universal functions for composition operators. Complex Variables Theory Appl. 27(1) (1995), 47–56.Google Scholar
Bernardes, N. C. Jr, Bonilla, A., Müller, V. and Peris, A.. Distributional chaos for linear operators. J. Funct. Anal. 265 (2013), 2143–2163.CrossRefGoogle Scholar
Bonet, J. and Peris, A.. Hypercyclic operators on non-normable Fréchet spaces. J. Funct. Anal. 159(2) (1998), 587–595.CrossRefGoogle Scholar
Bourdon, P. S.. Invariant manifolds of hypercyclic vectors. Proc. Amer. Math. Soc. 118(3) (1993), 845–847.CrossRefGoogle Scholar
Godefroy, G. and Shapiro, J. H.. Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2) (1991), 229–269.CrossRefGoogle Scholar
Grosse-Erdmann, K.-G. and Peris Manguillot, A.. Linear Chaos (Universitext). Springer, London, 2011.CrossRefGoogle Scholar
Herrero, D. A.. Limits of hypercyclic and supercyclic operators. J. Funct. Anal. 99(1) (1991), 179–190.CrossRefGoogle Scholar
Hou, B., Tian, G. and Shi, L.. Some dynamical properties for linear operators. Illinois J. Math. 53(3) (2009), 857–864.CrossRefGoogle Scholar
Hou, B., Tian, G. and Zhu, S.. Approximation of chaotic operators. J. Operator Theory 67(2) (2012), 469–493.Google Scholar
Li, T. Y. and Yorke, J. A.. Period three implies chaos. Amer. Math. Monthly 82(10) (1975), 985–992.CrossRefGoogle Scholar
Martínez-Giménez, F., Oprocha, P. and Peris, A.. Distributional chaos for operators with full scrambled sets. Math. Z. 274 (2013), 603–612.CrossRefGoogle Scholar
Meise, R. and Vogt, D.. Introduction to Functional Analysis. The Clarendon Press, Oxford University Press, New York, 1997.CrossRefGoogle Scholar
Piórek, J.. On the generic chaos in dynamical systems. Univ. Iagel. Acta Math. 25 (1985), 293–298.Google Scholar
Prǎjiturǎ, G. T.. Irregular vectors of Hilbert space operators. J. Math. Anal. Appl. 354(2) (2009), 689–697.CrossRefGoogle Scholar
Rudin, W.. Function Theory in Polydiscs. W. A. Benjamin, Inc., New York, 1969.Google Scholar
Rudin, W.. Function Theory in the Unit Ball of ℂn. Springer, Berlin, 1980.CrossRefGoogle Scholar
Snoha, L.. Generic chaos. Comment. Math. Univ. Carolin. 31(4) (1990), 793–810.Google Scholar
Snoha, L.. Dense chaos. Comment. Math. Univ. Carolin. 33(4) (1992), 747–752.Google Scholar