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Shadowing and structural stability for operators

Part of: Topological dynamics Smooth dynamical systems: general theory General theory of linear operators

Published online by Cambridge University Press:  10 January 2020

NILSON C. BERNARDES JR  and
ALI MESSAOUDI
NILSON 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 ncbernardesjr@gmail.com
ALI MESSAOUDI
Affiliation:
Departamento de Matemática, Universidade Estadual Paulista, Rua Cristóvão Colombo, 2265, Jardim Nazareth, São José do Rio Preto, SP, 15054-000, Brasil email ali.messaoudi@unesp.br
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Abstract

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_{0}(\mathbb{Z})$ and $\ell _{p}(\mathbb{Z})$ ($1\leq p<\infty$) that satisfy the shadowing property.

Keywords

linear operatorsshadowingexpansivityhyperbolicitystructural stability

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.) 37C20: Generic properties, structural stability 37B99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 961 - 980
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Copyright
© Cambridge University Press, 2020

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References

Alves, J. F.. Hyperbolic isomorphisms in Banach spaces, www.fc.up.pt/pessoas/jfalves/pub/senegal.pdf.Google Scholar
Andronov, A. and Pontrjagin, L.. Structurally stable systems. Dokl. Akad. Nauk SSSR 14 (1937), 247–250 (Russian).Google Scholar
Aoki, N.. Topological Dynamics (Topics in General Topology, 41) . North-Holland Mathematical Library, North-Holland, Amsterdam, 1989, pp. 625–740.Google Scholar
Bartoll, S., Martínez-Giménez, F. and Peris, A.. The specification property for backward shifts. J. Difference Equ. Appl. 18(4) (2012), 599–605.10.1080/10236198.2011.586636CrossRefGoogle Scholar
Bayart, F. and Matheron, É.. Dynamics of Linear Operators. Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Bayart, F. and Ruzsa, I. Z.. Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys. 35(3) (2015), 691–709.CrossRefGoogle Scholar
Bernardes, N. C. Jr, Bonilla, A., Müller, V. and Peris, A.. Distributional chaos for linear operators. J. Funct. Anal. 265(9) (2013), 2143–2163.Google Scholar
Bernardes, N. C. Jr, Bonilla, A., Müller, V. and Peris, A.. Li-Yorke chaos in linear dynamics. Ergod. Th. & Dynam. Sys. 35(6) (2015), 1723–1745.CrossRefGoogle Scholar
Bernardes, N. C. Jr, Bonilla, A. and Peris, A.. Mean Li-Yorke chaos in Banach spaces. J. Funct. Anal. 278(3) (2020), article 108343, 1–31.10.1016/j.jfa.2019.108343CrossRefGoogle Scholar
Bernardes, N. C. Jr, Cirilo, P. R., Darji, U. B., Messaoudi, A. and Pujals, E. R.. Expansivity and shadowing in linear dynamics. J. Math. Anal. Appl. 461(1) (2018), 796–816.CrossRefGoogle Scholar
Bowen, R.. 𝜔-limit sets for Axiom A diffeomorphisms. J. Differential Equations 18(2) (1975), 333–339.CrossRefGoogle Scholar
Cirilo, P. R., Gollobit, B. and Pujals, E. R.. Generalized hyperbolicity for linear operators, submitted.Google Scholar
Costakis, G. and Sambarino, M.. Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc. 132(2) (2004), 385–389.CrossRefGoogle Scholar
Eisenberg, M. and Hedlund, J. H.. Expansive automorphisms of Banach spaces. Pacific J. Math. 34(3) (1970), 647–656.CrossRefGoogle Scholar
Grosse-Erdmann, K.-G.. Hypercyclic and chaotic weighted shifts. Studia Math. 139(1) (2000), 47–68.10.4064/sm-139-1-47-68CrossRefGoogle Scholar
Grosse-Erdmann, K.-G. and Peris Manguillot, A.. Linear Chaos. Springer, London, 2011.CrossRefGoogle Scholar
Hartman, P.. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960), 610–620.CrossRefGoogle Scholar
Hedlund, J. H.. Expansive automorphims of Banach spaces, II. Pacific J. Math. 36(3) (1971), 671–675.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Mazur, M.. Hyperbolicity, expansivity and shadowing for the class of normal operators. Funct. Differ. Equ. 7(1–2) (2000), 147–156.Google Scholar
Megginson, R. E.. An Introduction to Banach Space Theory. Springer, New York, 1998.10.1007/978-1-4612-0603-3CrossRefGoogle Scholar
Moser, J. K.. On a theorem of Anosov. J. Differential Equations 5 (1969), 411–440.10.1016/0022-0396(69)90083-7CrossRefGoogle Scholar
Ombach, J.. The shadowing lemma in the linear case. Univ. Iagel. Acta Math. 31 (1994), 69–74.Google Scholar
Palis, J.. On the local structure of hyperbolic points in Banach spaces. An. Acad. Brasil. Ciênc. 40 (1968), 263–266.Google Scholar
Palmer, K.. Shadowing in Dynamical Systems—Theory and Applications (Mathematics and its Applications, 501) . Kluwer Academic Publishers, Dordrecht, 2000.10.1007/978-1-4757-3210-8CrossRefGoogle Scholar
Pilyugin, S. Yu.. Shadowing in Dynamical Systems (Lecture Notes in Mathematics, 1706) . Springer, Berlin, 1999.Google Scholar
Pugh, C. C.. On a theorem of P. Hartman. Amer. J. Math. 91 (1969), 363–367.CrossRefGoogle Scholar
Robbin, J. W.. Topological conjugacy and structural stability for discrete dynamical systems. Bull. Amer. Math. Soc. 78(6) (1972), 923–952.CrossRefGoogle Scholar
Salas, H. N.. Hypercyclic weighted shifts. Trans. Amer. Math. Soc. 347(3) (1995), 993–1004.CrossRefGoogle Scholar
Sinaĭ, Ja. G.. Gibbs measures in ergodic theory. Uspekhi Mat. Nauk 27(4(166)) (1972), 21–64 (in Russian).Google Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73(6) (1967), 747–817.CrossRefGoogle Scholar
Yosida, K.. Functional Analysis, 6th edn. Springer, New York, 1980.Google Scholar