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Shadowing and structural stability for operators

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

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 [email protected]
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 [email protected]
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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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