Expansivity and Shadowing in Linear Dynamics
By Nilson C. Bernardes, P. Cirilo, Udayan B. Darji, Ali Messaoudi, E. Pujals. 2016
87 citations recorded by the source corpus.
In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/db8f3db3dbae2337ec6a3722d95ea1f4d22407bd
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In Geometry and Topology
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Citations in the corpus
- Shadowing and structural stability for operators (2020, background): “This problem was finally solved by Bernardes et al [10], where examples of non-hyperbolic invertible operators with the shadowing property were obtained.”