Expansivity and Shadowing in Linear Dynamics

Abstract

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 ${\ell }_p$ ($1\le p<\infty$), 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.

References in corpus (5)

• Difference sets and frequently hypercyclic weighted shifts
• Recurrence properties of hypercyclic operators
• Lipschitz shadowing implies structural stability
• Some Consequences of the Shadowing Property in Low Dimensions
• Bounded Point Evaluations and Local Spectral Theory

Cited by in corpus (13)

• Shadowing for infinite dimensional dynamics and exponential trichotomies
• Shadowing for nonautonomous dynamics
• A generalized Grobman-Hartman theorem
• Li-Yorke Chaos for Composition Operators on $L^p$-Spaces
• A general approach to nonautonomous shadowing for nonlinear dynamics
• A variational principle for the metric mean dimension of level sets
• Linear Dynamics Induced by Odometers
• Conditional Lipschitz shadowing for ordinary differential equations
• Generalized hyperbolicity for linear operators
• Quasi-shadowing for partially hyperbolic dynamics on Banach spaces
• Generalized Hyperbolicity and Shadowing in $L^p$ spaces
• Entropy, pressure, ground states and calibrated sub-actions for linear dynamics
• Shift-like Operators on $L^p(X)$