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Expansivity and Shadowing in Linear Dynamics

Nilson C. Bernardes, Patricia R. Cirilo, Udayan B. Darji, Ali Messaoudi, Enrique R. Pujals

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 $\ell_p$ ($1 \leq p < \infty$), 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.

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 or (), 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.

Paper Structure

This paper contains 6 sections, 22 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Expansive behavior of operators
  3. Expansive weighted shifts
  4. Expansivity and spectrum
  5. Shadowing
  6. Positive shadowing

Key Result

Proposition 5

Let $T$ be an operator on a Banach space $X$. Then: If, in addition, $T$ is invertible, then:

Theorems & Definitions (69)

  • Definition 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Proposition 5
  • proof
  • Remark 6
  • Definition 7
  • proof
  • Remark 8
  • ...and 59 more