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Recurrence properties of hypercyclic operators

Juan Bès, Quentin Menet, Alfredo Peris, Yunied Puig de Dios

Abstract

We generalize the notions of hypercyclic operators, $\mathfrak{U}$-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets ${N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}$ for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.

Recurrence properties of hypercyclic operators

Abstract

We generalize the notions of hypercyclic operators, -frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called -frequent hypercyclicity. We then state an -Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the -frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.

Paper Structure

This paper contains 5 sections, 19 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. $\mathcal{A}$-frequently hypercyclic operators
  3. $\mathcal{A}$-Frequent Hypercyclicity Criterion
  4. Characterization of $\mathcal{A}$-frequently hypercyclic weighted shifts
  5. Banach densities and reiterative hypercyclicity

Key Result

Proposition 3

Let $X$ be a separable Banach space, $X\ne \{0\}$, let $\mathcal{A} \subset \mathcal{P}(\mathbb{Z}_+)$ be a non-trivial hereditarily upward family, and let $T\in \mathcal{L}(X)$. If $T$ is $\mathcal{A}$-frequently hypercyclic, then $\mathcal{A}$ is a hypercyclicity set.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Example 4
  • Definition 5
  • Theorem 6: Hypercyclicity Criterion Bes
  • Theorem 7: Frequent Hypercyclicity Criterion 4Bonilla
  • Remark 8
  • Theorem 9: $\mathcal{A}$-Frequent Hypercyclicity Criterion
  • ...and 27 more