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The Exact $\varepsilon$-Hypercyclicity Threshold

Geivison Ribeiro

TL;DR

This work solves Bayart's question by explicitly constructing a weighted shift T on the Hilbertian direct sum Z of copies of ℓ^2 that exhibits an exact ε-hypercyclicity threshold: T is δ-hypercyclic precisely for all δ in [ε,1). The construction hinges on a careful block-decomposition of finite-dimensional subspaces, a finite-approximation scheme via sets S_k, and a programmable sequence of weights A_n that yields simultaneous visiting properties while controlling growth. A dual argument shows that δ<ε cannot be achieved, establishing sharpness of the threshold. The result presents a first example of an operator with a sharp, problem-specific transition value for ε-hypercyclicity and opens directions for analogous thresholds in other notions and spaces.

Abstract

In this paper we give an affirmative answer to the problem proposed by Bayart in [J. Math. Anal. Appl. \textbf{529} (2024), 127278]: given $\varepsilon\in(0,1)$, there exists an operator which is $δ$-hypercyclic if and only if $δ\in[\varepsilon,1)$?

The Exact $\varepsilon$-Hypercyclicity Threshold

TL;DR

This work solves Bayart's question by explicitly constructing a weighted shift T on the Hilbertian direct sum Z of copies of ℓ^2 that exhibits an exact ε-hypercyclicity threshold: T is δ-hypercyclic precisely for all δ in [ε,1). The construction hinges on a careful block-decomposition of finite-dimensional subspaces, a finite-approximation scheme via sets S_k, and a programmable sequence of weights A_n that yields simultaneous visiting properties while controlling growth. A dual argument shows that δ<ε cannot be achieved, establishing sharpness of the threshold. The result presents a first example of an operator with a sharp, problem-specific transition value for ε-hypercyclicity and opens directions for analogous thresholds in other notions and spaces.

Abstract

In this paper we give an affirmative answer to the problem proposed by Bayart in [J. Math. Anal. Appl. \textbf{529} (2024), 127278]: given , there exists an operator which is -hypercyclic if and only if ?

Paper Structure

This paper contains 8 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Initial setting and operator
  3. Preliminary Results
  4. Decomposition of $F_k$
  5. Prescription of the weights
  6. Criterion and simultaneous approximation
  7. Lower barrier for every $\delta<\varepsilon$
  8. The main theorem

Key Result

Lemma 1

For every $u=(u_r)_{r=0}^{\infty}\in Z$,

Theorems & Definitions (20)

  • Definition 1
  • Lemma 1
  • proof
  • proof : Proof of (iii)
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • ...and 10 more