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Ultra hypercyclicity and its connection to mixing properties

Martin Liu, David Walmsley, James Xue

TL;DR

The paper investigates ultra hypercyclicity, a strengthening of hypercyclicity that requires the existence of a subsequence $(n_k)$ exhibiting a cover property on open sets. It proves ultra hypercyclicity implies weak mixing but does not imply mixing, and provides a complete weight-sequence characterization for ultra hypercyclic weighted backward shifts on $c_0$ and $\ell^p$, along with constructions showing the non-implications and the existence of both ultra hypercyclic and strongly hypercyclic shifts with distinct behaviors. It also introduces an ultra hypercyclicity criterion, and offers a correlated sufficient condition for strong hypercyclicity together with an explicit strongly hypercyclic but not ultra hypercyclic example, extending the known hierarchy among mixing properties in linear dynamics.

Abstract

Recently, two new topological properties for operators acting on a topological vector space were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity and compare it to strong hypercyclicity and hypermixing, as well as the classical notions of mixing, weak mixing, and hypercyclicity. We show that every ultra hypercyclic operator on Fréchet space must be weakly mixing, and that there exists a strongly hypercyclic operator which is not ultra hypercyclic. We also characterize, in terms of the weight sequence, the ultra hypercyclic weighted backward shifts on $c_0$ and $\ell^p$, $1\leq p<\infty$. Finally, we improve upon a necessary condition for strongly hypercyclic weighted backward shifts.

Ultra hypercyclicity and its connection to mixing properties

TL;DR

The paper investigates ultra hypercyclicity, a strengthening of hypercyclicity that requires the existence of a subsequence exhibiting a cover property on open sets. It proves ultra hypercyclicity implies weak mixing but does not imply mixing, and provides a complete weight-sequence characterization for ultra hypercyclic weighted backward shifts on and , along with constructions showing the non-implications and the existence of both ultra hypercyclic and strongly hypercyclic shifts with distinct behaviors. It also introduces an ultra hypercyclicity criterion, and offers a correlated sufficient condition for strong hypercyclicity together with an explicit strongly hypercyclic but not ultra hypercyclic example, extending the known hierarchy among mixing properties in linear dynamics.

Abstract

Recently, two new topological properties for operators acting on a topological vector space were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity and compare it to strong hypercyclicity and hypermixing, as well as the classical notions of mixing, weak mixing, and hypercyclicity. We show that every ultra hypercyclic operator on Fréchet space must be weakly mixing, and that there exists a strongly hypercyclic operator which is not ultra hypercyclic. We also characterize, in terms of the weight sequence, the ultra hypercyclic weighted backward shifts on and , . Finally, we improve upon a necessary condition for strongly hypercyclic weighted backward shifts.
Paper Structure (3 sections, 11 theorems, 35 equations, 1 table)

This paper contains 3 sections, 11 theorems, 35 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ultra hypercyclicity and weighted backward shifts
  3. Acknowledgments

Key Result

Theorem 2.1

Suppose $X$ is a first countable topological vector space and $T:X\to X$ is a surjective operator with right inverse map $S$. Then $T$ is ultra hypercyclic for $(n_k)_{k\geq 1}$ if and only if, for every non-zero vector $x\in X$ and any $y\in X$, there exists a sequence $(u_k)_{k\geq 1}$ in $X$ such

Theorems & Definitions (21)

  • Definition 1.1
  • Remark 1.2
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4: Ansari4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 11 more