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arxiv: 1907.05502 · v1 · pith:VAHI6XX5new · submitted 2019-07-11 · 🧮 math.FA · math.DS

mathcal{U}-Frequent hypercyclicity notions and related weighted densities

Romuald Ernst , C\'eline Esser , Quentin Menet This is my paper

Pith reviewed 2026-05-24 22:30 UTC · model grok-4.3

classification 🧮 math.FA math.DS
keywords hypercyclicityfrequent hypercyclicityreiterative hypercyclicityweighted densitieschaoslinear dynamicsBanach space operators
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The pith

Chaos does not imply U-frequent hypercyclicity with respect to any weighted upper density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines notions of hypercyclicity lying strictly between U-frequent hypercyclicity and reiterative hypercyclicity by introducing weighted upper densities that sit between the ordinary upper density and the upper Banach density. It shows that chaos implies reiterative hypercyclicity but none of the new weighted versions. It further proves that each of these hypercyclicity properties is preserved under passage to any finite product of the operator.

Core claim

Weighted upper densities define intermediate hypercyclicity notions between U-frequent hypercyclicity and reiterative hypercyclicity. Chaos implies reiterative hypercyclicity but fails to imply U-frequent hypercyclicity for any such weighted density. If an operator satisfies one of these properties then every n-fold product of the operator satisfies the same property.

What carries the argument

Weighted upper densities between the unweighted upper density and the upper Banach density, used to generate the intermediate hypercyclicity notions.

If this is right

  • If T is U-frequently hypercyclic then every n-fold product of T is U-frequently hypercyclic.
  • If T is reiteratively hypercyclic then every n-fold product of T is reiteratively hypercyclic.
  • The same product-preservation holds for each weighted U-frequent hypercyclicity notion introduced.
  • No chaotic operator can be U-frequently hypercyclic with respect to any of the weighted densities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation between chaos and the weighted notions shows that the choice of density controls whether hypercyclicity follows from chaos.
  • Product preservation may allow construction of new examples by taking products of known hypercyclic operators.
  • The weighted notions provide a scale that can be used to measure how far a given chaotic operator falls short of frequent hypercyclicity.

Load-bearing premise

The weighted upper densities satisfy the monotonicity and subadditivity properties that let the usual proofs for unweighted and Banach densities extend directly.

What would settle it

An explicit chaotic operator whose orbit is dense with respect to one of the weighted upper densities would falsify the claim that chaos implies none of them.

Figures

Figures reproduced from arXiv: 1907.05502 by C\'eline Esser, Quentin Menet, Romuald Ernst.

Figure 1
Figure 1. Figure 1: Ordering of densities when an bn & 0 and a, b ∈ S We define our weight sequence a by setting a0 = 1 and aj = 1 p j X−1 l=0 al when j ∈ [kp, kp+1[ and we claim that this sequence has the properties we are looking for. We begin by proving that a ∈ S. Indeed, akp = 1 p k Xp−1 l=0 al = 1 p  akp−1 + (p − 1)akp−1  = akp−1. Moreover, if k ∈ ]kp, kp+1[ then ak = 1 p k X−1 l=0 al = 1 p (ak−1 + pak−1) = p + 1 p ak… view at source ↗
read the original abstract

We study dynamical notions lying between $\mathcal{U}$-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies reiterative hypercyclicity, we show that chaos does not imply $\mathcal{U}$-frequent hypercyclicity with respect to any weighted upper density. Moreover, we show that if $T$ is $\mathcal{U}$-frequently hypercyclic (resp. reiteratively hypercyclic) then the n-fold product of $T$ is still $\mathcal{U}$-frequently hypercyclic (resp. reiteratively hypercyclic) and that this implication is also satisfied for each of the considered $\mathcal{U}$-frequent hypercyclicity notions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces intermediate notions of hypercyclicity lying between U-frequent hypercyclicity and reiterative hypercyclicity for linear operators on Banach spaces. These notions are defined via weighted upper densities positioned between the unweighted upper density and the upper Banach density. The authors prove that while every chaotic operator is reiteratively hypercyclic, chaos does not imply U-frequent hypercyclicity with respect to any weighted upper density satisfying the relevant monotonicity and subadditivity conditions. They further establish that U-frequent hypercyclicity (and each of the intermediate variants) is preserved under passage to any finite product of the operator.

Significance. If the proofs hold, the work sharpens the hierarchy of frequent hypercyclicity notions and isolates a clear separation between reiterative and U-frequent properties with respect to chaos. The product-stability result is a structural contribution that extends known facts for the endpoint notions. The paper supplies concrete counter-examples and positive preservation theorems that can be used in further studies of linear dynamical systems.

minor comments (2)
  1. The abstract states that the weighted densities are chosen to satisfy monotonicity and subadditivity so that standard arguments carry over, but the precise list of axioms imposed on the weights should be collected in a single definition or proposition early in the paper for easy reference.
  2. Notation for the various weighted densities (e.g., upper density with weight w) should be introduced uniformly and used consistently in all statements of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript, including the recommendation for minor revision. The report provides a concise summary of the main results but does not list any specific major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends standard hypercyclicity arguments to weighted upper densities satisfying monotonicity and subadditivity, positioning them between unweighted upper density and upper Banach density. The claims that chaos fails to imply the new notions and that products preserve the properties follow directly from adapting existing proofs for unweighted and Banach cases; no step reduces by definition to the new notions themselves, no parameters are fitted and relabeled as predictions, and no load-bearing self-citation chain is invoked. The derivation remains self-contained against external benchmarks of hypercyclicity theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records only the background notions mentioned; no free parameters, ad-hoc axioms, or invented entities are visible.

axioms (1)
  • domain assumption Standard properties of upper densities (monotonicity, subadditivity) hold for the weighted versions introduced.
    Required for the new notions to sit between U-frequent and reiterative hypercyclicity.

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Reference graph

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20 extracted references · 20 canonical work pages

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