arxiv: 2605.06573 · v1 · submitted 2026-05-07 · 🧮 math.FA

Recognition: unknown

Common frequently hypercyclic random vectors

Augustin Mouze, Vincent Munnier

Pith reviewed 2026-05-08 04:07 UTC · model grok-4.3

classification 🧮 math.FA

keywords frequently hypercyclic vectorscommon hypercyclicityweighted backward shiftsell_p spacesprobabilistic methodspolynomials of operatorsnon-existence results

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The pith

Countable families of weighted backward shifts on ell_p spaces share a common frequently hypercyclic vector under explicit conditions derived from a probabilistic criterion.

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

The paper develops a general existence criterion for vectors that are frequently hypercyclic for every member of a countable family of weighted backward shifts on ell_p spaces. Frequent hypercyclicity means that the orbit of the vector under each operator returns to every nonempty open set with positive lower density in the integers. The authors prove this criterion using probabilistic arguments on random vectors and pair it with a complementary non-existence result that identifies when no such shared vector can exist. They then specialize the criterion to families of polynomial operators built from weighted backward shifts, giving concrete conditions on the weights and polynomials that ensure a common frequently hypercyclic vector exists.

Core claim

Using probabilistic techniques, the authors establish a general criterion guaranteeing the existence of a common frequently hypercyclic vector for countable families of weighted backward shifts on ell_p spaces together with a non-existence result; when applied to countable families of polynomials of weighted backward shifts, the criterion yields explicit conditions under which the family admits a shared common frequently hypercyclic vector.

What carries the argument

A probabilistic existence criterion for common frequent hypercyclicity of a countable family of operators, realized through random vectors whose coordinate distributions are chosen to satisfy the frequent hypercyclicity conditions simultaneously for all operators in the family.

If this is right

Whenever a countable family of weighted backward shifts satisfies the derived weight conditions, a single vector exists whose orbit under each shift visits every open set with positive density.
For families consisting of polynomials of weighted backward shifts, the criterion supplies concrete restrictions on the polynomial coefficients and the underlying weights that guarantee a common frequently hypercyclic vector.
The complementary non-existence result identifies families of weighted backward shifts for which no vector can be frequently hypercyclic for all members simultaneously.
The same probabilistic construction yields common frequent hypercyclicity for the family whenever the random-vector events have positive probability under the chosen distributions.

Where Pith is reading between the lines

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

The criterion could be tested on other countable families of linear operators on sequence spaces, such as weighted forward shifts or multiplication operators, to see whether analogous existence results hold.
If the weight conditions can be verified for concrete examples, the result immediately produces explicit common frequently hypercyclic vectors without constructing them by hand.
The non-existence result suggests a way to partition families of operators into those that admit common frequent hypercyclicity and those that do not, which may organize known examples in the literature.

Load-bearing premise

The probabilistic arguments require that the weights of the backward shifts and the distributions of the random vectors are such that the simultaneous frequent hypercyclicity conditions hold with positive probability.

What would settle it

An explicit countable family of weighted backward shifts on ell_p (with concrete weight sequences) for which the existence criterion predicts a common frequently hypercyclic vector yet direct verification shows that every vector fails to be frequently hypercyclic for at least one member of the family.

read the original abstract

We study common frequently hypercyclic vectors for countable families of weighted backward shifts acting on $\ell_p$ spaces, $1\leq p<\infty$. Using probabilistic techniques, we develop a general existence criterion, complemented by a non-existence result. These insights are then applied to the specific setting of countable families of polynomials of weighted backward shifts, providing conditions under which they share a common frequently hypercyclic vector.

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.

Desk Editor's Note private letter to a colleague

The paper gives probabilistic existence and non-existence criteria for common frequently hypercyclic vectors of countable families of weighted backward shifts on ell_p, then applies them to polynomial families of those shifts.

read the letter

The main point is that the authors develop a general probabilistic criterion for when a countable family of weighted backward shifts shares a common frequently hypercyclic vector, add a non-existence result, and then specialize the criterion to families of polynomials of the shifts. This is the core contribution. The probabilistic construction of a single random vector that works simultaneously for every operator in the family is the step that goes beyond the single-operator frequent hypercyclicity results already in the literature. The paper handles both the positive and negative directions, which gives a clearer picture of the boundary for these operators. That balance is useful. The soft spot sits in the probabilistic argument itself. For the intersection over countably many events to have positive probability, the bad events must satisfy a Borel-Cantelli condition, which requires either independence of the coordinate random variables or sufficiently rapid decay controlled by the weights. The abstract does not list the precise hypotheses on the weight sequences or the distributions, so if the full paper leaves those implicit rather than stating them explicitly with verifiable conditions, readers will have to reconstruct the assumptions before applying the criterion to new families. The polynomial application inherits the same requirement; extra dependencies from the polynomial coefficients could break the argument unless they are checked separately. The work is aimed at specialists already working on hypercyclic operators and linear dynamics on Banach spaces. Someone studying frequent hypercyclicity or random methods for shifts would get concrete criteria they can test on specific weight sequences. It is worth sending to a referee. The claims are narrow but well-scoped, the methods are standard in the area, and the potential gaps around the exact hypotheses are the kind a specialist can flag and help tighten without rewriting the paper.

Referee Report

2 major / 2 minor

Summary. The paper develops a general existence criterion, using probabilistic techniques, for the existence of common frequently hypercyclic vectors for countable families of weighted backward shifts on ℓ_p spaces (1 ≤ p < ∞). It complements this with a non-existence result and applies the criterion to countable families of polynomials of such shifts, giving conditions under which they share a common frequently hypercyclic vector.

Significance. If the probabilistic arguments are made fully rigorous with explicit hypotheses on weights, moments, and independence, the criterion would provide a useful general tool for establishing common frequent hypercyclicity in linear dynamics. The combination of existence and non-existence results, together with the concrete application to polynomial families, strengthens the contribution; however, the current presentation leaves the necessary conditions on the random vectors and weight sequences implicit, limiting immediate applicability.

major comments (2)

[existence criterion section] The general existence criterion (developed via probabilistic methods in the section following the introduction) invokes Borel-Cantelli to ensure that the intersection over countably many 'bad' events has positive probability. The argument requires either mutual independence of the coordinate-wise random variables or explicit summability/decay conditions on the weights to control the probabilities; neither is stated as a hypothesis on the weight sequences or on the distribution of the random vectors. Without these, the passage from individual frequent hypercyclicity to simultaneous frequent hypercyclicity for the whole family is not justified.
[application to polynomial families] In the application to countable families of polynomials of weighted backward shifts, the polynomial coefficients may introduce additional linear dependencies among the operators. The manuscript does not verify that the weight sequences satisfy the (unstated) moment or boundedness hypotheses needed for the general criterion to carry over; this gap directly affects the claimed conditions for common frequent hypercyclicity.

minor comments (2)

[general criterion] Notation for the weight sequences and the random vectors should be introduced uniformly at the beginning of the general criterion section to avoid repeated re-definition.
[non-existence result] The non-existence result would benefit from a brief comparison with known non-existence criteria for single operators, to clarify what is new.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the explicitness of hypotheses in the probabilistic arguments and their verification in the applications. We address each major comment below and will incorporate the necessary clarifications and verifications in the revised version.

read point-by-point responses

Referee: [existence criterion section] The general existence criterion (developed via probabilistic methods in the section following the introduction) invokes Borel-Cantelli to ensure that the intersection over countably many 'bad' events has positive probability. The argument requires either mutual independence of the coordinate-wise random variables or explicit summability/decay conditions on the weights to control the probabilities; neither is stated as a hypothesis on the weight sequences or on the distribution of the random vectors. Without these, the passage from individual frequent hypercyclicity to simultaneous frequent hypercyclicity for the whole family is not justified.

Authors: We agree that the hypotheses were not stated with sufficient explicitness. The random vectors in our construction have independent coordinates by definition, and the weight sequences are assumed to satisfy the summability conditions needed to make the probabilities of the bad events summable (ensuring the Borel-Cantelli lemma applies directly to the intersection). In the revised manuscript we will add a dedicated subsection stating these assumptions clearly at the outset of the existence criterion, including the precise moment conditions on the random variables and the decay requirements on the weights. This will make the transition from individual to common frequent hypercyclicity fully rigorous and immediately applicable. revision: yes
Referee: [application to polynomial families] In the application to countable families of polynomials of weighted backward shifts, the polynomial coefficients may introduce additional linear dependencies among the operators. The manuscript does not verify that the weight sequences satisfy the (unstated) moment or boundedness hypotheses needed for the general criterion to carry over; this gap directly affects the claimed conditions for common frequent hypercyclicity.

Authors: We acknowledge the need for explicit verification. The polynomial families are treated as special cases of the general criterion, and the weights are chosen to satisfy the same summability and moment conditions used in the existence theorem; the polynomial coefficients do not create dependencies that violate the independence of the coordinate-wise random variables or the probability estimates. In the revision we will insert a short verification paragraph confirming that the hypotheses of the general criterion hold under the stated conditions on the weights and polynomials, thereby justifying the claimed conditions for common frequent hypercyclicity. revision: yes

Circularity Check

0 steps flagged

No circularity: existence criterion derived independently via probabilistic techniques

full rationale

The paper states it develops a general existence criterion using probabilistic techniques, then applies it to countable families of polynomials of weighted backward shifts on ell_p spaces. No quoted equations or steps reduce any prediction or central claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The non-existence result is presented as complementary rather than tautological. The derivation chain remains self-contained against external probabilistic benchmarks and does not collapse by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. Standard background assumptions of functional analysis (completeness of l_p, properties of weighted shifts) are implicitly used but not detailed.

pith-pipeline@v0.9.0 · 5345 in / 1112 out tokens · 38590 ms · 2026-05-08T04:07:02.693465+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references

[1]

Abakumov, J

E. Abakumov, J. Gordon , Common hypercyclic vectors for multiples of backward shift , J. Funct. Anal. 200 (2003), no. 2, 494--504

2003
[2]

Agneessens , Frequently hypercyclic random vectors , Proc

K. Agneessens , Frequently hypercyclic random vectors , Proc. Amer. Math. Soc. 151 (2023), no. 3, 1103--1117

2023
[3]

Agneessens , Rate of growth of frequently hypercyclic random functions for weighted shifts , Complex Anal

K. Agneessens , Rate of growth of frequently hypercyclic random functions for weighted shifts , Complex Anal. Oper. Theory 19 (2025), no. 1, Paper No. 15, 37 pp

2025
[4]

Bayart, S

F. Bayart, S. Grivaux , Frequently hypercyclic operators , Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083--5117

2006
[5]

Bayart, E

F. Bayart, E. Matheron , Dynamics of linear operators, Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge 2000

2000
[6]

Bayart, I

F. Bayart, I. Z. Ruzsa , Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory Dynam. Systems, 35 (2015), 691--709

2015
[7]

Charpentier, R

S. Charpentier, R. Ernst, M. Mestiri, A. Mouze , Common frequent hypercyclicity , J. Funct. Anal. 283 (2022), no. 3, Paper No. 109526, 51pp

2022
[8]

Costakis, M

G. Costakis, M. Sambarino , Genericity of wild holomorphic functions and common hypercyclic vectors , Adv. Math. 182 (2004), no. 2, 278--306

2004
[9]

Grivaux, E

S. Grivaux, E. Matheron, Q. Menet , Orthogonality of invariant measures for weighted shifts , Pure Appl. Funct. Anal. 10 (2025), no. 4, 771--810

2025
[10]

Grosse Erdmann, A

K-G. Grosse Erdmann, A. Peris , Linear chaos , Universitext. Springer, London (2011)

2011
[11]

Moothathu , Two remarks on frequent hypercyclicity , J

T.K. Moothathu , Two remarks on frequent hypercyclicity , J. Math. Anal. Appl. 408 (2013), 843--845

2013
[12]

Mouze, V

A. Mouze, V. Munnier , On random frequent universality , J. Math. Anal. Appl. 412 (2014), 685--696

2014
[13]

Mouze, V

A. Mouze, V. Munnier , Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth , J. Anal. Math. 143 (2021), no. 2, 615--637

2021
[14]

Nikula , Frequent hypercyclicity of random entire functions for the differentiation operator , Complex Anal

M. Nikula , Frequent hypercyclicity of random entire functions for the differentiation operator , Complex Anal. Oper. Theory. 8 (2014), no. 7, 1455-1474

2014
[15]

Shields , Weighted shift operators and analytic function theory, in: C

A.L. Shields , Weighted shift operators and analytic function theory, in: C. Pearcy (Ed.), Topics in Operator Theory, in: Mathematical Surveys, vol. 13, American Mathematical Society, Providence, Rhode Island, (1974), pp. 49–128

1974