Distributional and mean Li-Yorke chaos for weighted shifts on Fr\'echet sequence spaces

Jo\~ao V. A. Pinto

arxiv: 2603.02457 · v2 · submitted 2026-03-02 · 🧮 math.FA · math.DS

Distributional and mean Li-Yorke chaos for weighted shifts on Fr\'echet sequence spaces

Jo\~ao V. A. Pinto This is my paper

Pith reviewed 2026-05-15 16:19 UTC · model grok-4.3

classification 🧮 math.FA math.DS

keywords distributional chaosmean Li-Yorke chaosweighted backward shiftsFréchet sequence spacesKöthe sequence spacesfunctional analysisdynamical systems

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The pith

Weighted backward shifts on Fréchet sequence spaces are distributionally chaotic precisely when their weight sequences satisfy explicit growth conditions.

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

The paper provides characterizations of distributional chaos and mean Li-Yorke chaos specifically for weighted backward shifts on general Fréchet sequence spaces. These characterizations reduce the detection of chaotic behavior to conditions on the sequences of weights that define the operator. Applying the results to Köthe sequence spaces yields concrete criteria in terms of the defining matrix. Sympathetic readers care because such operators model many dynamical systems in infinite dimensions, and simple weight checks make it practical to identify chaotic examples without computing full orbits.

Core claim

A weighted backward shift on a Fréchet sequence space exhibits distributional chaos if and only if the weights allow for sufficiently rapid growth in certain seminorm evaluations along subsequences, and it exhibits mean Li-Yorke chaos under averaged versions of similar conditions. The paper derives these equivalences by relating the chaos definitions to the topology of the space and the action of the shift.

What carries the argument

The weighted backward shift operator, which maps a sequence to the shifted sequence scaled by the weight sequence at each position, acting continuously on the Fréchet topology.

If this is right

Criteria for Köthe spaces λ_p(A,J) follow directly by specializing the general conditions to the seminorms induced by the matrix A.
Both types of chaos can be decided from the behavior of finite products of consecutive weights.
The characterizations hold for both J = N and J = Z, covering one-sided and two-sided sequence spaces.
Mean Li-Yorke chaos is characterized separately but similarly to distributional chaos, using average densities rather than distributional measures.

Where Pith is reading between the lines

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

The same approach might apply to forward shifts or other weighted operators on these spaces.
These criteria could help construct explicit chaotic operators in applications to differential equations or signal processing.
Extensions to non-sequence Fréchet spaces would require different machinery.

Load-bearing premise

The underlying space must be a Fréchet sequence space and the operator a weighted backward shift whose weights meet the precise growth or summability conditions given in the characterizations.

What would settle it

A counterexample would be a weighted backward shift on a Fréchet sequence space where the weight products satisfy the stated inequalities but the orbits fail to be distributionally scrambled, or vice versa.

read the original abstract

In this paper, we give characterizations of distributional chaos and mean Li$-$Yorke chaos for weighted backward shifts acting on general Fr\'echet sequence spaces. As an application, we derive criteria for these two types of chaos in the setting of K\"othe sequence spaces $\lambda_p(A,J)$ for $p \in \{0\}\cup [1, \infty)$ and $J=\mathbb{N}$ or $J=\mathbb{Z}$.

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 necessary and sufficient conditions on the weights for distributional chaos and mean Li-Yorke chaos in backward shifts on Fréchet sequence spaces, with clean specialization to Köthe spaces.

read the letter

This paper gives if-and-only-if characterizations of distributional chaos and mean Li-Yorke chaos for weighted backward shifts on general Fréchet sequence spaces. It then specializes those to criteria on Köthe sequence spaces. The new part is handling the Fréchet case, where the topology comes from a countable family of seminorms rather than a single norm. The authors translate the chaos definitions into conditions on how the weights interact with those seminorms, and they prove both directions with direct estimates on the relevant distribution functions and average distances. That approach works cleanly for the general setting and carries over to the Köthe case without extra work. The proofs appear to rest on standard techniques for shifts, adapted to the seminorm family, so nothing looks forced. The conditions are stated explicitly in terms of limsup or summability involving the weights and the seminorm sequences. One soft spot is that the result stays inside this class of operators and spaces. It organizes the chaos notions for these shifts but does not connect to other types of operators or suggest how the ideas might apply more broadly. The paper is careful not to overclaim. This is for readers who already work with sequence spaces and linear dynamics, particularly those who care about precise criteria for chaos in non-normable topologies. A specialist in the area would find the characterizations useful for checking examples or building further results. I would send it to peer review. The central claims are verifiable through the estimates given, and the specialization to Köthe spaces adds concrete value without stretching the argument.

Referee Report

0 major / 3 minor

Summary. The manuscript gives if-and-only-if characterizations of distributional chaos and mean Li-Yorke chaos for weighted backward shifts on general Fréchet sequence spaces, expressed via growth or summability conditions on the weight sequence relative to the defining seminorms. These equivalences are obtained by direct estimates on the distributional functions and on the mean distances along orbits. The same conditions are then specialized to yield explicit criteria for the Köthe sequence spaces λ_p(A,J) when p ∈ {0} ∪ [1,∞) and J = ℕ or ℤ.

Significance. If the characterizations hold, the paper supplies necessary-and-sufficient conditions for two important forms of chaos on a broad class of non-normable Fréchet spaces, extending earlier Banach-space results. The direct estimates on distributional functions and orbit means constitute a clear technical strength, and the clean specialization to Köthe spaces makes the criteria immediately usable for concrete examples.

minor comments (3)

[Section 2] The definition of the Fréchet sequence space topology (seminorms p_k) is used repeatedly; a single displayed block collecting the standing assumptions on the seminorms would improve readability.
[Section 3] In the statement of the main characterization (likely Theorem 3.1 or 3.2), the phrase “for every ε > 0” appears in both directions; a brief parenthetical remark clarifying that the same ε works for the necessity and sufficiency parts would remove any possible ambiguity.
[Section 4] The application to λ_p(A,J) is stated for J = ℕ and J = ℤ; a short sentence indicating whether the proofs differ materially between the two cases would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives explicit if-and-only-if characterizations of distributional chaos and mean Li-Yorke chaos for weighted backward shifts on Fréchet sequence spaces directly from the operator action on sequences and the standard definitions of the chaos notions, using growth and summability estimates on the weight sequence relative to the seminorms. These equivalences are obtained via direct orbit estimates without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The specialization to Köthe spaces λ_p(A,J) follows by restriction of the same estimates. The derivation chain is therefore self-contained against the input definitions and does not collapse any claimed prediction or uniqueness result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

With only the abstract available, no specific free parameters, axioms, or invented entities can be identified; the work appears to rely on standard definitions of Fréchet and Köthe spaces from prior literature.

pith-pipeline@v0.9.0 · 5367 in / 1153 out tokens · 70113 ms · 2026-05-15T16:19:25.096436+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 12: B_w distributionally chaotic iff (A) dens(D)=1 lim n∈D w_{i-n}⋯w_{i-1}e_{i-n}=0 and (B) seminorm growth card condition on linear combinations
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 30 / 39: mean Li-Yorke chaotic iff liminf average d(w_{-k}⋯w_{-1}e_{-k},0)=0 and mean-sensitivity seminorm lower bound

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Bayart and ´E

F. Bayart and ´E. Matheron,Dynamics of Linear Operators, Cambridge University Press, Cambridge, 2009

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[2]

Bayart and I

F. Bayart and I. Ruzsa,Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory and Dynamical Systems35(2015), no. 3, 691–709. 20

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[3]

Berm´ udez, A

T. Berm´ udez, A. Bonilla, F. Mart´ ınez-Gim´ enez and A. Peris,Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl.373(2011), no. 1, 83–93

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Berm´ udez, A

T. Berm´ udez, A. Bonilla, V. M¨ uller and A. Peris,Ces` aro bounded operators in Banach spaces, J. Anal. Math.140(2020), no. 1, 187–206

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N. C. Bernardes Jr., A. Bonilla, V. M¨ uller and A. Peris,Distributional chaos for linear operators, J. Funct. Anal.265(2013), no. 9, 2143–2163

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[6]

N. C. Bernardes Jr., A. Bonilla, V. M¨ uller and A. Peris,Li-Yorke chaos in linear dynamics, Ergodic Theory and Dynamical Systems35(2015), no. 6, 1723–1745

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[7]

N. C. Bernardes Jr., A. Bonilla and A. Peris,Mean Li-Yorke chaos in Banach spaces, J. Funct. Anal.278(2020), no. 3, Paper No. 108343, 31 pp

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N. C. Bernardes Jr., A. Bonilla, A. Peris and X. Wu,Distributional chaos for operators on Banach spaces, J. Math. Anal. Appl.459(2018), no. 2, 797–821

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N. C. Bernardes Jr., A. Bonilla and J. V. A. Pinto,On the dynamics of weighted composition operators, J. Math. Anal. Appl.555(2026), no. 2, Paper No. 130171, 45 pp

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[10]

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Garcia-Ramos and L

F. Garcia-Ramos and L. Jin,Mean proximality and mean Li-Yorke chaos, Proc. Amer. Math. Soc.145(2017), no. 7, 2959-2969

work page 2017
[12]

Grosse-Erdmann and A

K. Grosse-Erdmann and A. Peris Manguillot,Linear Chaos, Springer, Berlin, 2011

work page 2011
[13]

Huang, J

W. Huang, J. Li and X. Ye,Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal.266(2014), no. 6, 3377-3394

work page 2014
[14]

Jiang and J

Z. Jiang and J. Li,Chaos for endomorphisms of completely metrizable groups and linear operators on Fr´ echet spaces, J. Math. Anal. Appl.543(2025), no. 2, Paper No. 129033, 51 pp

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K¨ othe,Topological Vector Spaces, I, Die Grundlehren der mathematischen Wis- senschaften, Band 159, Springer-Verlag, Berlin - Heidelberg - New York, 1969

G. K¨ othe,Topological Vector Spaces, I, Die Grundlehren der mathematischen Wis- senschaften, Band 159, Springer-Verlag, Berlin - Heidelberg - New York, 1969

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T. Y. Li and J. A. Yorke,Period three implies chaos, Amer. Math. Monthly82(1975), no. 10, 985-992

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[17]

T. Lu, X. Wu and P, Zhu,Uniform distributional chaos for weighted shift operators, Applied Mathematics Letters,26(2013), no. 1, 130-133

work page 2013
[18]

Mart´ ınez-Gim´ enez, P

F. Mart´ ınez-Gim´ enez, P. Oprocha and A. Peris,Distributional chaos for backward shifts, J. Math. Anal. Appl.351(2009), no. 2, 607-615

work page 2009
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Schweizer and J

B. Schweizer and J. Sm´ ıtal,Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc.344(1994), no. 2, 737-754

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Wu and P

X. Wu and P. Zhu,Li–Yorke chaos of backward shift operators on K¨ othe sequence spaces, Topology and its Applications,160(2013), no. 7, 924-929. 21 Jo˜ao V. A. Pinto Departamento de Matem´ atica Aplicada, Instituto de Matem´ atica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, RJ 21941-909, Brazil. e-mail address: joao.pinto@ufu.br 22

work page 2013

[1] [1]

Bayart and ´E

F. Bayart and ´E. Matheron,Dynamics of Linear Operators, Cambridge University Press, Cambridge, 2009

work page 2009

[2] [2]

Bayart and I

F. Bayart and I. Ruzsa,Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory and Dynamical Systems35(2015), no. 3, 691–709. 20

work page 2015

[3] [3]

Berm´ udez, A

T. Berm´ udez, A. Bonilla, F. Mart´ ınez-Gim´ enez and A. Peris,Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl.373(2011), no. 1, 83–93

work page 2011

[4] [4]

Berm´ udez, A

T. Berm´ udez, A. Bonilla, V. M¨ uller and A. Peris,Ces` aro bounded operators in Banach spaces, J. Anal. Math.140(2020), no. 1, 187–206

work page 2020

[5] [5]

N. C. Bernardes Jr., A. Bonilla, V. M¨ uller and A. Peris,Distributional chaos for linear operators, J. Funct. Anal.265(2013), no. 9, 2143–2163

work page 2013

[6] [6]

N. C. Bernardes Jr., A. Bonilla, V. M¨ uller and A. Peris,Li-Yorke chaos in linear dynamics, Ergodic Theory and Dynamical Systems35(2015), no. 6, 1723–1745

work page 2015

[7] [7]

N. C. Bernardes Jr., A. Bonilla and A. Peris,Mean Li-Yorke chaos in Banach spaces, J. Funct. Anal.278(2020), no. 3, Paper No. 108343, 31 pp

work page 2020

[8] [8]

N. C. Bernardes Jr., A. Bonilla, A. Peris and X. Wu,Distributional chaos for operators on Banach spaces, J. Math. Anal. Appl.459(2018), no. 2, 797–821

work page 2018

[9] [9]

N. C. Bernardes Jr., A. Bonilla and J. V. A. Pinto,On the dynamics of weighted composition operators, J. Math. Anal. Appl.555(2026), no. 2, Paper No. 130171, 45 pp

work page 2026

[10] [10]

N. C. Bernardes Jr. and F. M. Vasconcellos,Li-Yorke chaotic weighted composition operators, Rev. Real Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM119(2025), no. 1, Paper No. 13, 22 pp

work page 2025

[11] [11]

Garcia-Ramos and L

F. Garcia-Ramos and L. Jin,Mean proximality and mean Li-Yorke chaos, Proc. Amer. Math. Soc.145(2017), no. 7, 2959-2969

work page 2017

[12] [12]

Grosse-Erdmann and A

K. Grosse-Erdmann and A. Peris Manguillot,Linear Chaos, Springer, Berlin, 2011

work page 2011

[13] [13]

Huang, J

W. Huang, J. Li and X. Ye,Stable sets and mean Li-Yorke chaos in positive entropy systems, J. Funct. Anal.266(2014), no. 6, 3377-3394

work page 2014

[14] [14]

Jiang and J

Z. Jiang and J. Li,Chaos for endomorphisms of completely metrizable groups and linear operators on Fr´ echet spaces, J. Math. Anal. Appl.543(2025), no. 2, Paper No. 129033, 51 pp

work page 2025

[15] [15]

K¨ othe,Topological Vector Spaces, I, Die Grundlehren der mathematischen Wis- senschaften, Band 159, Springer-Verlag, Berlin - Heidelberg - New York, 1969

G. K¨ othe,Topological Vector Spaces, I, Die Grundlehren der mathematischen Wis- senschaften, Band 159, Springer-Verlag, Berlin - Heidelberg - New York, 1969

work page 1969

[16] [16]

T. Y. Li and J. A. Yorke,Period three implies chaos, Amer. Math. Monthly82(1975), no. 10, 985-992

work page 1975

[17] [17]

T. Lu, X. Wu and P, Zhu,Uniform distributional chaos for weighted shift operators, Applied Mathematics Letters,26(2013), no. 1, 130-133

work page 2013

[18] [18]

Mart´ ınez-Gim´ enez, P

F. Mart´ ınez-Gim´ enez, P. Oprocha and A. Peris,Distributional chaos for backward shifts, J. Math. Anal. Appl.351(2009), no. 2, 607-615

work page 2009

[19] [19]

Schweizer and J

B. Schweizer and J. Sm´ ıtal,Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc.344(1994), no. 2, 737-754

work page 1994

[20] [20]

Wu and P

X. Wu and P. Zhu,Li–Yorke chaos of backward shift operators on K¨ othe sequence spaces, Topology and its Applications,160(2013), no. 7, 924-929. 21 Jo˜ao V. A. Pinto Departamento de Matem´ atica Aplicada, Instituto de Matem´ atica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, RJ 21941-909, Brazil. e-mail address: joao.pinto@ufu.br 22

work page 2013