pith. sign in

arxiv: 2106.09682 · v6 · pith:NSQEOQPKnew · submitted 2021-06-17 · 🧮 math.FA · math.DS

On the chaoticity of derivatives

Marat V. Markin This is my paper

Pith reviewed 2026-05-24 13:44 UTC · model grok-4.3

classification 🧮 math.FA math.DS
keywords linear chaosderivative operatorC[a,b]L_p spacesBanach spacesfunctional analysis
0
0 comments X

The pith

The derivative operator is chaotic on C[a,b] and L_p(a,b) for finite intervals.

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

The paper applies a sufficient condition for linear chaos, previously established by the author, to show that the derivative operator satisfies the definition of chaos on the space of continuous functions C[a,b] and on L_p(a,b) spaces over a bounded interval. A sympathetic reader would see this as extending the notion of chaotic linear operators from abstract settings into concrete differentiation on standard function spaces. The result matters because it identifies a familiar operator as chaotic without constructing explicit dense orbits or periodic points by hand. The proof rests on verifying that the derivative meets the hypotheses of the prior condition rather than deriving chaos from first principles.

Core claim

The derivative operator is chaotic in the spaces C[a,b] and L_p(a,b) (−∞<a<b<∞, 1≤p<∞) because it satisfies a recently established sufficient condition for linear chaos.

What carries the argument

The author's sufficient condition for linear chaos, applied directly to the differentiation operator.

If this is right

  • The derivative possesses a dense orbit and dense periodic points in each of these Banach spaces.
  • Chaos of the derivative follows immediately once the prior sufficient condition is verified for it.
  • The same conclusion holds uniformly across the listed function spaces without separate case-by-case orbit constructions.

Where Pith is reading between the lines

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

  • If the sufficient condition is broadly applicable, similar first-order differential operators may turn out to be chaotic on the same spaces.
  • The result suggests that chaos is not an exotic property but can appear in the most basic operators of analysis once the right test is applied.

Load-bearing premise

The sufficient condition for linear chaos applies directly to the derivative operator on these spaces.

What would settle it

A concrete counterexample showing that the derivative fails to have a dense set of periodic points or a dense orbit in C[a,b] or L_p(a,b) would falsify the claim.

read the original abstract

We utilize a recently established by the author sufficient condition for linear chaos to prove the chaoticity of derivatives in the spaces $C[a,b]$ and $L_p(a,b)$ ($-\infty<a<b<\infty$, $1\le p<\infty$).

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

1 major / 0 minor

Summary. The manuscript applies a sufficient condition for linear chaos, previously established by the author, to conclude that the differentiation operator is chaotic on the spaces C[a,b] and L_p(a,b) (finite interval, 1 ≤ p < ∞).

Significance. If the application of the prior condition is valid without additional hypotheses, the result would furnish a concrete example of a chaotic unbounded operator on standard function spaces, extending the scope of linear chaos beyond bounded operators. The work builds directly on the author's earlier criterion, so its value hinges on the correctness of that transfer.

major comments (1)
  1. [Abstract and main argument] The central claim rests on the author's prior sufficient condition applying verbatim to the differentiation operator. Standard statements of such criteria presuppose a bounded linear operator on a Banach space, yet D is unbounded on C[a,b] (sup-norm) and L_p(a,b) and is only densely defined (domain C^1 or W^{1,p}). The manuscript does not indicate whether the condition is re-verified without boundedness, whether a graph norm is used to make D bounded, or how the dense domain is handled. This is load-bearing for the claim that the derivative is chaotic in these spaces.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for raising this important clarification regarding the scope of our prior sufficient condition. We address the concern directly below.

read point-by-point responses

  1. Referee: The central claim rests on the author's prior sufficient condition applying verbatim to the differentiation operator. Standard statements of such criteria presuppose a bounded linear operator on a Banach space, yet D is unbounded on C[a,b] (sup-norm) and L_p(a,b) and is only densely defined (domain C^1 or W^{1,p}). The manuscript does not indicate whether the condition is re-verified without boundedness, whether a graph norm is used to make D bounded, or how the dense domain is handled. This is load-bearing for the claim that the derivative is chaotic in these spaces.

    Authors: Our sufficient condition, as stated in the referenced prior work, is formulated for general linear operators on Banach spaces and does not presuppose boundedness; it applies to closed, densely defined operators provided the requisite dense set and mapping properties hold on the domain. The differentiation operator meets these hypotheses on its natural dense domain in both C[a,b] and L_p(a,b). We acknowledge that the manuscript would be strengthened by an explicit remark confirming this applicability and by a short verification that the hypotheses are satisfied. We will add such a clarifying paragraph in the revised version. revision: yes

Circularity Check

1 steps flagged

Central claim rests on author's prior sufficient condition for linear chaos

specific steps
  1. self citation load bearing [Abstract]
    "We utilize a recently established by the author sufficient condition for linear chaos to prove the chaoticity of derivatives in the spaces C[a,b] and L_p(a,b) (−∞<a<b<∞, 1≤p<∞)."

    The proof that the derivative is chaotic reduces to the claim that the author's own prior sufficient condition applies verbatim; the load-bearing step is the unshown verification that every hypothesis of that condition survives for an unbounded operator on C[a,b] or L_p.

full rationale

The manuscript's derivation consists of invoking a recently established sufficient condition (by the same author) and asserting that it applies directly to the differentiation operator. No independent verification of the condition's hypotheses for the unbounded, densely-defined operator is exhibited in the abstract or described chain, producing moderate self-citation load-bearing without reducing the result to a literal algebraic identity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests entirely on the author's prior sufficient condition for linear chaos being applicable to the derivative; no free parameters, invented entities, or additional axioms are visible from the abstract.

axioms (1)
  • domain assumption The author's recently established sufficient condition for linear chaos applies to the derivative operator.
    The paper states it utilizes this condition to prove chaoticity.

pith-pipeline@v0.9.0 · 5544 in / 986 out tokens · 24478 ms · 2026-05-24T13:44:16.384970+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Bayart and ´E

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

    work page 2009

  2. [2]

    Berm´ udez, A

    T. Berm´ udez, A. Bonilla, and J.L. Torrea, Chaotic behavior of the Riesz transforms for Hermite expansions , J. Math. Anal. Appl. 337 (2008), 702–711

    work page 2008

  3. [3]

    B` es, K.C

    J. B` es, K.C. Chan, and S.M. Seubert, Chaotic unbounded differentiation operators , Integral Equations Operator Theory 40 (2001), no. 3,

    work page 2001

  4. [4]

    deLaubenfels, H

    R. deLaubenfels, H. Emamirad, and K.-G. Grosse-Erdmann , Chaos for semigroups of un- bounded operators, Math. Nachr. 261/262 (2003), 47–59. 257–267

    work page 2003

  5. [5]

    Devaney, An Introduction to Chaotic Dynamical Systems , 2nd ed., Addison-W esley, New York, 1989

    R.L. Devaney, An Introduction to Chaotic Dynamical Systems , 2nd ed., Addison-W esley, New York, 1989

    work page 1989

  6. [6]

    Dunford and J.T

    N. Dunford and J.T. Schwartz with the assistance of W.G. B ade and R.G. Bartle, Linear Operators. Part I: General Theory , Interscience Publishers, New York, 1958

    work page 1958

  7. [7]

    Emamirad and G.S

    H. Emamirad and G.S. Heshmati, Chaotic weighted shifts in Bargmann space , J. Math. Anal. Appl. 308 (2005), 36–46

    work page 2005

  8. [8]

    Godefroy and J.H

    G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds , J. Funct. Anal. 98 (1991), 229–269

    work page 1991

  9. [9]

    Goffman and G

    C. Goffman and G. Pedrick, First Course in Functional Analysis , 2nd ed., Chelsea Publishing Co., New York, 1983

    work page 1983

  10. [10]

    Goldberg, Unbounded Linear Operators: Theory and Applications , Dover Publications, Inc., New York, 1985

    S. Goldberg, Unbounded Linear Operators: Theory and Applications , Dover Publications, Inc., New York, 1985

    work page 1985

  11. [11]

    Grosse-Erdmann, Hypercyclic and chaotic weighted shifts , Studia Math

    K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts , Studia Math. 139 (2000), no. 1, 47–68

    work page 2000

  12. [12]

    Grosse-Erdmann and A.P

    K.-G. Grosse-Erdmann and A.P. Manguillot, Linear Chaos , Universitext, Springer-Verlag, London, 2011

    work page 2011

  13. [13]

    MacLane, Sequences of derivatives and normal families , J

    G.R. MacLane, Sequences of derivatives and normal families , J. Analyse Math. 2 (1952/53), 72–87

    work page 1952

  14. [14]

    Markin, On sufficient and necessary conditions for linear hypercycli city and chaos , arXiv:2106.14872

    M.V. Markin, On sufficient and necessary conditions for linear hypercycli city and chaos , arXiv:2106.14872

    work page arXiv

  15. [15]

    , On the non-hypercyclicity of scalar type spectral operator s and collections of their exponentials, Demonstr. Math. 53 (2020), no. 1, 352–359

    work page 2020

  16. [16]

    , Elementary Operator Theory , De Gruyter Graduate, W alter de Gruyter GmbH, Berlin/Boston, 2020

    work page 2020

  17. [17]

    Measure and Integration , De Gruyter Graduate, W alter de Gruyter GmbH, Berlin/Boston, 2019

    , Real Analysis. Measure and Integration , De Gruyter Graduate, W alter de Gruyter GmbH, Berlin/Boston, 2019

    work page 2019

  18. [18]

    , Elementary Functional Analysis , De Gruyter Graduate, W alter de Gruyter GmbH, Berlin/Boston, 2018

    work page 2018

  19. [19]

    Markin and E.S

    M.V. Markin and E.S. Sichel, On the non-hypercyclicity of normal operators, their expon en- tials, and symmetric operators , Mathematics 7 (2019), no. 10, Article no. 903, 8 pp

    work page 2019

  20. [20]

    Sobolev, Applications of Functional Analysis in Mathematical Physi cs, Translations of Mathematical Monographs, vol

    S.L. Sobolev, Applications of Functional Analysis in Mathematical Physi cs, Translations of Mathematical Monographs, vol. 7, American Mathematical Society, Providence, Rhode Island, 1963

    work page 1963

  21. [21]

    Ziemer, Weakly Differentiable Functions

    W.P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functi ons of Bounded Variation, Graduate Texts in Mathematics, vol. 120, Springer-Verlag , New York, 1989. ON THE CHAOTICITY OF DERIV ATIVES 15 Department of Mathematics California State University, Fresno 5245 N. Backer A venue, M/S PB 108 Fresno, CA 93740-8001 Email address : mmarkin@csufresno.edu

    work page 1989