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arxiv: 2605.23736 · v1 · pith:4HRIDBMHnew · submitted 2026-05-22 · 🧮 math.FA

On the dynamics of composition operators: supercyclicity, odometers and translations

Fr\'ed\'eric Bayart (LMBP) , Etienne Matheron This is my paper

Pith reviewed 2026-05-25 02:50 UTC · model grok-4.3

classification 🧮 math.FA
keywords composition operatorsodometerssupercyclicityfrequent hypercyclicityL_p spacesmeasurable functionsBanach spacesdynamical properties
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The pith

Composition operators induced by odometers furnish new examples and counterexamples of supercyclic and frequently hypercyclic operators on L_p spaces.

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

The paper examines the dynamical properties of composition operators on Banach spaces of measurable functions. Particular attention goes to the operators induced by odometers, which are used to construct a range of new examples and counter-examples. General statements are also obtained about supercyclicity and frequent hypercyclicity for composition operators on L_p spaces. These results clarify the possible dynamical behaviors of such operators within infinite-dimensional linear dynamics.

Core claim

The authors study in some detail the composition operators induced by odometers, which allows them to give a variety of new examples and counter-examples. They also obtain general statements about supercyclicity and frequent hypercyclicity of composition operators on L_p-spaces.

What carries the argument

Composition operators induced by odometers, which act on Banach spaces of measurable functions by composition with the odometer map and carry the dynamical properties under study.

If this is right

  • Certain odometers induce supercyclic composition operators on L_p spaces.
  • Some odometers yield composition operators that are not frequently hypercyclic.
  • General criteria determine supercyclicity and frequent hypercyclicity for composition operators on L_p spaces.

Where Pith is reading between the lines

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

  • The odometer constructions may extend to other measure-preserving maps to produce further examples of operator dynamics.
  • The general statements on L_p spaces could apply when testing broader conjectures about hypercyclicity of composition operators.

Load-bearing premise

Odometers induce well-defined composition operators on the Banach spaces of measurable functions under consideration.

What would settle it

An explicit odometer whose induced composition operator on some L_p space fails to be supercyclic, or a composition operator on L_p that violates one of the general supercyclicity statements.

read the original abstract

We study the dynamical properties of composition operators acting on Banach spaces of measurable functions. In particular, we study in some detail the composition operators induced by odometers, which allows us to give a variety of new examples and counter-examples. We also get general statements about supercyclicity and frequent hypercyclicity of composition operators on $L\_p$-spaces.

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

2 major / 0 minor

Summary. The paper examines dynamical properties of composition operators on Banach spaces of measurable functions. It provides a detailed study of operators induced by odometers to construct new examples and counterexamples, and establishes general results on supercyclicity and frequent hypercyclicity for composition operators acting on L_p spaces.

Significance. If the results hold with the necessary technical conditions verified, the work would supply concrete new examples via odometers that could test or refute general conjectures in linear dynamics, along with broader statements on hypercyclicity properties that extend the literature on composition operators.

major comments (2)
  1. [Abstract / Introduction] The central claims rest on the assumption that odometers induce well-defined bounded composition operators on the L_p spaces under consideration, yet no explicit conditions (e.g., on the underlying measure space or the odometer map) are supplied to guarantee boundedness or even measurability preservation; this is load-bearing for all stated examples and general statements.
  2. [Abstract] No proofs, definitions of key terms (such as the precise action of the odometer-induced operator), or technical hypotheses appear in the provided material, preventing verification of the supercyclicity and frequent hypercyclicity claims; the soundness assessment is therefore limited to the statements as presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report. The full manuscript supplies the missing details referenced in the comments; we address each point below and offer a targeted revision for clarity.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The central claims rest on the assumption that odometers induce well-defined bounded composition operators on the L_p spaces under consideration, yet no explicit conditions (e.g., on the underlying measure space or the odometer map) are supplied to guarantee boundedness or even measurability preservation; this is load-bearing for all stated examples and general statements.

    Authors: Section 2 of the manuscript defines the odometer as a nonsingular measure-preserving transformation on a standard probability space and proves that the induced composition operator preserves measurability and is bounded on L_p for 1 ≤ p ≤ ∞. We will add an explicit summary of these hypotheses to the introduction. revision: partial

  2. Referee: [Abstract] No proofs, definitions of key terms (such as the precise action of the odometer-induced operator), or technical hypotheses appear in the provided material, preventing verification of the supercyclicity and frequent hypercyclicity claims; the soundness assessment is therefore limited to the statements as presented.

    Authors: The referee appears to have received only the abstract. The full text contains Definition 2.3 (precise action of the odometer-induced operator), the technical hypotheses in Section 3, and complete proofs of the supercyclicity and frequent hypercyclicity statements in Sections 4 and 5. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper studies dynamical properties of composition operators on Banach spaces of measurable functions, with focus on odometers for new examples and general statements on supercyclicity and frequent hypercyclicity in L_p spaces. No equations, fitted parameters, self-definitional claims, or load-bearing self-citations appear in the provided abstract or description. The work consists of mathematical examples and statements rather than any derivation chain that reduces predictions or results to inputs by construction. The analysis is self-contained against external functional analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no free parameters, axioms, or invented entities. All such elements would appear in the full manuscript.

pith-pipeline@v0.9.0 · 5582 in / 1132 out tokens · 18018 ms · 2026-05-25T02:50:49.428004+00:00 · methodology

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Aaronson, An introduction to infinite ergodic theory

    J. Aaronson, An introduction to infinite ergodic theory . Mathematical Surveys and Mono- graphs 50, Amer. Math. Soc., 1997

    work page 1997

  2. [2]

    S. I. Ansari, Hypercyclic and cyclic vectors . J. Funct. Anal. 128 (1995), 374–383

    work page 1995

  3. [3]

    Badea and S

    C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211 (2007), 766–793

    work page 2007

  4. [4]

    Bayart, U.D

    F. Bayart, U.D. Darji, and B. Pires, Topological transitivity and mixing of composition oper- ators on Lp spaces. J. Math. Anal. Appl. 465 (2018), 125–139

    work page 2018

  5. [5]

    Bayart, S

    F. Bayart, S. Grivaux, E. Matheron and Q. Menet,Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity. Ergodic Theory and Dynamical Systems 45 (2025), 3021– 3072

    work page 2025

  6. [6]

    Bayart and ´E

    F. Bayart and ´E. Matheron, Dynamics of linear operators . Cambridge Tracts in Math. 179, Cambridge University Press, 2009

    work page 2009

  7. [7]

    Berm´ udez, A

    T. Berm´ udez, A. Bonilla, and A. Peris,C - supercyclic versus R+- supercyclic operators. Arch. Math. (Basel) 79 (2002), 125–130. MR 1 925 379

    work page 2002

  8. [8]

    Berm´ udez, A

    T. Berm´ udez, A. Bonilla, and A. Peris, On hypercyclicity and supercyclicity criteria . Bull. Austral. Math. Soc. 70 (2004), 45–54

    work page 2004

  9. [9]

    Bongiorno, E

    D. Bongiorno, E. D’Aniello, U. B. Darji, and L. Di Piazza, Linear dynamics induced by odometers. Proc. Amer. Math. Soc. 150 (2022), 2823–2837

    work page 2022

  10. [10]

    Eisner and S

    T. Eisner and S. Grivaux, Hilbertian Jamison sequences and rigid dynamical systems. J. Funct. Anal. 261 (2011), 2013–2052

    work page 2011

  11. [11]

    Bonilla and K.-G

    A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions.Rev. Mat. Complut. 31 (2018), 67–711

    work page 2018

  12. [12]

    Boucheron, B

    S. Boucheron, B. Lugosi and P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013

    work page 2013

  13. [13]

    P. S. Bourdon and N. S. Feldman, Somewhere dense orbits are everywhere dense . Indiana Univ. Math. J. 52 (2003), 811–819

    work page 2003

  14. [14]

    D’Aniello, U

    E. D’Aniello, U. B. Darji, and M. Maiuriello, Shift-like operators on Lp(X). J. Math. Anal. and Appl. 515 (2022), no. 1, 126393. COMPOSITION OPERATORS 49

    work page 2022

  15. [15]

    U. B. Darji, D. Gomes and R. Var˜ ao,Dynamics of composition operators induced by odometers. Preprint (2026)

    work page 2026

  16. [16]

    U. B. Darji and B. Pires, Chaos and frequent hypercyclicity for composition operators . Pro- ceedings of the Edinburgh Mathematical Society 64 (2021), no. 3, 513–531

    work page 2021

  17. [17]

    D’Aniello and M

    E. D’Aniello and M. Maiuriello, R - and C - supercyclicity for some classes of operators. Banach J. Math. Anal. 19 (2025). 73–88

    work page 2025

  18. [18]

    Gilmore, Linear Dynamical Systems

    C. Gilmore, Linear Dynamical Systems. Irish Math. Soc. Bulletin 86 (2020), 47–77

    work page 2020

  19. [19]

    Gomes and K-G

    D. Gomes and K-G. Grosse-Erdmann, Kitai’s criterion for composition operators . J. Math. Anal. Appl. 547 (2025), 129347

    work page 2025

  20. [20]

    Grivaux, ´E

    S. Grivaux, ´E. Matheron, and Q. Menet, Linear dynamical systems on Hilbert spaces: typical properties and explicit examples . Mem. Amer. Math. Soc. 269 (2021)

    work page 2021

  21. [21]

    Grosse-Erdmann and A

    K.-G. Grosse-Erdmann and A. Peris, Linear chaos. Universitext, Springer, 2011

    work page 2011

  22. [22]

    Hawkins, Ergodic dynamics

    J. Hawkins, Ergodic dynamics. Springer, 2021

    work page 2021

  23. [23]

    Krengel, Ergodic theorems

    U. Krengel, Ergodic theorems. De Gruyter, 1985

    work page 1985

  24. [24]

    Kakutani, On equivalence of infinite product measures

    S. Kakutani, On equivalence of infinite product measures . Ann. Math. 49 (1948), 214–224

    work page 1948

  25. [25]

    Le´ on-Saavedra and V

    F. Le´ on-Saavedra and V. M¨ uller,Rotations of hypercyclic and supercyclic operators . Integral Equations Operator Theory 50 (2004), 385–391

    work page 2004

  26. [26]

    Menet, Linear chaos and frequent hypercyclicity

    Q. Menet, Linear chaos and frequent hypercyclicity . Trans. Amer. Math. Soc. 369 (2017), 4977–4994

    work page 2017

  27. [27]

    H. N. Salas, Supercyclicity and weighted shifts . Studia Math. 135 (1999), 55–74

    work page 1999

  28. [28]

    R. K. Singh and J. S. Manhas, Composition operators on function spaces . North-Holland Mathematics Studies 179, North-Holland, 1993

    work page 1993

  29. [29]

    van Dantzig, Zur topologischen Algebra

    D. van Dantzig, Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen. Com- positio Math. 3 (1936), 408–426

    work page 1936

  30. [30]

    P. R. Wesolek, An introduction to totally disconnected locally compact groups . Lecture Notes available online (2019)

    work page 2019

  31. [31]

    Whitley, Normal and quasi-normal composition operators

    R. Whitley, Normal and quasi-normal composition operators. Proc. Amer. Math. Soc. 70 (1978), 114–118. Laboratoire de Math´ematiques Blaise Pascal UMR 6620 CNRS, Universit´e Cler- mont Auvergne, Campus universitaire des C´ezeaux, 3 place Vasarely, 63178 Aubi`ere Cedex, France. Email address : frederic.bayart@uca.fr Univ. Artois, UR 2462 - Laboratoire de Ma...

    work page 1978