---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '22406'
abstract:
- lang: eng
  text: "It is known that for a uniform morphic sequence \U0001D496 =⟨\U0001D462\U0001D45B⟩∞\r\n\U0001D45B=0
    and an algebraic number \U0001D6FD such that |\U0001D6FD| >1, the number [[\U0001D496]]\U0001D6FD
    :=∑∞\r\n\U0001D45B=0(\U0001D462\U0001D45B/\U0001D6FD\U0001D45B) either lies in
    ℚ⁡(\U0001D6FD) or is transcendental. In this paper, we show a similar rational–transcendental
    dichotomy for sequences defined by irreducible Pisot morphisms on binary alphabets.
    Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete
    spectrum), we generalise the latter result to arbitrary finite alphabets. In certain
    cases, we are able to show transcendence of [[\U0001D496]]\U0001D6FD outright.
    In particular, for \U0001D458 ≥2, if \U0001D496 is the k-Bonacci word, then [[\U0001D496]]\U0001D6FD
    is transcendental."
acknowledgement: "We thank the anonymous referee for identifying an error in an earlier\r\nversion
  of the paper. We gratefully acknowledge support from UKRI Frontier Research\r\nGrant
  EP/X033813/1, ERC grant DynAMiCS (101167561) and DFG grant 389792660 as\r\npart
  of TRR 248. J.O. is also affiliated with Keble College, Oxford as an Emmy Network\r\nfellow."
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Pavol
  full_name: Kebis, Pavol
  id: 2e0132b3-4e98-11ef-b275-cf7281c2802a
  last_name: Kebis
- first_name: FLORIAN
  full_name: LUCA, FLORIAN
  last_name: LUCA
- first_name: JOEL
  full_name: OUAKNINE, JOEL
  last_name: OUAKNINE
- first_name: ANDREW
  full_name: SCOONES, ANDREW
  last_name: SCOONES
- first_name: JAMES
  full_name: WORRELL, JAMES
  last_name: WORRELL
citation:
  ama: Kebis P, LUCA F, OUAKNINE J, SCOONES A, WORRELL J. Transcendence for Pisot
    morphic words over an algebraic base. <i>Ergodic Theory and Dynamical Systems</i>.
    2026:1-22. doi:<a href="https://doi.org/10.1017/etds.2026.10324">10.1017/etds.2026.10324</a>
  apa: Kebis, P., LUCA, F., OUAKNINE, J., SCOONES, A., &#38; WORRELL, J. (2026). Transcendence
    for Pisot morphic words over an algebraic base. <i>Ergodic Theory and Dynamical
    Systems</i>. Cambridge University Press. <a href="https://doi.org/10.1017/etds.2026.10324">https://doi.org/10.1017/etds.2026.10324</a>
  chicago: Kebis, Pavol, FLORIAN LUCA, JOEL OUAKNINE, ANDREW SCOONES, and JAMES WORRELL.
    “Transcendence for Pisot Morphic Words over an Algebraic Base.” <i>Ergodic Theory
    and Dynamical Systems</i>. Cambridge University Press, 2026. <a href="https://doi.org/10.1017/etds.2026.10324">https://doi.org/10.1017/etds.2026.10324</a>.
  ieee: P. Kebis, F. LUCA, J. OUAKNINE, A. SCOONES, and J. WORRELL, “Transcendence
    for Pisot morphic words over an algebraic base,” <i>Ergodic Theory and Dynamical
    Systems</i>. Cambridge University Press, pp. 1–22, 2026.
  ista: Kebis P, LUCA F, OUAKNINE J, SCOONES A, WORRELL J. 2026. Transcendence for
    Pisot morphic words over an algebraic base. Ergodic Theory and Dynamical Systems.,
    1–22.
  mla: Kebis, Pavol, et al. “Transcendence for Pisot Morphic Words over an Algebraic
    Base.” <i>Ergodic Theory and Dynamical Systems</i>, Cambridge University Press,
    2026, pp. 1–22, doi:<a href="https://doi.org/10.1017/etds.2026.10324">10.1017/etds.2026.10324</a>.
  short: P. Kebis, F. LUCA, J. OUAKNINE, A. SCOONES, J. WORRELL, Ergodic Theory and
    Dynamical Systems (2026) 1–22.
das_tickbox: '0'
date_created: 2026-07-27T05:53:25Z
date_published: 2026-07-10T00:00:00Z
date_updated: 2026-08-03T06:17:50Z
day: '10'
ddc:
- '000'
department:
- _id: ToHe
- _id: GradSch
doi: 10.1017/etds.2026.10324
external_id:
  arxiv:
  - '2405.05279'
has_accepted_license: '1'
keyword:
- balanced-pair algorithm
- Cobham’s conjecture
- k-Bonacci words
- Pisot conjecture
- subspace theorem
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1017/etds.2026.10324
mathsc:
- 11J81
- 37B10
- 11J87
month: '07'
oa: 1
oa_version: Published Version
page: 1-22
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
  eissn:
  - 1469-4417
  issn:
  - 0143-3857
publication_status: epub_ahead
publisher: Cambridge University Press
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Transcendence for Pisot morphic words over an algebraic base
tmp:
  image: /images/cc_by.png
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  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
_id: '18112'
abstract:
- lang: eng
  text: 'It is conjectured that the only integrable metrics on the two-dimensional
    torus are Liouville metrics. In this paper, we study a deformative version of
    this conjecture: we consider integrable deformations of a non-flat Liouville metric
    in a conformal class and show that for a fairly large class of such deformations,
    the deformed metric is again Liouville. The principal idea of the argument is
    that the preservation of rational invariant tori in the foliation of the phase
    space forces a linear combination on the Fourier coefficients of the deformation
    to vanish. Showing that the resulting linear system is non-degenerate will then
    yield the claim. Since our method of proof immediately carries over to higher
    dimensional tori, we obtain analogous statements in this more general case. To
    put our results in perspective, we review existing results about integrable metrics
    on the torus.'
acknowledgement: I am very grateful to Vadim Kaloshin for suggesting the topic, his
  guidance during this project, and many helpful comments on an earlier version of
  the manuscript. Moreover, I would like to thank Comlan Edmond Koudjinan and Volodymyr
  Riabov for interesting discussions. Partial financial support by the ERC Advanced
  Grant ‘RMTBeyond’ No. 101020331 is gratefully acknowledged. This project received
  funding from the European Research Council (ERC) ERC Grant No. 885707.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Sven Joscha
  full_name: Henheik, Sven Joscha
  id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
  last_name: Henheik
  orcid: 0000-0003-1106-327X
citation:
  ama: Henheik SJ. Deformational rigidity of integrable metrics on the torus. <i>Ergodic
    Theory and Dynamical Systems</i>. 2025;45(2):467-503. doi:<a href="https://doi.org/10.1017/etds.2024.48">10.1017/etds.2024.48</a>
  apa: Henheik, S. J. (2025). Deformational rigidity of integrable metrics on the
    torus. <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press.
    <a href="https://doi.org/10.1017/etds.2024.48">https://doi.org/10.1017/etds.2024.48</a>
  chicago: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on
    the Torus.” <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University
    Press, 2025. <a href="https://doi.org/10.1017/etds.2024.48">https://doi.org/10.1017/etds.2024.48</a>.
  ieee: S. J. Henheik, “Deformational rigidity of integrable metrics on the torus,”
    <i>Ergodic Theory and Dynamical Systems</i>, vol. 45, no. 2. Cambridge University
    Press, pp. 467–503, 2025.
  ista: Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus.
    Ergodic Theory and Dynamical Systems. 45(2), 467–503.
  mla: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the
    Torus.” <i>Ergodic Theory and Dynamical Systems</i>, vol. 45, no. 2, Cambridge
    University Press, 2025, pp. 467–503, doi:<a href="https://doi.org/10.1017/etds.2024.48">10.1017/etds.2024.48</a>.
  short: S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503.
corr_author: '1'
date_created: 2024-09-22T22:01:43Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2026-07-29T13:18:16Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1017/etds.2024.48
ec_funded: 1
external_id:
  isi:
  - '001308182000001'
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- access_level: open_access
  checksum: 650fe115d998fe0ac3a8d0c7519447c8
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  creator: dernst
  date_created: 2025-01-13T08:51:40Z
  date_updated: 2025-01-13T08:51:40Z
  file_id: '18828'
  file_name: 2025_ErgodicTheory_Henheik.pdf
  file_size: 659100
  relation: main_file
  success: 1
file_date_updated: 2025-01-13T08:51:40Z
has_accepted_license: '1'
intvolume: '        45'
isi: 1
issue: '2'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 467-503
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
  call_identifier: H2020
  grant_number: '885707'
  name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
  eissn:
  - 1469-4417
  issn:
  - 0143-3857
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
related_material:
  record:
  - id: '19540'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Deformational rigidity of integrable metrics on the torus
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2025'
...