Skip to content
DraconiticResearch
Search Sources API
186,546 studies31 sourcesUpdated 54 min ago
PreprintarXiv licencearXiv

Quantitative Rigidity of pseudo-rotations on the two-torus and Sarnak's conjecture

  • Yinshan Chang
  • Jian Wang
  • Junchang Zhou
Open at sourcePDFDOICite

Abstract

We establish quantitative rigidity results for pseudo-rotations of the two-torus under a (C,δ)-deviation condition relative to their rotation vectors. The main ingredient is a quantitative free-disk estimate that converts bounds on orbit deviation into explicit control of the distance between the iterates and the identity map. Under such a (C,δ)-deviation condition, we show that Hölder continuous super-Liouvillean irrational pseudo-rotations are C0-rigid with an exponential decay rate and that Ck semi-irrational pseudo-rotations of strong non-Brjuno type exhibit Ck−1-rigidity with a superpolynomial decay rate. Moreover, under this deviation condition and sufficiently large irrationality measure, we show that Hölder continuous skew products on 𝕋2 over circle rotations are C0-rigid with a polynomial decay rate. As a consequence, all these classes satisfy Sarnak's conjecture.

Related

  1. Analytic pseudo-rotations
    Pierre Berger
    Annals of MathematicsPeer-reviewedLicence not statedFull text5 citedMathematical physicsOpenAlex
  2. On Sarnak--Strömbergsson conjecture
    Senping Luo, Juncheng Wei
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  3. Non-rigid disk pseudo-rotations in C1 and C1,β
    Ziran Liu, Jian Wang, Peizheng Yu
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  4. Quantitative Rigidity of Ricci Nonnegative Manifolds via the Regular Green Constant
    Zhelei Huang, Bijia Luo, Jie Zhou
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  5. On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces
    Zakariae Aznay, Fadil Chabbabi
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{chang2026quantitative,
  title = {{Quantitative Rigidity of pseudo-rotations on the two-torus and Sarnak's conjecture}},
  author = {Yinshan Chang and Jian Wang and Junchang Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.26976},
  eprint = {2608.26976},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26976},
}
RIS
TY  - UNPB
TI  - Quantitative Rigidity of pseudo-rotations on the two-torus and Sarnak's conjecture
AU  - Yinshan Chang
AU  - Jian Wang
AU  - Junchang Zhou
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2608.26976
UR  - https://arxiv.org/abs/2608.26976
AB  - We establish quantitative rigidity results for pseudo-rotations of the two-torus under a (C,δ)-deviation condition relative to their rotation vectors. The main ingredient is a quantitative free-disk estimate that converts bounds on orbit deviation into explicit control of the distance between the iterates and the identity map. Under such a (C,δ)-deviation condition, we show that Hölder continuous super-Liouvillean irrational pseudo-rotations are C^0-rigid with an exponential decay rate and that C^k semi-irrational pseudo-rotations of strong non-Brjuno type exhibit C^k−1-rigidity with a superpolynomial decay rate. Moreover, under this deviation condition and sufficiently large irrationality measure, we show that Hölder continuous skew products on 𝕋^2 over circle rotations are C^0-rigid with a polynomial decay rate. As a consequence, all these classes satisfy Sarnak's conjecture.
ER  -