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PreprintCC BY 4.0arXiv

On Aubry's completeness conjecture

  • Tianqi Shi
  • Jinxin Xue
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Abstract

In this paper, we prove Aubry's completeness stating conjecture that for a twist map the graph of rotation numbers as a function of the cohomology classes is a purely singularly continuous function (called complete devil's staircase by Aubry) when the set of all minimal configurations is uniformly hyperbolic. Such a phenomenon is crucial for characterizing the chain of atoms being an insulator for the Frenkel-Kontorova model, and can be considered as the analogue of the phase locking phenomenon in critical circle maps as well as the fractional quantum Hall effect. In contrast, in the presence of a positive measure set of KAM tori, we prove that the devil's staircase is incomplete.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026aubrys,
  title = {{On Aubry's completeness conjecture}},
  author = {Tianqi Shi and Jinxin Xue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.00305},
  eprint = {2605.00305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.00305},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Aubry's completeness conjecture
AU  - Tianqi Shi
AU  - Jinxin Xue
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2605.00305
UR  - https://arxiv.org/abs/2605.00305
AB  - In this paper, we prove Aubry's completeness stating conjecture that for a twist map the graph of rotation numbers as a function of the cohomology classes is a purely singularly continuous function (called complete devil's staircase by Aubry) when the set of all minimal configurations is uniformly hyperbolic. Such a phenomenon is crucial for characterizing the chain of atoms being an insulator for the Frenkel-Kontorova model, and can be considered as the analogue of the phase locking phenomenon in critical circle maps as well as the fractional quantum Hall effect. In contrast, in the presence of a positive measure set of KAM tori, we prove that the devil's staircase is incomplete.
ER  -