Real-analytic standard families and rotational invariant curves

Abstract

In this paper we study one-parameter families of exact standard maps of the form f̂_k,g(x̂,ŷ) =(x̂+ŷ+kg(x̂) bmod 1,ŷ+kg(x̂)), where k is a real parameter and g:𝕋 → ℝ is real-analytic, 1-periodic and has zero mean. The most famous example is when g(x̂)=sin(2πx̂)/2π. In this case, the closed subset A_{sin(2π·)/(2π)}⊂ ℝ of values of k for which f̂_{k,sin(2π·)/(2π)} has rotational invariant curves is conjectured to be a closed interval. Much has been done towards this conjecture, both numerically and analytically. But a complete proof has not yet been achieved. Our approach here goes in the opposite direction: we construct a function θ:𝕋 → ℝ satisfying the above conditions, such that A_θ is not an interval: It contains at least 2 different connected components, which are non-degenerate closed intervals.

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BibTeX RIS

BibTeX

@misc{addaszanata2026realanalytic,
  title = {{Real-analytic standard families and rotational invariant curves}},
  author = {Salvador Addas-Zanata},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28449},
  eprint = {2610.28449},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28449},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Real-analytic standard families and rotational invariant curves
AU  - Salvador Addas-Zanata
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28449
UR  - https://arxiv.org/abs/2610.28449
AB  - In this paper we study one-parameter families of exact standard maps of the form f̂_k,g(x̂,ŷ) =(x̂+ŷ+kg(x̂) bmod 1,ŷ+kg(x̂)), where k is a real parameter and g:𝕋 → ℝ is real-analytic, 1-periodic and has zero mean. The most famous example is when g(x̂)=sin(2πx̂)/2π. In this case, the closed subset A_sin(2π·)/(2π)⊂ ℝ of values of k for which f̂_k,sin(2π·)/(2π) has rotational invariant curves is conjectured to be a closed interval. Much has been done towards this conjecture, both numerically and analytically. But a complete proof has not yet been achieved. Our approach here goes in the opposite direction: we construct a function θ:𝕋 → ℝ satisfying the above conditions, such that A_θ is not an interval: It contains at least 2 different connected components, which are non-degenerate closed intervals.
ER  -

Abstract

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