{
  "abstract": "For any $t\\in\\mathbb{R}$, consider the Affine twist map $\\rm{Aff_t}:\\mathbb{T}^2\\to\\mathbb{T}^2$ given by $$\\rm{Aff_t}(x,y)=(x+y \\text{ mod 1}, y+t \\text{ mod 1}).$$ The map $\\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of $\\rm{Aff_t}$ to the vertical cylinder is reduced to $\\{t\\}.$ Now, for $k\\in\\mathbb{R},$ define $$f_{k,t}(x,y):=(x+y+k\\sin (2πx)\\text{ mod 1},y+k\\sin (2πx)+t\\text{ mod 1}).$$ We show that: 1) when $t=p/q$ for integers $p$ and $q>0$, KAM theory implies the existence of a constant $k_{p/q}>0$ such that for $|k|<k_{p/q}$, the vertical rotation set of an adequate lift of $f_{k,p/q}$ is just $\\{p/q\\}.$ 2) when $t$ is irrational, for any $k\\neq 0$ the vertical rotation set of the adequate lift of $f_{k,t}$ is a non-degenerate interval which contains $t$ in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \\cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for $f_{k,t}$, for all non-zero values of $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30665",
  "author_total": 1,
  "authors": [
    {
      "name": "Salvador Addas Zanata"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2p7fnwtvcs5mb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30665",
  "fulltext_url": null,
  "id": "2p7fnwtvcs5mb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30665",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214592933",
  "pdf_url": "https://arxiv.org/pdf/2609.30665",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2p7fnwtvcs5mb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30665",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30665"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On Standard perturbations of the Affine twist map",
  "title_text": "On Standard perturbations of the Affine twist map",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2p7fnwtvcs5mb",
  "venue": "arXiv",
  "versions": []
}