{
  "abstract": "Passing from a prime modulus to a higher prime power can change both the lengths and the number of cycles of a polynomial map. We determine these changes for Chebyshev polynomials $T_n$ modulo $p^k$, for every degree $n\\ge2$, every odd prime $p$, and every $k\\ge1$. The results apply whether or not $T_n$ permutes the residue classes. We give explicit formulas for the number of fixed points, all possible cycle lengths, and the number of cycles of each length. The fixed-point formula involves four greatest common divisors and a correction specific to $p=3$. We explain this exception by showing how distinct rational fixed points become congruent modulo $3$. When $p\\mid n$, each cycle modulo $p$ corresponds to exactly one cycle of the same length modulo every $p^k$. When $p\\nmid n$, we determine how many cycles arise from each cycle modulo $p$ and when longer cycles first appear. The proofs combine the classical relation between Chebyshev polynomials and power maps with $p$-adic arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.04417",
  "author_total": 2,
  "authors": [
    {
      "name": "Chatchawan Panraksa"
    },
    {
      "name": "Aram Tangboonduangjit"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/01ywx3pg0tn0j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.04417",
  "fulltext_url": null,
  "id": "01ywx3pg0tn0j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.04417",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7160561739",
  "pdf_url": "https://arxiv.org/pdf/2605.04417",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/01ywx3pg0tn0j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2605.04417",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.04417"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers",
  "title_text": "Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/01ywx3pg0tn0j",
  "venue": "arXiv",
  "versions": []
}