{
  "abstract": "This paper generalizes previous results to any Parry polynomials $p_n(z)$ \\cite{Pa} of finite degree $n\\geq 2$. Let $β>1$ be the unique positive root of $p_n$ and consider the transfer operator $\\mathscr{P}$ induced by the map $[0,1]\\ni x\\mapsto βx-\\lfloor βx\\rfloor \\in [0,1)$. Suppose all other roots of $p_n$ have magnitude less than one. Then for any smooth function $F$ on $[0,1]$ we prove two term asymptotics in $L^\\infty$ of the form \\begin{equation*} \\big(\\mathscr{P}^r(F)\\big)(\\cdot)= u(\\cdot) \\int_0^1 F(x)\\: \\mathrm{d} x +β^{-r}v(\\cdot) \\big (F(1)-F(0)\\big )+o(β^{-r}),\\quad r\\geq 1, \\end{equation*} where $u$ is piecewise constant and $\\mathscr{P}u=u$, while $v$ is piecewise equal to a polynomial of degree one and $\\mathscr{P}v= β^{-1}v$. When $\\mathscr{P}$ is restricted to functions of bounded variation $BV([0,1])$, it is known that $\\vert z \\vert\\leq β^{-1}$ contains its essential spectrum. Thus the contribution coming from $v$ in the above estimate is not guaranteed if $F$ is not smooth enough. Furthermore, we show that when $\\mathscr{P}$ is restricted to $BV([0,1])$ then any $\\vert z \\vert<β^{-1}$ is an eigenvalue with infinite multiplicity. Moreover, we construct a meromorphic extension of $(\\mathscr{P}-z)^{-1}$ to the domain $\\vert z \\vert>β^{-1}$, where the finite rank poles are numbers $ζ$ with $β^{-1}<\\vert ζ\\vert\\leq 1$ such that $βζ$ is a root of $p_n(z)$. If such poles $ζ$ other than $1$ exist, then we construct eigenvectors of $\\mathscr{P}^*$ in $BV^*$ obeying $\\mathscr{P}^*\\hat{U}=ζ\\hat{U}$, where $\\hat{U}(F)$ is given by series involving the values of $F$ taken in some specially chosen, finer and finer partitions of $[0,1]$. When $F$ is either a polynomial or a characteristic function like $χ_{[0,x]}$, then $\\hat{U}(F)$ can be explicitly computed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27844",
  "author_total": 3,
  "authors": [
    {
      "name": "Horia D. Cornean"
    },
    {
      "name": "Ira W. Herbst"
    },
    {
      "name": "Kasper S. Sørensen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fykrpg56g642p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27844",
  "fulltext_url": null,
  "id": "fykrpg56g642p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27844",
  "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"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223096758",
  "pdf_url": "https://arxiv.org/pdf/2610.27844",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fykrpg56g642p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27844",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27844"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators",
  "title_text": "Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/fykrpg56g642p",
  "venue": "arXiv",
  "versions": []
}