{
  "abstract": "For a permutation of a finite set, we define a landscape by summing a discounted weight over the times each point returns to itself. The landscape value is determined by the point's least period. We relate the trace of the associated resolvent to fixed-point and exact-period counts, and show that the associated determinant is the reciprocal of the Artin-Mazur dynamical zeta function. We apply this framework to the discrete cat map modulo $N$. For every prime $p\\ne5$, all nonzero points have period equal to the order of the cat-map matrix modulo $p$, yielding explicit formulas for the landscape trace and determinant. We prove a period-lifting formula at odd primes $p\\ne5$, derive exact fixed-point and period counts modulo $5^a$, and combine prime-power results for composite moduli using the Chinese remainder theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24857",
  "author_total": 1,
  "authors": [
    {
      "name": "Aryaman Chandra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/88aqdbn6w1ctg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24857",
  "fulltext_url": null,
  "id": "88aqdbn6w1ctg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24857",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7171737408",
  "pdf_url": "https://arxiv.org/pdf/2607.24857",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/88aqdbn6w1ctg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2607.24857",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24857"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Arithmetic Landscape Functions of a Discrete Cat Map",
  "title_text": "Arithmetic Landscape Functions of a Discrete Cat Map",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/88aqdbn6w1ctg",
  "venue": "arXiv",
  "versions": []
}