{
  "abstract": "Let $\\mathbb{Z}_p$ be the ring of $p$-adic integers with respect to a prime $p$ and let $d$ be a positive integer. For each $\\mathbf{z}=(z_1, z_2,..., z_d) \\in \\mathbb{Z}_p^d$, let $T_{\\mathbf{z}}: \\mathbb{Z}^d \\times \\mathbb{Z}_p \\to \\mathbb{Z}_p$ be an adding-type $\\mathbb{Z}^d$-action on $\\mathbb{Z}_p$ defined by $T^{\\mathbf{n}}_{\\mathbf{z}}(x):=x+ \\sum_{i=1}^d n_i z_i$ for $\\mathbf{n}=(n_1,n_2, \\cdots, n_d) \\in \\mathbb{Z}^d$ and $x \\in \\mathbb{Z}_p$. Under some mild assumptions on $\\mathbf{z}$, the action $T_{\\mathbf{z}}$ is a free $\\mathbb{Z}^d$-odometer (by odometer, we mean a minimal and equicontinuous action on a Cantor space). In this paper, we derive a necessary condition for continuous orbit equivalence between such $\\mathbb{Z}^d$-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these $\\mathbb{Z}^d$-odometers. It turns out that such a speedup of $T_{\\mathbf{z}}$ is again an adding-type $p$-adic $\\mathbb{Z}^d$-odometer $T_{\\mathbf{w}}$ for some $\\mathbf{w} \\in \\mathbb{Z}_p^d$. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free $\\mathbb{Z}^d$-odometers) which are not continuously orbit equivalent to the original ones and hence disproves a conjecture by Johnson and McClendon. Our result also indicates that continuous orbit equivalence is a rare phenomenon for free minimal constant speedups of $p$-adic $\\mathbb{Z}^d$-odometers of adding type when $d \\geqslant 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01450",
  "author_total": 1,
  "authors": [
    {
      "name": "Changhua Jiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8rcvqhe1akt22.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01450",
  "fulltext_url": null,
  "id": "8rcvqhe1akt22",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01450",
  "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|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7206162932",
  "pdf_url": "https://arxiv.org/pdf/2609.01450",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8rcvqhe1akt22.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01450",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01450"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On free minimal constant speedups violating continuous orbit equivalence in $p$-adic $\\mathbb{Z}^d$-odometers of adding type",
  "title_text": "On free minimal constant speedups violating continuous orbit equivalence in p-adic ℤ^d-odometers of adding type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/8rcvqhe1akt22",
  "venue": "arXiv",
  "versions": []
}