{
  "abstract": "Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $τ$-topology, the inclusion $K\\hookrightarrow E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $μ_K$ on $E(X,G)$, called its Haar decompression. Haar decompression gives an exact ergodicity description. For an arbitrary flow $(X,G)$, amenability of its Ellis flow is equivalent to hereditary amenability of all finite powers $X^n$. In the tame setting, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. Moreover, for the minimal left ideal $\\mathcal{M}$ in $E(X,G)$ containing $K$, $μ_K$ is $G$-invariant if and only if $(\\mathcal{M},G)$ is amenable; when this holds, $μ_K$ is the unique ergodic measure on $\\mathcal{M}$ and $\\mathcal{M}=\\overline{K}$. Then, we conclude that the ergodic measures of $E(X,G)$ are precisely the Haar decompressions associated with amenable minimal left ideals. We also prove that every minimal tame amenable flow is uniquely ergodic and that every ergodic invariant measure on a tame flow has minimal support. Consequently, every ergodic measure on a tame ambit arises by evaluating a suitable Haar decompression.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24381",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Max Hoffmann"
    },
    {
      "name": "Krzysztof Krupiński"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3a3csswtsyaeb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24381",
  "fulltext_url": null,
  "id": "3a3csswtsyaeb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24381",
  "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"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7171510703",
  "pdf_url": "https://arxiv.org/pdf/2607.24381",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3a3csswtsyaeb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2607.24381",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24381"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Haar decompression and amenability of Ellis flows",
  "title_text": "Haar decompression and amenability of Ellis flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/3a3csswtsyaeb",
  "venue": "arXiv",
  "versions": []
}