Haar decompression and amenability of Ellis flows

[2607.24381] Haar decompression and amenability of Ellis flows Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2607.24381 (math) [Submitted on 27 Jul 2026 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Haar decompression and amenability of Ellis flows Authors: Daniel Max Hoffmann , Krzysztof Krupiński View PDF HTML (experimental) 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 $\tau$-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 $\mu_K$ 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 $K$, $\mu_K$ is $G$-invariant if and only if $(\mathcal{M},G)$ is amenable; when this holds, $\mu_K$ 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. Comments: 39 pages Subjects: Dynamical Systems (math.DS) ; Logic (math.LO) MSC classes: Primary: 37B02, Secondary: 37A15, 54H15 Cite as: arXiv:2607.24381 [math.DS] (or arXiv:2607.24381v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2607.24381 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Daniel Max Hoffmann [ view email ] [v1] Mon, 27 Jul 2026 12:57:42 UTC (56 KB) [v2] Fri, 28 Aug 2026 11:19:10 UTC (61 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-07 Change to browse by: math math.LO References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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