Haar decompression and amenability of Ellis flows
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↪ 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 μK 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 Xn. 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 ℳ in E(X,G) containing K, μK is G-invariant if and only if (ℳ,G) is amenable; when this holds, μK is the unique ergodic measure on ℳ and ℳ=K̅. 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.
Related
BibTeX
@misc{hoffmann2026haar,
title = {{Haar decompression and amenability of Ellis flows}},
author = {Daniel Max Hoffmann and Krzysztof Krupiński},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.24381},
eprint = {2607.24381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24381},
}
RIS
TY - UNPB
TI - Haar decompression and amenability of Ellis flows
AU - Daniel Max Hoffmann
AU - Krzysztof Krupiński
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2607.24381
UR - https://arxiv.org/abs/2607.24381
AB - 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↪ 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 ℳ in E(X,G) containing K, μ_K is G-invariant if and only if (ℳ,G) is amenable; when this holds, μ_K is the unique ergodic measure on ℳ and ℳ=K̅. 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.
ER -