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Joint Point-Distality and structure theory for commuting actions

  • Eli Glasner
  • Chunlin Liu
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Abstract

Let G and H act commutatively and minimally on a compact metrizable space X. We prove that if the two subactions are point-distal, equivalently HPI, then the action generated by them is again point-distal. We also construct a common highly proximal extension on which both subactions are strictly HPI. As consequences, transitivity of mixed products and linear iterates upgrades to minimality in the point-distal category. We then compare the canonical structure of the two subactions with that of the joint action. The maximal highly proximal operations coincide, whereas the maximal joint isometric factor over a common factor is the meet of the two subaction-isometric factors. This gives a recursive description of the joint Furstenberg tower. Finally, we construct commuting minimal distal homeomorphisms of 𝕋3 whose canonical Furstenberg towers are different, showing that the meet formula can be strict.

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Cite

BibTeXRIS
BibTeX
@misc{glasner2026joint,
  title = {{Joint Point-Distality and structure theory for commuting actions}},
  author = {Eli Glasner and Chunlin Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06032},
  eprint = {2609.06032},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06032},
}
RIS
TY  - UNPB
TI  - Joint Point-Distality and structure theory for commuting actions
AU  - Eli Glasner
AU  - Chunlin Liu
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06032
UR  - https://arxiv.org/abs/2609.06032
AB  - Let G and H act commutatively and minimally on a compact metrizable space X. We prove that if the two subactions are point-distal, equivalently HPI, then the action generated by them is again point-distal. We also construct a common highly proximal extension on which both subactions are strictly HPI. As consequences, transitivity of mixed products and linear iterates upgrades to minimality in the point-distal category. We then compare the canonical structure of the two subactions with that of the joint action. The maximal highly proximal operations coincide, whereas the maximal joint isometric factor over a common factor is the meet of the two subaction-isometric factors. This gives a recursive description of the joint Furstenberg tower. Finally, we construct commuting minimal distal homeomorphisms of 𝕋^3 whose canonical Furstenberg towers are different, showing that the meet formula can be strict.
ER  -