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PreprintCC BY 4.0arXiv

The conjugacy and flip conjugacy problem for Cantor minimal systems

  • Konrad Deka
  • Kosma Kasprzak
  • Philipp Kunde
  • Dominik Kwietniak
  • Felipe García-Ramos
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Abstract

We prove that topological conjugacy and flip conjugacy of minimal homeomorphisms of a Cantor space are complete analytic relations and hence are not Borel. We also prove that mutual reducibility by injective continuous graph homomorphisms and topological graph isomorphism are complete analytic relations on the space of nonempty compact graphs on a fixed Cantor vertex space with continuous chromatic number two. Both relations remain complete analytic on the larger space of nonempty compact graphs on the same Cantor vertex space with continuous chromatic number two or three.

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Cite

BibTeXRIS
BibTeX
@misc{deka2026conjugacy,
  title = {{The conjugacy and flip conjugacy problem for Cantor minimal systems}},
  author = {Konrad Deka and Kosma Kasprzak and Philipp Kunde and Dominik Kwietniak and Felipe García-Ramos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14656},
  eprint = {2609.14656},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14656},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The conjugacy and flip conjugacy problem for Cantor minimal systems
AU  - Konrad Deka
AU  - Kosma Kasprzak
AU  - Philipp Kunde
AU  - Dominik Kwietniak
AU  - Felipe García-Ramos
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14656
UR  - https://arxiv.org/abs/2609.14656
AB  - We prove that topological conjugacy and flip conjugacy of minimal homeomorphisms of a Cantor space are complete analytic relations and hence are not Borel. We also prove that mutual reducibility by injective continuous graph homomorphisms and topological graph isomorphism are complete analytic relations on the space of nonempty compact graphs on a fixed Cantor vertex space with continuous chromatic number two. Both relations remain complete analytic on the larger space of nonempty compact graphs on the same Cantor vertex space with continuous chromatic number two or three.
ER  -