Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 12 min ago
PreprintarXiv licencearXiv

The Borel complexity of conjugacy for Cantor minimal systems

  • Xinan Dai
  • Wenhao Deng
  • Yingdong Shi
  • Tailin Wu
  • Yuchen Yang
Open at sourcePDFDOICite

Abstract

We prove that conjugacy of minimal homeomorphisms of the Cantor space is Borel bireducible with isomorphism of countable graphs, answering the Cantor minimal case of a question of Foreman. We obtain the lower bound by encoding countably based profinite groups. Finite quotient homomorphisms are represented by factor maps between a common family of minimal subshifts. Amalgamation makes the resulting inverse limit independent, up to conjugacy, of the quotient presentation. Conversely, finite-stage factorization recovers the group from any conjugacy of these inverse limits.

Related

  1. The conjugacy and flip conjugacy problem for Cantor minimal systems
    Konrad Deka, Kosma Kasprzak, Philipp Kunde +2
    arXivPreprintCC BY 4.0Full textDynamical systemsarXiv
  2. The Complexity of Conjugacy, Orbit Equivalence, and von Neumann Equivalence of Actions of Nonamenable Groups
    Eusebio Gardella, Martino Lupini
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text6 citedMathematical physicsOpenAlex
  3. Borel complexity for product of trees and commuting partial maps
    Koichi Oyakawa
    arXivPreprintarXiv licenceFull textAlgebra & number theoryGeometry & topologyarXiv
  4. Inequivalent complexity criteria for free boundary minimal surfaces
    Alessandro Carlotto, Giada Franz
    Advances in MathematicsPeer-reviewedLicence not statedFull text1 citedGeneral mathematicsHAL
  5. The Complexity of Minimal Positive Realization
    Zhaobo Liu
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv

Cite

BibTeXRIS
BibTeX
@misc{dai2026borel,
  title = {{The Borel complexity of conjugacy for Cantor minimal systems}},
  author = {Xinan Dai and Wenhao Deng and Yingdong Shi and Tailin Wu and Yuchen Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12031},
  eprint = {2609.12031},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12031},
}
RIS
TY  - UNPB
TI  - The Borel complexity of conjugacy for Cantor minimal systems
AU  - Xinan Dai
AU  - Wenhao Deng
AU  - Yingdong Shi
AU  - Tailin Wu
AU  - Yuchen Yang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.12031
UR  - https://arxiv.org/abs/2609.12031
AB  - We prove that conjugacy of minimal homeomorphisms of the Cantor space is Borel bireducible with isomorphism of countable graphs, answering the Cantor minimal case of a question of Foreman. We obtain the lower bound by encoding countably based profinite groups. Finite quotient homomorphisms are represented by factor maps between a common family of minimal subshifts. Amalgamation makes the resulting inverse limit independent, up to conjugacy, of the quotient presentation. Conversely, finite-stage factorization recovers the group from any conjugacy of these inverse limits.
ER  -