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Embeddings of Relative Dimension Groups for Zero-Dimensional Dynamical Systems

  • Maryam Hosseini
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Abstract

Let π:(X,T)→(Y,S) be a factor map between zero-dimensional dynamical systems, and let X0 and Y0 be quasi-sections with π(X0)=Y0. We study the homomorphism π*:K0(Y|Y0,S)→ K0(X|X0,T), [f]↦[f∘π], between the corresponding relative dimension groups, and identify it with the homomorphism induced by the premorphism between Bratteli--Vershik realizations associated with π. Generalizing a result of Glasner and Weiss for minimal Cantor systems, we show that if Y0 is quasi-simple, then π* is an injective unital order-preserving homomorphism, and that if X0=π−1(Y0), then π* is a unital order embedding; the latter condition is characterized in terms of the premorphism. Examples show that quasi-simplicity cannot be dropped, and that without the condition X0=π−1(Y0) the map π* need not be an order embedding.

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Cite

BibTeXRIS
BibTeX
@misc{hosseini2026embeddings,
  title = {{Embeddings of Relative Dimension Groups for Zero-Dimensional Dynamical Systems}},
  author = {Maryam Hosseini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28813},
  eprint = {2610.28813},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28813},
}
RIS
TY  - UNPB
TI  - Embeddings of Relative Dimension Groups for Zero-Dimensional Dynamical Systems
AU  - Maryam Hosseini
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28813
UR  - https://arxiv.org/abs/2610.28813
AB  - Let π:(X,T)→(Y,S) be a factor map between zero-dimensional dynamical systems, and let X_0 and Y_0 be quasi-sections with π(X_0)=Y_0. We study the homomorphism π^*:K^0(Y|Y_0,S)→ K^0(X|X_0,T), [f]↦[f∘π], between the corresponding relative dimension groups, and identify it with the homomorphism induced by the premorphism between Bratteli--Vershik realizations associated with π. Generalizing a result of Glasner and Weiss for minimal Cantor systems, we show that if Y_0 is quasi-simple, then π^* is an injective unital order-preserving homomorphism, and that if X_0=π^−1(Y_0), then π^* is a unital order embedding; the latter condition is characterized in terms of the premorphism. Examples show that quasi-simplicity cannot be dropped, and that without the condition X_0=π^−1(Y_0) the map π^* need not be an order embedding.
ER  -