Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0402309

Minimal Dynamical Systems and Approximate Conjugacy

Huaxin Lin·Hiroki Matui
View original source ↗Read source document ↗Export metadata ↓

Abstract

Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via $K$-theory and $C^*$-algebras. For example, it is shown that two Cantor minimal systems are approximately $τ$-conjugate if and only if they are orbit equivalent and have the same periodic spectrum. It is also shown that two such systems are approximately $K$-conjugate if and only if the corresponding crossed product $C^*$-algebras have the same scaled ordered $K$-theory. Consequently, two Cantor minimal systems are approximately $K$-conjugate if and only if the associated transformation $C^*$-algebras are isomorphic. Incidentally, this approximate $K$-conjugacy coincides with Giordano, Putnam and Skau's strong orbit equivalence for the Cantor minimal systems.

Explore related subjects

math.OA

Keep this discovery

BibTeXRIS

Huaxin Lin, Hiroki Matui. 2004-02-19. Minimal Dynamical Systems and Approximate Conjugacy. https://arxiv.org/abs/math/0402309

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the II$_{1}$ Factors of Fuchsian Groups

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, $L(\Gamma)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.

math.OA↗
arXiv2026

On AF- and type I-ideals in certain crossed product C$^\ast$-algebras

We study locally finite-dimensional ideals in crossed products of totally disconnected spaces by free actions of the integers and in uniform Roe algebras of exact discrete groups. In the first case, we present a dynamical description of the largest locally finite-dimensional ideal, which turns out to coincide with the intersection of all maximal ideals. In the latter case, we provide a coarse geometric characterization of the locally finite-dimensional compact ideals. Moreover, we show that for crossed products of totally disconnected spaces by free actions of exact groups, the largest type I-ideal is locally finite-dimensional. In the case of uniform Roe algebras, we provide coarse geometric conditions for compact ideals guaranteeing that the ideal is type I and admits an embedding of a UHF-algebra, respectively.

math.OA↗
arXiv2026

Continuous family of compact quantum metric space structures from cocycle twisted crossed product $\textrm{C}^{\ast}$-algebras

We establish the existence of a three-parameter family of compact quantum metric space structures on cocycle twisted crossed products by discrete groups. We are mainly interested in the case where the acting group has exponential/subexponential growth. We prove that the family is jointly continuous with respect to the parameters when the acting group is exact. We obtain quantitative upper and lower bounds for the associated metric dimensions. In particular, the bounds are helpful to prove the failure of lower semicontinuity of the metric dimension with respect to the quantum Gromov-Hausdorff distance. We also prove invariance of metric dimension under zero quantum Gromov-Hausdorff distance.

math.OA↗