Given a semigroup of local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. Under suitable hypotheses we may also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymor…

2006-10-10math.DSmath.OA

Continuous orbit equivalence of semigroup actions

In this paper, we consider semigroup actions of discrete countable semigroups on compact spaces by surjective local homeomorphisms. We introduce notions of continuous one-sided orbit equivalence and continuous orbit equivalence for semigroup actions, and characterize them in terms of the corresponding semi-groupoids and transformation groupoids respectively. Finally, we consider the case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of group actions.

2021-09-27math.DSmath.OA

Groupoid models for the C*-algebra of labelled spaces

We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.

2018-11-30math.OA

交换局部同胚的C*-代数的理想结构

Ideal structure of C*-algebras of commuting local homeomorphisms

该研究利用平展群胚中的双截和谐族概念，确定了含所有2-图C*-代数的一大类可分群胚C*-代数的本原理想空间与理想格，统一了相关领域的已知结果。

2023-12-17math.OA

The equivalence relations of local homeomorphisms and Fell algebras

We study the groupoid C*-algebras associated to the equivalence relation induced by a quotient map on a locally compact Hausdorff space. This C*-algebra is always a Fell algebra, and if the quotient space is Hausdorff, it is a continuous-trace algebra. We show that the C*-algebra of a locally compact, Hausdorff and principal groupoid is a Fell algebra if and only if the groupoid is one of these relations, extending a theorem of Archbold and Somerset about etale groupoids. The C*-algebras of these relations are, up to Morita equivalence, precisely the Fell algebras with trivial Dixmier-Douady i…

2012-07-11math.OA

KMS states on $C^$-algebras associated to a family of $$-commuting local homeomorphisms

We consider a family of $*$-commuting local homeomorphisms on a compact space, and build a compactly aligned product system of Hilbert bimodules (in the sense of Fowler). This product system has a Nica-Toeplitz algebra and a Cuntz-Pimsner algebra. Both algebras carry a gauge action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus. We study the KMS states of these dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica-Toeplitz algebra. To study KMS states for sm…

2018-04-17math.OA

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KMS states on C*-algebras associated to local homeomorphisms

For every Hilbert bimodule over a C*-algebra, there are natural gauge actions of the circle on the associated Toeplitz algebra and Cuntz-Pimsner algebra, and hence natural dynamics obtained by lifting these gauge actions to actions of the real line. We study the KMS states of these dynamics for a family of bimodules associated to local homeomorphisms on compact spaces. For inverse temperatures larger than a certain critical value, we find a large simplex of KMS states on the Toeplitz algebra, and we show that all KMS states on the Cuntz-Pimsner algebra have inverse temperature at most this cri…

2014-03-11math.OA

A local characterization for the Cuntz semigroup of AI-algebras

We give a local characterization for the Cuntz semigroup of AI-algebras building upon Shen's characterization of dimension groups. Using this result, we provide an abstract characterization for the Cuntz semigroup of AI-algebras.

2021-02-26math.OA

Local homeomorphisms that *-commute with the shift

Exel and Renault proved that a sliding block code on a one-sided shift space coming from a progressive block map is a local homeomorphism. We provide a counterexample showing that the converse does not hold. We use this example to generalize the notion of progressive to a property of block maps we call weakly progressive, and we prove that a sliding block code coming from a weakly progressive block map is a local homeomorphism. We also introduce the notion of a regressive block map and prove that a sliding block code *-commutes with the shift map if and only if it comes from a regressive block…

2010-10-27math.DSmath.OA

与满射局部同胚和低复杂度子位移相关的维数

Dimensions associated with surjective local homeomorphisms and subshifts with low complexity

本文证明了具有有限左特殊元素的满射非周期子位移的Cuntz-Pimsner代数核维数有限，并推广到局部同胚，证明其相关维数及动态渐近维数均有限。

2023-11-09math.DSmath.OA

Local Unitary Cocycles of E_0-semigroups

This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E_0-semigroup. The gauge group of a spatial E_0-semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E_0-semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k, then the gauge group action is transitive on the set of k+1-tuples of appropriately normalized independent units. An action of the gauge group having this…

2008-05-19math.OA

局部同胚的非游荡集与熵

Nonwandering sets and the entropy of local homeomorphisms

该研究针对局部紧豪斯多夫空间上局部同胚诱导的动力系统，引入非游荡集与游荡集概念，证明系统熵为非游荡集与游荡集闭包上熵的最大值，并给出子移位等应用示例。

2024-10-10math.DSmath.OA

通过群胚与 C*-代数研究局部同胚的共轭性

Conjugacy of local homeomorphisms via groupoids and C*-algebras

本文利用群胚与 C*-代数的同构刻画局部紧豪斯多夫空间上部分定义局部同胚系统的拓扑共轭性，推广了 Matsumoto 等人的近期结果。

2022-08-09math.DSmath.OA

Entropy of local homeomorphisms with applications to infinite alphabet shift spaces

In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu-Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew's definition of entropy via covers and Bowen's definition of entropy via separated sets. We propose a definition of factor map between Deaconu-Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bo…

2023-01-23math.DSmath.OA

局部同胚的动力学比较

Dynamical comparison for local homeomorphisms

Universität Münster（明斯特大学）

本文证明了紧可度量化空间上极小满射非单射局部同胚对应的Deaconu–Renault群胚的动力学比较性质，由此得到相关C*-代数的性质、验证了Matui AH猜想，还证明了极小群作用的经典小边界性质。

2026-10-16math.DSmath.OA

Sturmian subshifts and their C*-algebras

This paper investigates the structure of C*-algebras built from one-sided Sturmian subshifts. They are parametrised by irrationals in the unit interval and built from a local homeomorphism associated to the subshift. We provide an explicit construction and description of this local homeomorphism. The C*-algebras are *-isomorphic exactly when the systems are conjugate, and they are Morita equivalent exactly when the defining irrationals are equivalent (this happens precisely when the systems are flow equivalent). Using only elementary dynamical tools, we compute the dynamic asymptotic dimension…

2021-07-22math.DSmath.OA

满射局部同胚与一族分离图之间的对应关系

A correspondence between surjective local homeomorphisms and a family of separated graphs

本文通过双色Bratteli分离图建立满射局部同胚动力系统与分离图子类（l-图）的双射对应，推广有限型移位，并将相关Steinberg与C*代数表示为广义有限移位代数的余极限。

2024-01-17math.DSmath.OAmath.RA