A correspondence between surjective local homeomorphisms and a family of separated graphs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913229993672704 |
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| author | Ara, Pere Claramunt, Joan |
| author_facet | Ara, Pere Claramunt, Joan |
| contents | We present a graph-theoretic model for dynamical systems $(X,σ)$ given by a surjective local homeomorphism $σ$ on a totally disconnected compact metrizable space $X$. In order to make the dynamics appear explicitly in the graph, we use two-colored Bratteli separated graphs as the graphs used to encode the information. In fact, our construction gives a bijective correspondence between such dynamical systems and a subclass of separated graphs which we call $l$-diagrams. This construction generalizes the well-known shifts of finite type, and leads naturally to the definition of a generalized finite shift. It turns out that any dynamical system $(X,σ)$ of our interest is the inverse limit of a sequence of generalized finite shifts. We also present a detailed study of the corresponding Steinberg and $C^*$ algebras associated with the dynamical system $(X,σ)$, and we use the above approximation of $(X,σ)$ to write these algebras as colimits of the associated algebras of the corresponding generalized finite shifts, which we call generalized finite shift algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03807 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A correspondence between surjective local homeomorphisms and a family of separated graphs Ara, Pere Claramunt, Joan Operator Algebras Dynamical Systems Rings and Algebras We present a graph-theoretic model for dynamical systems $(X,σ)$ given by a surjective local homeomorphism $σ$ on a totally disconnected compact metrizable space $X$. In order to make the dynamics appear explicitly in the graph, we use two-colored Bratteli separated graphs as the graphs used to encode the information. In fact, our construction gives a bijective correspondence between such dynamical systems and a subclass of separated graphs which we call $l$-diagrams. This construction generalizes the well-known shifts of finite type, and leads naturally to the definition of a generalized finite shift. It turns out that any dynamical system $(X,σ)$ of our interest is the inverse limit of a sequence of generalized finite shifts. We also present a detailed study of the corresponding Steinberg and $C^*$ algebras associated with the dynamical system $(X,σ)$, and we use the above approximation of $(X,σ)$ to write these algebras as colimits of the associated algebras of the corresponding generalized finite shifts, which we call generalized finite shift algebras. |
| title | A correspondence between surjective local homeomorphisms and a family of separated graphs |
| topic | Operator Algebras Dynamical Systems Rings and Algebras |
| url | https://arxiv.org/abs/2303.03807 |