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

Pere Ara; Joan Claramunt

doi:10.3934/dcds.2023143

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2024, Volume 44, Issue 5: 1178-1266. Doi: 10.3934/dcds.2023143

This issue Previous Article Transmission of fast solitons for the NLS with an external potential Next Article Response solutions for finite smooth harmonic oscillators with quasi-periodic forcing

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

Pere Ara^{1, 2,} and
Joan Claramunt^3, ,

1.
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Spain
2.
Centre de Recerca Matemàtica, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Spain
3.
Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Av. de la Universidad 30, 28911 Leganés (Madrid), Spain

^*Corresponding author: Joan Claramunt
^*Corresponding author: Joan Claramunt

Received: April 2023

Revised: October 2023

Early access: December 2023

Published: May 2024

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Abstract

We present a graph-theoretic model for dynamical systems $(X, \sigma)$ given by a surjective local homeomorphism $\sigma$ 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 $(X, \sigma)$ 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 $(X, \sigma)$ , and we use the above approximation of $(X, \sigma)$ to write these algebras as colimits of the associated algebras of the corresponding generalized finite shifts, which we call generalized finite shift algebras.

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Mathematics Subject Classification: Primary: 37B10, 46L55; Secondary: 16S88.

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The first three levels of an $l$ -diagram

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Construction of the edges of $(F, D)$

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First two levels of the $l$ -diagram associated with $(X, \varphi)$

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First two levels of the $l$ -diagram associated with $(X^+, \sigma)$

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Construction of the map $\sigma$

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The generalized finite shift $(E, C)$

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Relation between $(\overline{e}, e)$ and $\overline{e}'$ via red edges

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Visual explanation of the formula $D_{2j}A_{2j} = P_{2j+2} A_{2j+2}P_{2j+1}^T$

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The generalized finite shift graph $(E, C)$

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The different possible local configurations for the graph $(E, C)$

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Description of the shift map $\sigma$ on $X \subseteq \Omega(E, C)$ . Here the configuration $\xi$ is given by $\xi = \{1, \alpha_0^{-1}, \beta_1^{-1}, \beta_0\alpha_0^{-1}, \beta_1\alpha_0^{-1}, \beta_0\beta_1^{-1}, \alpha_0\beta_1^{-1}, \dots\}$ , which has local configuration at $1$ given by $\{\alpha_0^{-1}, \beta_1^{-1}\}$ . The shifted configuration $\sigma(\xi)$ has local configuration at $1$ given by $\{\alpha_0^{-1}, \beta_0^{-1}\}$

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Description of the shift map $\tau$ on $\mathfrak X$ . Here the configuration $\xi$ is given by $\xi = \{1, b, a^{-1}, b^{-1}, ab, a^{-1}b, a^{-2}, b^{-1}a^{-1}, b^{-2}, a^{-1}b^{-1}, \dots\}$ , which has local configuration at $1$ given by $\{b, a^{-1}, b^{-1}\}$ . The shifted configuration $\sigma(\xi)$ has local configuration at $1$ given by $\{a, a^{-1}, b^{-1}\}$ . The picture conveys the same idea of a river basin present for instance in [30,p. 5041]. From any vertex there is only one way forward, and this fact enables us to define the shift map

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