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 of the (groupoid of the) local homeomorphism to be one, and by a result of Guentner, Willett, and Yu, it follows that the nuclear dimension of the C*-algebras is one.

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and \(C^*\)-algebras. This significantly generalises recent work of Matsumoto and of the second- and third-named authors.

We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

Given an ample groupoid \(G\) with compact unit space, we study the canonical representation of the topological full group \([[G]]\) in the full groupoid \(C^*\)-algebra \(C^*(G)\). In particular, we show that the image of this representation generates \(C^*(G)\) if and only if \(C^*(G)\) admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for \(C^*\)-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of nonamenable and countable groups on the Cantor set.

A one-sided shift of finite type \((\mathsf{X}_A, \sigma_A)\) determines on the one hand a Cuntzâ€“Krieger algebra \(\mathcal{O}_A\) with a distinguished abelian subalgebra \(\mathcal{D}_A\) and a certain completely positive map \(\tau_A\) on \(\mathcal{O}_A\). On the other hand, \((\mathsf{X}_A, \sigma_A)\) determines a groupoid \(\mathcal{G}_A\) together with a certain homomorphism \(\varepsilon_A\) on \(\mathcal{G}_A\). We show that each of these two sets of data completely characterizes the one-sided conjugacy class of \(\mathsf{X}_A\). This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

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##### C*-algebras from symbolic dynamical systems, YMC*A, University of Münster, Germany (Aug 2021).

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##### Between symbolic dynamics and C*-algebras, 2 talks, Glasgow Working Seminar, University of Glasgow, United Kingdom (May 2020).

##### The conjugacy problem of symbolic dynamics, an operator algebraic approach OANCG Seminar, University of Wollongong, Australia (Nov 2020).

##### What's your problem? Virtual OANCG Seminar, University of Wollongong, Australia (May 2020).

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##### Investigating symbolic dynamics using C*-algebras, AustMS, Macquarie University, Australia (Dec. 2017).

##### One-sided and two-sided dynamics and Cuntz-Krieger algebras, AMSSC 2017, University of Wollongong, Australia (Dec. 2017).

##### C*-algebras associated to symbolic dynamics, DUC*S, Aarhus University, Denmark (Jan. 2017).

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