arXiv subjects

Bastián Espinoza

Publications and source records attributed to Bastián Espinoza.

8 recordsLinked to original sources

arXiv2026

Continuous eigenvalues of minimal subshifts via S-adic representations and coboundaries

We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems. These characterizations rely on a description of the system in terms of $S$-adic structures (i.e. infinite compositions of morphisms) satisfying natural mild conditions, such as recognizability and primitivity. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations involve sequences of letter coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between letter coboundaries and extension graphs, and we provide several sets of sufficient conditions ensuring the triviality of letter coboundaries. These results are applied, among other settings, to linear involutions and to the Thue--Morse system in the rational base $3/2$. We also illustrate the versatility of the notion of letter coboundaries in the context of bounded symbolic discrepancy. In particular, we recover a simple characterization of letter balance for primitive substitutive subshifts. Finally, we refine known descriptions of the possible continuous eigenvalues in terms of the measures of the bases of the towers provided by the $S$-adic representation.

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arXiv2026

Topological weak mixing for certain linear involutions of hyperelliptic type

A linear involution is an injective piecewise isometry defined on a pair of disjoint intervals. They are defined by a combinatorial data given by a generalized permutation and a length vector. As it was done for interval exchange transformations, it is conjectured that, except for some combinatorial data, a typical linear involution is measure-theoretically weakly mixing. In this direction, one may first explore the question of topological weak mixing for topological models of linear involutions. In this article we prove topological weak mixing for the natural symbolic codings of typical linear involutions defined by some generalized permutations of hyperelliptic type. We consider the generalized permutation \[ σ_{s,r} = \begin{pmatrix} 0 & A & 1 & 2 & \cdots & s & A & s+1 & s+2 & \cdots & s+r \\ s+r & \cdots & s+2 & s+1 & B & s & \cdots & 2 & 1 & B & 0 \end{pmatrix}. \] We prove that the natural symbolic coding of a typical linear involution defined by a generalized permutation in the Rauzy class of $σ_{s,r}$ is topologically weakly mixing, provided that it has at least a simple letter and its associated genus is sufficiently large.

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arXiv2024

Eigenvalues and the stabilized automorphism group

We study the stabilized automorphism group of minimal and, more generally, certain transitive dynamical systems. Our approach involves developing new algebraic tools to extract information about the rational eigenvalues of these systems from their stabilized automorphism groups. In particular, we prove that if two minimal system have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues. Using these tools, we also extend Schmieding's result on the recovery of entropy from the stabilized automorphism group to include irreducible shifts of finite type.

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arXiv2023

The Jacobs--Keane theorem from the $\cS$-adic viewpoint

In the light of recent developments of the ${\mathcal S}$-adic study of subshifts, we revisit, within this framework, a well-known result on Toeplitz subshifts due to Jacobs--Keane giving a sufficient combinatorial condition to ensure discrete spectrum.We show that the notion of coincidences, originally introduced in the '$70$s for the study of the discrete spectrum of substitution subshifts, together with the ${\mathcal S}$-adic structure of the subshift allow to go deeper in the study of Toeplitz subshifts. We characterize spectral properties of the factor maps onto the maximal equicontinuous topological factors by means of coincidences density.We also provide an easy to check necessary and sufficient condition to ensure unique ergodicity for constant length ${\mathcal S}$-adic subshifts.

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arXiv2023

The structure of low complexity subshifts

An idea that became unavoidable to study zero entropy symbolic dynamics is that the dynamical properties of a system induce in it a combinatorial structure. An old problem addressing this intuition is finding a structure theorem for linear-growth complexity subshifts using the S-adic formalism. It is known as the S-adic conjecture and motivated several influential results in the theory. In this article, we completely solve the conjecture by providing an S-adic structure for this class. Our methods extend to nonsuperlinear-complexity subshifts. An important consequence of our main results is that these complexity classes gain access to the S-adic machinery. We show how this provides a unified framework and simplified proofs of several known results, including the pioneering 1996 Cassaigne's Theorem.

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arXiv2022

Symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank

This paper studies several aspects of symbolic ({\em i.e.}\ subshift) factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $π^{-1}(y)$ of factor maps $π\colon(X,T)\to(Y,S)$ between minimal $\mathcal{S}$-adic subshifts of finite alphabet rank and show that they have the same finite cardinality for all $y$ in a residual subset of $Y$. Finally, we prove that the number of symbolic factors (up to conjugacy) of a fixed subshift of finite topological rank is finite, thus extending Durand's similar theorem on linearly recurrent subshifts [D00].

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arXiv2022

On symbolic factors of $\mathcal{S}$-adic subshifts of finite topological rank

This paper studies several aspects of symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $π^{-1}(y)$ of factor maps $π\colon(X,T)\to(Y,T)$ between minimal $\mathcal{S}$-adic subshifts of finite alphabet rank and show that they have the same finite cardinality for all $y$ in a residual subset of $Y$. Finally, we prove that the number of symbolic factors (up to conjugacy) of a fixed subshift of finite topological rank is finite, thus extending Durand's similar theorem on linearly recurrent subshifts.

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arXiv2021

On the automorphism group of minimal S-adic subshifts of finite alphabet rank

It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $\mathbb{Z}$ [DDPM15, CK15]. In this article we extend this result to a broader class proving that the automorphism group of a minimal S-adic subshift of finite alphabet rank is virtually $\mathbb{Z}$. The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.

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