arXiv subjects

Álvaro Bustos-Gajardo

Publications and source records attributed to Álvaro Bustos-Gajardo.

5 recordsLinked to original sources

arXiv2026

Cobham's theorem for the Gaussian integers

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $α=-m+i $- and $β=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $α$ and $β$ are multiplicatively independent Gaussian integers, and at least one of $α$, $β$ is not an $n$-th root of an integer, then any $α$- and $β$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $α$-automatic for any root of an integer $α$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

math.NT↗

arXiv2025

Natural extensions of embeddable semigroup actions

Semigroup actions and their invertible extensions are discussed. First, we develop a theory of natural extensions for continuous actions of countable, embeddable semigroups. Second, we demonstrate that not every surjective such action of a semigroup, which embeds into a group and generates it, can be extended to an action of said group, and that this phenomenon is specific to non-reversible semigroups. Furthermore, we characterize the free group on a semigroup (the group together with the embedding) as the unique pair that always admits such an extension, showing that both the choice of the receiving group and the embedding are crucial for this construction. Next, we prove that the classical notion of a natural extension -- requiring all other invertible extensions to factor through it -- only works in the context of compact extensions of left reversible semigroup actions and fails outside of it, thus providing a characterization of left reversibility. We finish by briefly studying topological dynamical properties of the natural extension in the amenable case.

math.DS↗

arXiv2025

Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts

In this work, we consider $\mathbb{Z}^d$-shifts generated by digit substitutions. For such a shift $\mathbb{X}$, we study the elements of the normaliser of $\mathbb{Z}^d$ in the group of self homeomorphisms (called extended symmetries) whose local maps guaranteed by the generalised Curtis--Hedlund--Lyndon theorem have radius-zero. Using the formalism of minimal sets developed by Lemańczyk, Müllner and Yassawi, we provide an algorithm to compute elements of $\mathcal{N}(\mathbb{X})$ that preserve the hierarchical structure. We also investigate the interaction of extended symmetries with (i) the height lattice and (ii) the irregular fibres over the maximal equicontinuous factor. Towards (ii), we introduce the notion of derived substitutions to provide a complete description of the irregular fibres, extending a result by Coven, Quas and Yassawi in the one-dimensional case.

math.DS↗

arXiv2025

Extensibility and denseness of periodic semigroup actions

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.

math.DS↗

arXiv2023

Torsion-free $S$-adic shifts and their spectrum

In this work we study $S$-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion-free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We illustrate our results throughout with examples that explain the subtleties that can arise.

math.DS↗