Álvaro Bustos

The Open University (UK) · Department of Mathematics and Statistics

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-rec...

In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic and dynamical properties of these subshifts and recall known results regarding their symmetry groups. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended sy...

The sets of $k$-free integers in general quadratic number fields are studied, with special emphasis on (extended) symmetries and their impact on the topological dynamical systems induced by such integers. We establish correspondences between number-theoretic and dynamical quantities, and use symmetries and entropy to distinguish the systems.

In this contribution, we discuss the automorphism group, i.e., the centralizer of the shift action inside the group of self-homeomorphisms of a subshift, together with the extended symmetry group (the corresponding normalizer) of certain Zd subshifts with a hierarchical structure, like bijective substitutive subshifts and the Robinson tiling.

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to check them. Moreover, we show that, for any finite group $G$ and any subgroup $P$ of the $d$-dimensional hyp...

Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our f...

Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centraliser and normaliser of the shift dynamical system, which are natural topological invariants. Here, our f...

Automorphism groups are intrincate conjugacy invariants for subshifts, which can reveal important features of the dynamical structure of a shift action. One important case is the study of automorphism groups when the underlying subshift has a very rigid structure, e.g. substitutive subshifts or aperiodic constructions with large-scale self-similari...