Decomposition theorem for invertible substitutions on three-letter alphabet

On the group theoretical background of assigning stepwise mutations onto phylogenies

In a recent paper, Klaere et al. modeled the impact of substitutions on arbitrary branches of a phylogenetic tree on an alignment site by the so-called One Step Mutation (OSM) matrix. By utilizing the concept of the OSM matrix for the four-state nucleotide alphabet, Nguyen et al. presented an efficient procedure to compute the minimal number of substitutions needed to translate one alignment site into another.The present paper delivers a proof for this computation.Moreover, we provide several mathematical insights into the generalization of the OSM matrix to multistate alphabets.The constructi…

On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata

This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.

Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution

There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-menti…

生成自由群的自相似群的动力学与阻碍

Freeness Obstructions for Self-Similar Groups Acting on Rooted Trees

该研究证明自相似群自由性的两个一般阻碍，推导自动机群相关结论，明确自由性与双可逆性的关联，回答Grigorchuk问题并扩展Sidki的相关结果。

Homoclinically expansive actions and a Garden of Eden theorem for harmonic models

Let $Γ$ be a countable Abelian group and $f \in \Z[Γ]$, where $\Z[Γ]$ denotes the integral group ring of $Γ$. Consider the Pontryagin dual $X_f$ of the cyclic $\Z[Γ]$-module $\Z[Γ]/\Z[Γ] f$ and suppose that $f$ is weakly expansive (e.g., $f$ is invertible in $\ell^1(Γ)$, or, when $Γ$ is not virtually $\Z$ or $\Z^2$, $f$ is well-balanced) and that $X_f$ is connected. We prove that if $τ\colon X_f \to X_f$ is a $Γ$-equivariant continuous map, then $τ$ is surjective if and only if the restriction of $τ$ to each $Γ$-homoclinicity class is injective. We also show that this equivalence remains valid…