In this article we study automorphisms of Toeplitz subshifts. Such groups are
abelian and any finitely generated torsion subgroup is finite and cyclic. When
the complexity is non superlinear, we prove that the automorphism group is,
modulo a finite cyclic group, generated by a unique root of the shift. In the
subquadratic complexity case, we show that the automorphism group modulo the
torsion is generated by the roots of the shift map and that the result of the
non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct
examples of minimal Toeplitz subshifts with complexity bounded by $C
n^{1+\epsilon}$ whose automorphism groups are not finitely generated. Finally,
we observe the coalescence and the automorphism group give no restriction on
the complexity since we provide a family of coalescent Toeplitz subshifts with
positive entropy such that their automorphism groups are arbitrary finitely
generated infinite abelian groups with cyclic torsion subgroup (eventually
restricted to powers of the shift)

Donoso, Sebastián

Durand, Fabien

Maass, Alejandro

Petite, Samuel

English

arXiv

International audienceIn this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non-superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non-superlinear case is optimal. Namely, for any ε > 0 we construct examples of minimal Toeplitz subshifts with complexity bounded by Cn 1+ε whose automorphism groups are not finitely generated. Finally, we observe the coalescence and the automorphism group give no restriction on the complexity since we provide a family of coalescent Toeplitz subshifts with positive entropy such that their automorphism groups are arbitrary finitely generated infinite abelian groups with cyclic torsion subgroup (eventually restricted to powers of the shift)

Donoso, Sebastian

Hal-Diderot

On automorphism groups of Toeplitz subshifts *

Sebastian Donoso

Fabien Durand

Alejandro Maass

Samuel Petite

Crossref

Discrete Analysis

On automorphism groups of Toeplitz subshifts

Archive Ouverte en Sciences de l'Information et de la Communication

On automorphism groups of Toeplitz subshifts, Discrete Analysis 2017:11, 19 pp.

A discrete dynamical system is a space $X$ with some kind of structure, together with a map $\sigma\colon X\to X$ that preserves  the structure. (For instance, if $X$ is a topological space, then one asks for $\sigma$ to be continuous, and if $X$ is a differentiable manifold, then one asks for it to be a diffeomorphism.) Given such a system, one studies the structure of the orbits $x, \sigma x, \sigma^2x, \dots$ that are obtained by iterating the map $\sigma$. A particularly interesting subfield of dynamics is _symbolic dynamics_, where $X$ is a space of bi-infinite sequences over a finite alphabet $A$, $X$ is closed under the left shift, and $\sigma$ is that left shift. One also asks for $X$ to be closed in the topological sense: we take the discrete topology on $A$ and the product topology on $A^{\mathbb Z}$, of which $X$ is a subset. A system $(X,\sigma)$ is called a _shift space_. Such spaces can encode interesting combinatorial information, and that has led to a very fruitful interplay between combinatorics and dynamical systems.

An _automorphism_ of the system $(X, \sigma)$ is a homeomorphism $\phi\colon X\to X$  that commutes with $\sigma$, and ${\rm Aut}(X, \sigma)$ denotes the group (under composition) of automorphisms of the system. The _complexity_ of a shift system ${\rm Aut}(X, \sigma)$ is the map $p\colon\mathbb N\to \mathbb N$ that counts the number of blocks of length $n$ appearing in the sequences $x\in X$.  If the complexity is linear, then the automorphism group is understood for any shift ${\rm Aut}(X, \sigma)$, but beyond linear, the problem becomes complicated. For example, under mild assumptions on the shift ${\rm Aut}(X, \sigma)$, the automorphism group is not finitely generated and it contains isomorphic copies of all finite groups, countably many copies of $\mathbb Z$, and the free group on any finite number of generators. Thus while ${\rm Aut}(X, \sigma)$ is always countable, in general it can be quite complicated and difficult to compute. However, for several reasons it is desirable to do so: for example, it gives a useful invariant.

This paper continues recent work on automorphism groups for various classes of shift spaces, computing the automorphism group for the class of Toeplitz shifts, a large class of shift systems frequently used to provide counterexamples in symbolic dynamics.    A sequence $x\in A^{\mathbb Z}$ is _Toeplitz_ if every finite block in $x$ appears periodically, and a shift space $(X, \sigma)$ is a _Toeplitz shift_ if $X$ is the orbit closure of some Toeplitz sequence. (It is not hard to construct Toeplitz sequences that are not periodic. For one example, take $x_n$ to be the parity of $k$, where $k$ is maximal such that $2^k|n$.) This rigid structure on $X$ implies that ${\rm Aut}(X, \sigma)$ is Abelian, and this is the starting point for the classification of the automorphism groups of Toeplitz shifts.

The authors start with Toeplitz shifts of subquadratic complexity, showing that the automorphism group is spanned by the roots of the shift map $\sigma$ modulo the torsion subgroup $T$ of ${\rm Aut}(X, \sigma)$. More generally, they show that if ${\rm Aut}(X, \sigma)/\langle\sigma\rangle$ is a periodic group, then the automorphism group is spanned by $T$ and the roots of the shift $\sigma$ (that is, the automorphisms $\phi$ such that $\phi^n=\sigma$ for some $n$). Under the further assumption that $T$ is trivial, they show that the automorphism group is either infinite cyclic or is not finitely generated.  This method leads to examples of  Toeplitz shifts whose complexity is arbitrarily close to linear, in the sense that for every $\varepsilon > 0$ the complexity satisfies the inequality $p(n)\leq Cn^{1+\varepsilon}$ for some constant $C=C_\varepsilon > 0$, such that the automorphism group is not finitely generated. Note that this result cannot be extended to linear complexity, where it is known that the automorphism group is always finitely generated.

In the opposite regime, that of high complexity, the authors show that the automorphism group need not be large. Given any infinite and finitely generated Abelian group $G$ with cyclic torsion, they construct a Toeplitz shift with positive entropy (meaning that the complexity function grows exponentially) whose automorphism group  is exactly $G$

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On automorphism groups of Toeplitz subshifts

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Archive Ouverte en Sciences de l'Information et de la Communication

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