Two asymptotic configurations on a full $\mathbb {Z}^d$-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb {Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $\mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.

We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.

In this paper, we construct a uniformly recurrent infinite word of low complexity without uniform frequencies of letters. This shows the optimality of a bound of Boshernitzan, which gives a sufficient condition for a uniformly recurrent infinite word to admit uniform frequencies.

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the $\alpha $-free words for rational $\alpha $, $2 < \alpha \leq 7/3$), is decidable. As a consequence, many results previously obtained about this class through tedious case-based proofs can now be proved “automatically,” using a decision procedure, and new claims can be proved or disproved simply by restating them as logical formulas.

We study the measurable dynamical properties of the interval map generated by the model-case erasing substitution $\rho $, defined by

$$ \begin{align*} \rho(00)=\text{empty word},\quad \rho(01)=1,\quad \rho(10)=0,\quad \rho(11)=01. \end{align*} $$

Given $\beta \in (1,2]$ , let $T_{\beta }$ be the $\beta $ -transformation on the unit circle $[0,1)$ such that $T_{\beta }(x)=\beta x\pmod 1$ . For each $t\in [0,1)$ , let $K_{\beta }(t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^{n}_{\beta }(x): n\ge 0\}$ never hits the open interval $(0,t)$ . Kalle et al [Ergod. Th. & Dynam. Sys. 40(9) (2020) 2482–2514] proved that the Hausdorff dimension function $t\mapsto \dim _{H} K_{\beta }(t)$ is a non-increasing Devil’s staircase. So there exists a critical value $\tau (\beta )$ such that $\dim _{H} K_{\beta }(t)>0$ if and only if $t<\tau (\beta )$ . In this paper, we determine the critical value $\tau (\beta )$ for all $\beta \in (1,2]$ , answering a question of Kalle et al (2020). For example, we find that for the Komornik–Loreti constant $\beta \approx 1.78723$ , we have $\tau (\beta )=(2-\beta )/(\beta -1)$ . Furthermore, we show that (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps, with $\tau (1+)=0$ and $\tau (2)=1/2$ ; and (iii) there exists an open set $O\subset (1,2]$ , whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, convex, and strictly decreasing on each connected component of O. Consequently, the dimension $\dim _{H} K_{\beta }(t)$ is not jointly continuous in $\beta $ and t. Our strategy to find the critical value $\tau (\beta )$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.

We consider a subshift of finite type on q symbols with a union of t cylinders based at words of identical length p as the hole. We explore the relationship between the escape rate into the hole and a rational function, $r(z)$, of correlations between forbidden words in the subshift with the hole. In particular, we prove that there exists a constant $D(t,p)$ such that if $q>D(t,p)$, then the escape rate is faster into the hole when the value of the corresponding rational function $r(z)$ evaluated at $D(t,p)$ is larger. Further, we consider holes which are unions of cylinders based at words of identical length, having zero cross-correlations, and prove that the escape rate is faster into the hole with larger Poincaré recurrence time. Our results are more general than the existing ones known for maps conjugate to a full shift with a single cylinder as the hole.

The Thue–Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors w within this sequence, or more precisely, the sequence of gaps between consecutive occurrences. This gap sequence is morphic; we prove that it is not automatic as soon as the length of w is at least $2$ , thereby answering a question by J. Shallit in the affirmative. We give an explicit method to compute the discrepancy of the number of occurrences of the block $\mathtt {01}$ in the Thue–Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base- $2$ transducer.

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux–Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal {S}$ -adic representation where the morphisms in $\mathcal {S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those $\mathcal {S}$ -adic representations, heading towards an $\mathcal {S}$ -adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness.

Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$ . In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma )$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma )$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma )$ is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [ 10 ] and Schmeling in [ 36 ] in the context of $\beta $ -transformations.

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by

$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$

$E_{\unicode[STIX]{x1D6FD}}$

$t\in [0,1)$

$t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$

$E_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$E_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$E_{\unicode[STIX]{x1D6FD}}$

$E_{2}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$

$K_{\unicode[STIX]{x1D6FD}}(t)$

$t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$

$\unicode[STIX]{x1D6FD}\in (1,2)$

Suppose $(X,\unicode[STIX]{x1D70E})$ is a subshift, $P_{X}(n)$ is the word complexity function of $X$, and $\text{Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_{X}(n)=o(n^{2}/\log ^{2}n)$, then $\text{Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_{X}(n)=o(n^{2})$, then $\text{Aut}(X)$ can never contain a non-abelian free monoid (and, in particular, can never contain a non-abelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a monoid.

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are ε-far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with ‘constant’ query complexity, depending only on ε and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques is often enormous and impractical. It remains a major open problem if better bounds hold.

Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ε query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.

In 2006 Brown asked the following question in the spirit of Ramsey theory: given a non-periodic infinite word $x=x_{1}x_{2}x_{3}\ldots$ with values in a set $\mathbb{A}$, does there exist a finite colouring $\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow C$ relative to which $x$ does not admit a $\unicode[STIX]{x1D711}$-monochromatic factorization, i.e. a factorization of the form $x=u_{1}u_{2}u_{3}\ldots$ with $\unicode[STIX]{x1D711}(u_{i})=\unicode[STIX]{x1D711}(u_{\!j})$ for all $i,j\geqslant 1$? Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniformly recurrent words and for various classes of uniformly recurrent words including Sturmian words and fixed points of strongly recognizable primitive substitutions. In this paper we give a complete and optimal affirmative answer to this question by showing that if $x=x_{1}x_{2}x_{3}\ldots$ is an infinite non-periodic word with values in a set $\mathbb{A}$, then there exists a $2$-colouring $\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow \{0,1\}$ such that for any factorization $x=u_{1}u_{2}u_{3}\ldots$ we have $\unicode[STIX]{x1D711}(u_{i})\neq \unicode[STIX]{x1D711}(u_{\!j})$ for some $i\neq j$.