We investigate tameness of Toeplitz shifts. By introducing the notion of extended Bratteli–Vershik diagrams, we show that such shifts with finite Toeplitz rank are tame if and only if there are at most countably many orbits of singular fibres over the maximal equicontinuous factor. The ideas are illustrated using the class of substitution shifts. A body of elaborate examples shows that the assumptions of our results cannot be relaxed.

In this paper we continue the study of automorphism groups of constant-length substitution shifts and also their topological factors. We show that, up to conjugacy, all roots of the identity map are letter-exchanging maps, and all other non-trivial automorphisms arise from twisted compressions of another constant-length substitution. We characterize the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant-length substitution shift is topologically conjugate to a constant-length substitution shift via a letter-to-letter code.

We show that spacetime diagrams of linear cellular automata $\unicode[STIX]{x1D6F7}:\,\mathbb{F}_{p}^{\mathbb{Z}}\rightarrow \mathbb{F}_{p}^{\mathbb{Z}}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions that are eventually constant. Each automatic spacetime diagram defines a $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant subset of $\mathbb{F}_{p}^{\mathbb{Z}}$, where $\unicode[STIX]{x1D70E}$ is the left shift map, and if the initial condition is not eventually periodic, then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measures on $\mathbb{F}_{3}^{\mathbb{Z}}$. Finally, given a linear cellular automaton $\unicode[STIX]{x1D6F7}$, we construct a nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measure on $\mathbb{F}_{p}^{\mathbb{Z}}$ for all but finitely many $p$.

We investigate different notions of recognizability for a free monoid morphism $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{B}}^{\ast }$. Full recognizability occurs when each (aperiodic) point in ${\mathcal{B}}^{\mathbb{Z}}$ admits at most one tiling with words $\unicode[STIX]{x1D70E}(a)$, $a\in {\mathcal{A}}$. This is stronger than the classical notion of recognizability of a substitution $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{A}}^{\ast }$, where the tiling must be compatible with the language of the substitution. We show that if $|{\mathcal{A}}|=2$, or if $\unicode[STIX]{x1D70E}$’s incidence matrix has rank $|{\mathcal{A}}|$, or if $\unicode[STIX]{x1D70E}$ is permutative, then $\unicode[STIX]{x1D70E}$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé [Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci.99(2) (1992), 327–334] and Bezuglyi et al [Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys.29(1) (2009), 37–72], by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an $S$-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$-adic shift it generates, and the measurable Bratteli–Vershik dynamical system that it defines.

Many dynamical systems can be naturally represented as Bratteli–Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X is linearly recurrent, then we show how to represent X using a two-dimensional subshift of finite type Y; each ‘row’ in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y-admissible configuration can then be recoded as the space-time diagram of a one-dimensional cellular automaton Φ; in this way X is embedded in Φ (i.e. X is conjugate to a subsystem of Φ). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

Let ${\mathbb{M}}={\mathbb{N}}^D$ be the positive orthant of a $D$-dimensional lattice and let $({\mathcal{G}},+)$ be a finite abelian group. Let ${\mathfrak{G}} \subseteq{\mathcal{G}}^{\mathbb{M}}$ be a subgroup shift, and let $\mu$ be a Markov random field whose support is ${\mathfrak{G}}$. Let $\Phi: {\mathfrak{G}} \longrightarrow {\mathfrak{G}}$ be a linear cellular automaton. Under broad conditions on ${\mathcal{G}}$, we show that the Cesaro average $N^{-1} \sum_{n=0}^{N-1} \Phi^n (\mu)$ converges to a measure of maximal entropy for the shift action on ${\mathfrak{G}}$.

Let ${\mathbb{M}}={\mathbb{Z}}^D$ be a $D$-dimensional lattice, and let $({\mathcal{A}},+)$ be an abelian group. ${\mathcal{A}}^{\mathbb{M}}$ is then a compact abelian group under componentwise addition. A continuous function $\Phi:{\mathcal{A}}^{\mathbb{M}}\longrightarrow{\mathcal{A}}^{\mathbb{M}}$ is called a linear cellular automaton if there is a finite subset ${\mathbb{F}}\subset{\mathbb{M}}$ and non-zero coefficients $\varphi_{\textsf{f}}\in{\mathcal{Z}}$ so that, for any ${\bf{a}}\in{\mathcal{A}}^{\mathbb{M}}\, \Phi({\bf{a}}) = \sum_{{\textsf{f}}\in{\mathbb{F}}}\varphi_{\textsf{f}}\cdot\sigma^{{\textsf{f}}}({\bf{a}})$. Suppose that $\mu$ is a probability measure on ${\mathcal{A}}^{\mathbb{M}}$ whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on $\Phi$ and $\mu$) under which $\Phi$asymptotically randomizes$\mu$, meaning that $\mathrm{wk}^*-\lim_{{\mathbb{J}}\ni j\rightarrow\infty} \Phi^j\mu = \eta$, where $\eta$ is the Haar measure on ${\mathcal{A}}^{\mathbb{M}}$, and ${\mathbb{J}}\subset{\mathbb{N}}$ has Cesàro density one. In the case when $\Phi=1+\sigma$ and ${\mathcal{A}}=({{\mathbb{Z}}_{/p}})^s$ ($p$ prime), we provide a condition on $\mu$ that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by $1+\sigma$.

If $\mathbb{M}$ is a monoid and $\mathcal{A}$ is an Abelian group, then $\mathcal{A}^{\mathbb{M}}$ is a compact Abelian group; a linear cellular automaton (LCA) is a continuous endomorphism $\mathfrak{F}:\mathcal{A}^{\mathbb{M}}\longrightarrow \mathcal{A}^{\mathbb{M}}$ that commutes with all shift maps. If $\mathfrak{F}$ is diffusive, and $\mu$ is a harmonically mixing (HM) probability measure on $\mathcal{A}^{\mathbb{M}}$, then the sequence $\{\mathfrak{F}^N \mu\}_{N=1}^{\infty}$ weak* converges to the Haar measure on $\mathcal{A}^{\mathbb{M}}$ in density. Fully supported Markov measures on $\mathcal{A}^{\mathbb{Z}}$ are HM and non-trivial LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ are diffusive when $\mathcal{A}=\mathbb{Z}_{/p}$ is a prime cyclic group.

In the present work, we provide sufficient conditions for diffusion of LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ when $\mathcal{A}=\mathbb{Z}_{/n}$ is any cyclic group or when $\mathcal{A}=(\mathbb{Z}_{/p^r})^J$ (p prime). We also show that any fully supported Markov random field on $\mathcal{A}^{(\mathbb{Z}^D)}$ is HM (where $\mathcal{A}$ is any Abelian group).

Let $r \geq 1$; the Følner sequences $\{F^{n,1}\}_{n=1}^{\infty}, \{F^{n,2}\}_{n=1}^{\infty}, \dotsc, \{F^{n,r}\}_{n=1}^{\infty}$ satisfy the bounded intersection property if there is a constant p such that, for any $n\in\mathbb{N}$ and $1\leq i\leq r$, each $F^{n,i}$ can intersect no more than p disjoint translates of $F^{n,1}, F^{n,2},\dotsc, F^{n,r}$. They have comparable magnitudes if $0<\varliminf_{n}|F^{n,i}|/|F^{n,j}|<\infty$ for $1\leq i,j\leq r$. Suppose that G is a countable Abelian group with an element of infinite order, and let $\mathcal{X}$ be a mixing finite (or $\beta$-local, with $\beta>1/2$) rank action of G on a probability space. Suppose that the Følner sequences of $\mathcal{X}$ satisfy the bounded intersection property, and have comparable magnitudes. Then $\mathcal{X}$ is mixing of all orders. We follow Ryzhikov's joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of $\mathcal{X}$ is necessarily product measure.

Let \mathbb{M} be a monoid (e.g. \mathbb{N}, \mathbb{Z}, or \mathbb{M}^D), and \mathcal{A} an abelian group. \mathcal{A}^\mathbb{M} is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism \mathfrak{F}:\mathcal{A}^\mathbb{M}\longrightarrow\mathcal{A}^\mathbb{M} that commutes with all shift maps. Let \mu be a (possibly non-stationary) probability measure on \mathcal{A}^\mathbb{M}; we develop sufficient conditions on \mu and \mathfrak{F} so that the sequence \{\mathfrak{F}^N\mu\}_{N=1}^\infty weak* converges to the Haar measure on \mathcal{A}^\mathbb{M} in density (and thus, in Cesàro average as well). As an application, we show that, if \mathcal{A}=\mathbb{Z}_{/p} (p prime), \mathfrak{F} is any ‘non-trivial’ LCA on \mathcal{A}^{(\mathbb{Z}^D)}, and \mu belongs to a broad class of measures (including most Bernoulli measures (for D \geq 1) and ‘fully supported’ N-step Markov measures (when D=1)), then \mathfrak{F}^N\mu weak* converges to the Haar measure in density.

Suppose $G$ is a countable, Abelian group with an element of infinite order and let $\text{ }\!\!\chi\!\!\text{ }$ be a mixing rank one action of $G$ on a probability space. Suppose further that the Følner sequence $\{{{F}_{n}}\}$ indexing the towers of $\text{ }\!\!\chi\!\!\text{ }$ satisfies a “bounded intersection property”: there is a constant $p$ such that each $\{{{F}_{n}}\}$ can intersect no more than $p$ disjoint translates of $\{{{F}_{n}}\}$. Then $\text{ }\!\!\chi\!\!\text{ }$ is mixing of all orders. When $G\,=\,\mathbf{Z}$, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of $\text{ }\!\!\chi\!\!\text{ }$ is necessarily product measure. This method generalizes Ryzhikov’s technique.