We prove that for any non-degenerate dendrite D, there exist topologically mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$ such that the natural extensions (as known as shift homeomorphisms) $\sigma _F$ and $\sigma _f$ are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map f does not depend on the dendrite D and can be selected so that the inverse limit $\underleftarrow {\lim } (D,F)$ is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and Hénon maps to natural extensions of dendrite maps.

Given a minimal action $\alpha $ of a countable group on the Cantor set, we show that the alternating full group $\mathsf {A}(\alpha )$ is non-amenable if and only if the topological full group $\mathsf {F}(\alpha )$ is $C^*$ -simple. This implies, for instance, that the Elek–Monod example of non-amenable topological full group coming from a Cantor minimal $\mathbb {Z}^2$ -system is $C^*$ -simple.

Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$ for every $f\in \mathcal {C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of $\mu (n)$, one can put any bounded sequence $a_{n}$ such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence $a_{n}$ but $(X,T)$ does not.

In this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy $h(X)> 0$, then the entropies of the SFT subsystems of X are dense in the interval $[0, h(X)]$. In fact, we prove a ‘relative’ version of the same result: if X is a G-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs Z for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic G-shifts.

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.

We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, a concept recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion, we count the number of orbits for the action of the subgroup of fixed points (respectively periodic) points on the set of infinite fixed (respectively periodic) points. Finally, we study the dynamics of infinite points: for automorphisms and some endomorphisms, defined in a precise way, fitting a classification given by Delgado and Ventura, we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic.

Every topological group G has, up to isomorphism, a unique minimal G-flow that maps onto every minimal G-flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K. Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $G/K\ltimes K$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $2^{\mathbb {N}}$ , $M(\mathbb {Z})$ , or $M(\mathbb {Z})\times 2^{\mathbb {N}}.$

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg–Zimmer tower of a measurably distal system and showing that at each step there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.

We show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule $\Sigma \subset A^G$ and every injective G-equivariant uniformly continuous module homomorphism $\tau \colon \! \Sigma \to B^G$, a subshift $\Delta \subset \Sigma $ is of finite type, respectively sofic, if and only if so is the image $\tau (\Delta )$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.

For a subshift $(X, \sigma _{X})$ and a subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on X, we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim _{n\rightarrow \infty }(1/{n})\int \log f_{n}\, d\kern-1pt\mu =\int h \,d\kern-1pt\mu $ for every invariant measure $\mu $ on X. For this purpose, we first we study necessary and sufficient conditions for ${\mathcal F}$ to be an asymptotically additive sequence in terms of certain properties for periodic points. For a factor map $\pi : X\rightarrow Y$ , where $(X, \sigma _{X})$ is an irreducible shift of finite type and $(Y, \sigma _{Y})$ is a subshift, applying our results and the results obtained by Cuneo [Additive, almost additive and asymptotically additive potential sequences are equivalent. Comm. Math. Phys. 37 (3) (2020), 2579–2595] on asymptotically additive sequences, we study the existence of h with regard to a subadditive sequence associated to a relative pressure function. This leads to a characterization of the existence of a certain type of continuous compensation function for a factor map between subshifts. As an application, we study the projection $\pi \mu $ of an invariant weak Gibbs measure $\mu $ for a continuous function on an irreducible shift of finite type.

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb {N} \to \mathbb {N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty $ , that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$ . This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

Let $(X_k)_{k\geq 0}$ be a stationary and ergodic process with joint distribution $\mu $ , where the random variables $X_k$ take values in a finite set $\mathcal {A}$ . Let $R_n$ be the first time this process repeats its first n symbols of output. It is well known that $({1}/{n})\log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$ -spectrum defined as

provided the limit exists. We consider the case where $(X_k)_{k\geq 0}$ is distributed according to the equilibrium state of a potential with summable variation, and we prove that

where $P((1-q)\varphi )$ is the topological pressure of $(1-q)\varphi $ , the supremum is taken over all shift-invariant measures, and $q_\varphi ^*$ is the unique solution of $P((1-q)\varphi ) =\sup _\eta \int \varphi \,d\eta $ . Unexpectedly, this spectrum does not coincide with the $L^q$ -spectrum of $\mu _\varphi $ , which is $P((1-q)\varphi )$ , and it does not coincide with the waiting-time $L^q$ -spectrum in general. In fact, the return-time $L^q$ -spectrum coincides with the waiting-time $L^q$ -spectrum if and only if the equilibrium state of $\varphi $ is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of $({1}/{n})\log R_n$ .

We show that every isometric action on a Cantor set is conjugate to an inverse limit of actions on finite sets; and that every faithful isometric action by a finitely generated amenable group is residually finite.

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math. 397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys. 39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math. 308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

We show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.

We show that the universal minimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation-invariant Boolean algebras of subsets of the group satisfying a higher-order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.

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 present a new sufficient criterion to prove that a non-sofic half-synchronized subshift is direct prime. The criterion is based on conjugacy invariant properties of Fischer graphs of half-synchronized shifts. We use this criterion to show as a new result that all n-Dyck shifts are direct prime, and we also give new proofs of direct primeness of non-sofic beta-shifts and non-sofic S-gap shifts. We also construct a class of non-sofic synchronized direct prime subshifts which additionally admit reversible cellular automata with all directions sensitive.

We prove that for any transitive subshift X with word complexity function $c_n(X)$ , if $\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$ , then the quotient group ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ of the automorphism group of X by the subgroup generated by the shift $\sigma $ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if ${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$ , then $\mathrm {Aut}(X,\sigma )$ is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing $f: \mathbb {N} \rightarrow \mathbb {N}$ , there exists a minimal subshift X with ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ isomorphic to G and ${c_n(X)}/{nf(n)} \rightarrow 0$ .

We investigate quantitative aspects of the locally embeddable into finite groups (LEF) property for subgroups of the topological full group of a two-sided minimal subshift over a finite alphabet, measured via the LEF growth function. We show that the LEF growth of may be bounded from above and below in terms of the recurrence function and the complexity function of the subshift, respectively. As an application, we construct groups of previously unseen LEF growth types, and exhibit a continuum of finitely generated LEF groups which may be distinguished from one another by their LEF growth.