Due to a result by Glasner and Downarowicz [Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. 48(1) (2016), 321–338], it is known that a minimal system is mean equicontinuous if and only if it is an isomorphic extension of its maximal equicontinuous factor. The majority of known examples of this type are almost automorphic, that is, the factor map to the maximal equicontinuous factor is almost one-to-one. The only cases of isomorphic extensions which are not almost automorphic are again due to Glasner and Downarowicz, who in the same article provide a construction of such systems in a rather general topological setting. Here, we use the Anosov–Katok method to provide an alternative route to such examples and to show that these may be realized as smooth skew product diffeomorphisms of the two-torus with an irrational rotation on the base. Moreover – and more importantly – a modification of the construction allows to ensure that lifts of these diffeomorphisms to finite covering spaces provide novel examples of finite-to-one topomorphic extensions of irrational rotations. These are still strictly ergodic and share the same dynamical eigenvalues as the original system, but show an additional singular continuous component of the dynamical spectrum.

We introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$. A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Röver’s group $V(\Gamma )$, and others of Nekrashevych’s ‘simple groups of dynamical origin’).

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $\bar {d}$ metric ($\bar {d}$-approachable shift spaces). The class of $\bar {d}$-approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar {d}$-approachability, together with a closely connected notion of $\bar {d}$-shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys. 43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $\bar {d}$-approachability and $\bar {d}$-shadowing. Here, we study further properties and connections between $\bar {d}$-shadowing and $\bar {d}$-approachability. We prove that $\bar {d}$-shadowing implies $\bar {d}$-stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar {d}$-shadowing property the Hausdorff pseudodistance ${\bar d}^{\mathrm {H}}$ between shift spaces induced by $\bar {d}$ is the same as the Hausdorff distance between their simplices of invariant measures with respect to the Hausdorff distance induced by Ornstein’s metric $\bar {d}$ between measures. We prove that without $\bar {d}$-shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the $\bar {d}$-shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that $\bar {d}$-shadowing indeed generalizes the specification property.

We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$, there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2) 199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math. 384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

Given a countable group G and a G-flow X, a probability measure $\mu $ on X is called characteristic if it is $\mathrm {Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal G-flow, for any group G, which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups $\{\Delta _i: i\in I\}$, when is there a faithful G-flow for which every $\Delta _i$ acts minimally?

Pavlov [Adv. Math. 295 (2016), 250–270; Nonlinearity 32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

We define a notion of substitution on colored binary trees that we call substreetution. We show that a point fixed by a substreetution may (or not) be almost periodic, and thus the closure of the orbit under the $\mathbb {F}_{2}^{+}$-action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measures on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.

Let G be a countable residually finite group (for instance, ${\mathbb F}_2$) and let $\overleftarrow {G}$ be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every $r\geq 1$, we construct a Toeplitz G-subshift $(X,\sigma ,G)$, which is an almost one-to-one extension of $\overleftarrow {G}$, having r ergodic measures $\nu _1, \ldots ,\nu _r$ such that for every $1\leq i\leq r$, the measure-theoretic dynamical system $(X,\sigma ,G,\nu _i)$ is isomorphic to $\overleftarrow {G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups); however, we point out the differences and obstructions that could appear when the acting group is not amenable.

A hyperbolic group G acts by homeomorphisms on its Gromov boundary. We show that if $\partial G$ is a topological n–sphere, the action is topologically stable in the dynamical sense: any nearby action is semi-conjugate to the standard boundary action.

We characterize measure-theoretic sequence entropy pairs of continuous actions of abelian groups using mean sensitivity. This addresses an open question of Li and Yu [On mean sensitive tuples. J. Differential Equations 297 (2021), 175–200]. As a consequence of our results, we provide a simpler characterization of Kerr and Li’s independence sequence entropy pairs ($\mu $-IN-pairs) when the measure is ergodic and the group is abelian.

Assume $G\prec H$ are groups and ${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$. We apply these results in the model theoretic context. Namely, assume G is a group definable in a model M and $M\prec ^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups $S_{ext,G}(M)$ and $S_{ext,G}(N)$. Assuming every minimal left ideal in $S_{ext,G}(N)$ is a group we prove that the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$.

In this paper, we give necessary conditions for an $N$-expansive homeomorphism of a compact metric space to be nonchaotic in the Li–Yorke sense. As application we give a partial answer to a conjecture in [2].

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.

Let $\Sigma $ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of probability measure preserving ergodic minimal profinite actions for the fundamental group of $\Sigma $ that are topologically free but not essentially free, a property that we call allostery. Moreover, the invariant random subgroups we obtain are pairwise distincts.

We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of ${\mathcal B}$-free subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz ${\mathcal B}$-free systems. Bull. Pol. Acad. Sci. Math. 65(2) (2017), 139–152]. Additionally, we provide an example of a ${\mathcal B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 of S. Ferenczi et al [Sarnak’s conjecture: what’s new. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Eds. S. Ferenczi, J. Kułaga-Przymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163–235].

In this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

Let G be a countably infinite discrete amenable group. It should be noted that a G-system $(X,G)$ naturally induces a G-system $(\mathcal {M}(X),G)$, where $\mathcal {M}(X)$ denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map $\pi : (X,G)\to (Y,G)$ between two G-systems induces a factor map $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$. It turns out that $\widetilde {\pi }$ is open if and only if $\pi $ is open. When Y is fully supported, it is shown that $\pi $ has relative uniformly positive entropy if and only if $\widetilde {\pi }$ has relative uniformly positive entropy.

Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $\mu $-sequence entropy tuple, the $\mu $-mean sensitive tuple, and the $\mu $-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.

In this paper, we study almost proximal extensions of minimal flows. Let $\pi : (X,T)\rightarrow (Y,T)$ be an extension of minimal flows. Then $\pi $ is called an almost proximal extension if there is some $N\in {\mathbb N}$ such that the cardinality of any almost periodic subset in each fiber is not greater than N. When $N=1$, $\pi $ is proximal. We will give the structure of $\pi $ and give a dichotomy theorem: any almost proximal extension of minimal flows is either almost finite to one, or almost all fibers contain an uncountable strongly scrambled subset. Using the category method, Glasner and Weiss showed the existence of proximal but not almost one-to-one extensions [On the construction of minimal skew products. Israel J. Math. 34 (1979), 321–336]. In this paper, we will give explicit such examples, and also examples of almost proximal but not almost finite to one extensions.

We give a characterization of inter-model sets with Euclidean internal space. This characterization is similar to previous results for general inter-model sets obtained independently by Baake, Lenz and Moody, and Aujogue. The new ingredients are two additional conditions. The first condition is on the rank of the abelian group generated by the set of internal differences. The second condition is on a flow on a torus defined via the address map introduced by Lagarias. This flow plays the role of the maximal equicontinuous factor in the previous characterizations.