Inspired by a twist map theorem of Mather. we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map g to the k-fold cover. For each irrational in the rotation set’s interior, the collection of the k-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball with the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool used is a generalization of a construction of Hedlund and Morse that generates symbolic analogs of these k-fold well-ordered invariant sets.

We exhibit, for arbitrary $\epsilon> 0$, subshifts admitting weakly mixing (probability) measures with word complexity p satisfying $\limsup p(q) / q < 1.5 + \epsilon $. For arbitrary $f(q) \to \infty $, said subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often. We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q = \infty $ and that this is optimal for rank-ones.

We look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$. In the case of $\mathbb {F}_n\times \mathbb {Z}$, the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

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 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.

In this work, we explore the dynamical implications of a spectral sequence analysis of a filtered chain complex associated to a non-singular Morse–Smale (NMS) flow $\varphi $ on a closed orientable $3$-manifold $M^3$ with no heteroclinic trajectories connecting saddle periodic orbits. We introduce the novel concepts of cancellations and reductions of pairs of periodic orbits based on Franks’ morsification and Smale’s cancellation theorems. The main goal is to establish an algebraic-dynamical correspondence between the unfolding of this spectral sequence associated to $\varphi $ and a family of flows obtained by cancelling and reducing pairs of periodic orbits of $\varphi $ on $M^3$. This correspondence is achieved through a spectral sequence sweeping algorithm (SSSA), which determines the order in which these cancellations and reductions of periodic orbits occur, producing a family of NMS flows that reaches a core flow when the spectral sequence converges.

Given a locally finite graph $\Gamma $, an amenable subgroup G of graph automorphisms acting freely and almost transitively on its vertices, and a G-invariant activity function $\unicode{x3bb} $, consider the free energy $f_G(\Gamma ,\unicode{x3bb} )$ of the hardcore model defined on the set of independent sets in $\Gamma $ weighted by $\unicode{x3bb} $. Under the assumption that G is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|\unicode{x3bb} \|_\infty < \unicode{x3bb} _c(\Delta )$, there exists a randomized $\epsilon $-additive approximation scheme for $f_G(\Gamma ,\unicode{x3bb} )$ that runs in time $\mathrm {poly}((1+\epsilon ^{-1})\lvert \Gamma /G \rvert )$, where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $-regular tree. In addition, if G has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. However, we observe that if $\|\unicode{x3bb} \|_\infty> \unicode{x3bb} _c(\Delta )$, there is no efficient approximation scheme, unless $\mathrm {NP} = \mathrm {RP}$. This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

For linear nonuniform cellular automata (NUCA) which are local perturbations of linear CA over a group universe G and a finite-dimensional vector space alphabet V over an arbitrary field k, we investigate their Dedekind finiteness property, also known as the direct finiteness property, i.e., left or right invertibility implies invertibility. We say that the group G is $L^1$-surjunctive, resp. finitely $L^1$-surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in addition k is finite. In parallel, we introduce the ring $D^1(k[G])$ which is the Cartesian product $k[G] \times (k[G])[G]$ as an additive group but the multiplication is twisted in the second component. The ring $D^1(k[G])$ contains naturally the group ring $k[G]$ and we obtain a dynamical characterization of its stable finiteness for every field k in terms of the finite $L^1$-surjunctivity of the group G, which holds, for example, when G is residually finite or initially subamenable. Our results extend known results in the case of CA.

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.

We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$-algebras associated to them, in terms of the pair of original graphs.

Given a subshift $\Sigma $ of finite type and a finite set S of finite words, let $\Sigma \langle S\rangle $ denote the subshift of $\Sigma $ that avoids S. We establish a general criterion under which we can bound the entropy perturbation $h(\Sigma ) - h(\Sigma \langle S\rangle )$ from above. As an application, we prove that this entropy difference tends to zero with a sequence of such sets $S_1, S_2,\ldots $ under various assumptions on the $S_i$.

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.

We provide an explicit $\mathcal {S}$-adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts.

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 investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé’s theorem and actually of a generalization to a more general class of morphisms due to Berthé et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931]. We actually prove the result of Berthé et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.

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.

In this paper, we address the problem of computing the topological entropy of a map $\psi : G \to G$, where G is a Lie group, given by some power $\psi (g) = g^k$, with k a positive integer. When G is abelian, $\psi $ is an endomorphism and its topological entropy is given by $h(\psi ) = \dim (T(G)) \log (k)$, where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math. 234 (2019), 55–80]. However, when G is not abelian, $\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\psi ) = \dim (T)\log (k)$, where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

We prove an explicit characterization of the points in Thurston’s Master Teapot, which can be implemented algorithmically to test whether a point in $\mathbb {C}\times \mathbb {R}$ belongs to the complement of the Master Teapot. As an application, we show that the intersection of the Master Teapot with the unit cylinder is not symmetrical under reflection through the plane that is the product of the imaginary axis of $\mathbb {C}$ and $\mathbb {R}$.

We study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate $(\delta _n)$ of the critical orbit such that, for almost every non-regular parameter a, the set of n such that $\vert F^n(0;a) \vert < \delta _n$ is infinite. In particular, when $\delta _n = n^{-1}$, this extends an earlier result by Avila and Moreira [Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2) 161(2) (2005), 831–881].