In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

We prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).

Katok [Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173] conjectured that every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c \in [0, h_{\mathrm {top}}(f))$, there exists an ergodic measure $\mu $ of f satisfying $h_{\mu }(f)=c$. In this paper, we obtain a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for basic sets of flows that characterize the refined roles of ergodic measures in the invariant ones. In this process, we establish a ‘multi-horseshoe’ entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain the same result for singular hyperbolic attractors.

We characterize the stabilized automorphism group for odometers and Toeplitz subshifts, and then prove an invariance property of the stabilized automorphism group of these dynamical systems. Namely, we prove the isomorphism invariance of the primes for which the p-adic valuation of the period structure tends to infinity. A particular case of interest is that for torsion-free odometers, the stabilized automorphism group is a full isomorphism invariant.

In this paper, we first give a necessary and sufficient condition for a factor code with an unambiguous symbol to admit a subshift of finite type restricted to which it is one-to-one and onto. We then give a necessary and sufficient condition for the standard factor code on a spoke graph to admit a subshift of finite type restricted to which it is finite-to-one and onto. We also conjecture that for such a code, the finite-to-one and onto property is equivalent to the existence of a stationary Markov chain that achieves the capacity of the corresponding deterministic channel.

We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.

For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.

We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G, and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike V) does not admit a proper action on a CAT$(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci. 84 (2017), 288–301].

We prove results about subshifts with linear (word) complexity, meaning that $\limsup \frac {p(n)}{n} < \infty $, where for every n, $p(n)$ is the number of n-letter words appearing in sequences in the subshift. Denoting this limsup by C, we show that when $C < \frac {4}{3}$, the subshift has discrete spectrum, that is, is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with $C = \frac {3}{2}$ which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether $C = \frac {5}{3}$ was the minimum possible among such subshifts; our results show that the infimum in fact lies in $[\frac {4}{3}, \frac {3}{2}]$. All results are consequences of a general S-adic/substitutive structure proved when $C < \frac {4}{3}$.

We consider the attractor $\Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $\Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.

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.

In this paper, we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar é–Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

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 prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

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.

We prove a generalization of Krieger’s embedding theorem, in the spirit of zero-error information theory. Specifically, given a mixing shift of finite type X, a mixing sofic shift Y, and a surjective sliding block code $\pi : X \to Y$, we give necessary and sufficient conditions for a subshift Z of topological entropy strictly lower than that of Y to admit an embedding $\psi : Z \to X$ such that $\pi \circ \psi $ is injective.

Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

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.