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.

Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties, such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that nontrivially satisfy the conditions for having this equivalence and a nonempty range of inverse temperatures where uniqueness holds.

Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d \geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets $\mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.

For $\mathscr {B} \subseteq \mathbb {N} $, the $ \mathscr {B} $-free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $, have their analogues for $ X_{\eta } $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21(3) (2021), Paper No. 2140008]. A central assumption in our work is that $\eta ^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta } $) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta } $ from above and below.

We generalize to a broader class of decoupled measures a result of Ziv and Merhav on universal estimation of the specific cross (or relative) entropy, originally for a pair of multilevel Markov measures. Our generalization focuses on abstract decoupling conditions and covers pairs of suitably regular g-measures and pairs of equilibrium measures arising from the “small space of interactions” in mathematical statistical mechanics.

We obtain the following embedding theorem for symbolic dynamical systems. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G that has no global period, meaning that the shift action is faithful on Y. If the topological entropy of X is strictly less than that of Y and Y contains at least one factor of X, then X embeds into Y. This result partially extends the classical result of Krieger when $G = \mathbb {Z}$ and the results of Lightwood when $G = \mathbb {Z}^d$ for $d \geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.

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.

In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.

We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $. We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.

Bratteli–Vershik models of compact, invertible zero-dimensional systems have been well studied. We take up such a study for polygonal billiards on the hyperbolic plane, thus considering these models beyond zero-dimensions. We describe the associated Bratteli models and show that these billiard dynamics can be described by Vershik maps.

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.