Abstract An avoshift is a subshift where for each set C from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on C to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets C ) by, for example, cellwise quasigroup shifts, totally extremally permutive (TEP) subshifts, and subshifts of finite type (SFTs) with a safe symbol. In this paper, we concentrate on avoshifts on polycyclic groups, when the sets C are what we call ‘inductive intervals’. We show that then, avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.

Abstract For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

Abstract Given a weakly almost additive sequence of continuous functions with bounded variation ${\mathcal {F}}=\{\log f_n\}_{n=1}^{\infty }$ on a subshift X over finitely many symbols, we study properties of a function f on X such that $\lim _{n\to \infty }({1}/{n})\int \log f_n\,d\mu =\int f\,d\mu $ for every invariant measure $\mu $ on X . Under some conditions, we construct a function f on X explicitly, and study a relation between the property of ${\mathcal {F}}$ and some particular types of f . As applications, we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.

Abstract In this paper, we study the quantitative recurrence properties in the case of $\mathbb {Z}$ -extension of Axiom A flows on a Riemannian manifold. We study the asymptotic behavior of the first return time to a small neighborhood of the starting point. We establish results of almost everywhere convergence, and of convergence in distribution with respect to any probability measure absolutely continuous with respect to the infinite invariant measure. In particular, our results apply to geodesic flows on the $\mathbb {Z}$ -cover of compact smooth surfaces of negative curvature.

Abstract In this article, we prove that the set of well-approximable points $W_\varphi (z) = \{x \in X : d (f^n x, z ) < \varphi (n) \mathrm {\ for\ infinite\ } n \in \mathbb {N}^+\}$ in the shrinking targets problems is distributional chaotic of type 1 for systems with a weak form of the exponential specification property. We apply it to transitive Anosov systems, $\beta $ -shifts, etc.

Abstract In this paper, we prove a quantitative equidistribution theorem for polynomial sequences in a nilmanifold, where the average is taken along spheres instead of cubes. To be more precise, let $\Omega \subseteq \mathbb {Z}^{d}$ be the preimage of a sphere $\mathbb {F}_{p}^{d}$ under the natural embedding from $\mathbb {Z}^{d}$ to $\mathbb {F}_{p}^{d}$ . We show that if a rational polynomial sequence $(g(n)\Gamma )_{n\in \Omega }$ is not equidistributed on a nilmanifold $G/\Gamma $ , then there exists a non-trivial horizontal character $\eta $ of $G/\Gamma $ such that $\eta \circ g \,\mod \mathbb {Z}$ vanishes on $\Omega $ .

Abstract Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal {S}$ -adic subshifts. We obtain the uncountable case by showing that any topological dynamical system with at most countably many asymptotic components has zero topological entropy. We also construct systems that have arbitrarily high subexponential word complexity, but only one asymptotic component. We deduce that within any strong orbit equivalence class, there exists a minimal subshift whose automorphism group is isomorphic to $\mathbb {Z}$ .

Abstract In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. We use these disintegration statements to prove new results on the Diophantine properties of these measures.

Abstract Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias [Meyer’s concept of quasicrystal and quasiregular sets. Comm. Math. Phys. 179 (2) (1996), 365–376] provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset X of $\mathbb {R}^n$ is relatively dense and $X - X$ is uniformly discrete, then X is an approximate lattice. We prove two generalizations of Lagarias’ theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.

Abstract We prove that when the Aubry set for a Lipschitz continuous potential is a subshift of finite type, then the pressure function converges exponentially fast to its asymptote as the temperature goes to 0. The speed of convergence turns out to be the unique eigenvalue for the matrix whose entries are the costs between the different irreducible pieces of the Aubry set. For a special case of Walters potential, we show that perturbations of that potential that go faster to zero than the pressure do not change the selection, neither for the subaction nor for the limit measure, a zero temperature.