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.

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.

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.

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.

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.

We study the ergodic properties of the translation surface $X_{\unicode{x3bb} ,\mu }$ formed by gluing two flat tori along a slit with holonomy $(\unicode{x3bb} ,\mu ) \in \mathbb {R}^2$. Extending the dichotomy result of Cheung, Hubert, and Masur for the case $\mu = 0$, we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set $\operatorname {\mathrm {NE}}(X_{\unicode{x3bb} ,\mu },\omega )$ of non-ergodic directions is either $0$ or $\frac 12$. This dichotomy is completely characterized by the Pérez–Marco condition expressed in terms of best approximation denominators. As a corollary, we obtain that the Pérez–Marco condition for best approximation denominators is norm-independent.

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.

Polynomial Julia sets with tree structure, typically Hubbard trees, play an important role in holomorphic dynamics. In this paper, we study the dynamics of ${f_\alpha (z)=z^2+\alpha \bar {z}}$ for $\alpha $ being real and the Julia sets being trees. We show that all such $\alpha $ form an interval $[1,4]$. This answers a question of G. Sienra. We further show that $f_\alpha $ exhibits non-trivial dynamics on the Fatou set which equals to the escaping set.

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 $\mathbb {Z}$.

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 $.

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.

We solve the dry ten Martini problem for the unitary almost-Mathieu operator with Diophantine frequencies in the non-critical regime.

Let $\gamma _{n}= O (\log ^{-c}n)$ and let $\nu $ be the infinite product measure whose nth marginal is Bernoulli $(1/2+\gamma _{n})$. We show that $c=1/2$ is the threshold, above which $\nu $-almost every point is simply Poisson generic in the sense of Peres and Weiss, and below which this can fail. This provides a range in which $\nu $ is singular with respect to the uniform product measure, but $\nu $-almost every point is simply Poisson generic.

In this paper, we build some ergodic theorems involving the function $\Omega $, where $\Omega (n)$ denotes the number of prime factors of a natural number n counted with multiplicities. As a combinatorial application, it is shown that for any $k\in \mathbb {N}$ and every $A\subset \mathbb {N}$ with positive upper Banach density, there are $a,d\in \mathbb {N}$ such that $a,a+d,\ldots, a+kd,a+\Omega(d)\in A.$

Piecewise contractions (PCs) are piecewise smooth maps that decrease the distance between pairs of points in the same domain of continuity. The dynamics of a variety of systems is described by PCs. During the last decade, much effort has been devoted to proving that in parameterized families of one-dimensional PCs, the $\omega $-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor $\omega $-limit sets. In this article, we extend these results to the multi-dimensional case. More precisely, we provide criteria to show that an arbitrary family $\{f_{\mu }\}_{\mu \in U}$ of locally bi-Lipschitz piecewise contractions $f_\mu :X\to X$ defined on a compact metric space X is asymptotically periodic for Lebesgue almost every parameter $\mu $ running over an open subset U of the M-dimensional Euclidean space $\mathbb {R}^M$. As a corollary of our results, we prove that piecewise affine contractions of $\mathbb {R}^d$ defined in generic polyhedral partitions are asymptotically periodic.