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.

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 $\mathbb {Z}$.

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.

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.

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 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.

In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.

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 $a,a+d,\ldots, a+kd,a+\Omega(d)\in A.$

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.