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We show the existence of singular inner functions that are cyclic in some Besov-type spaces of analytic functions over the unit disc. Our sufficient condition is stated only in terms of the modulus of smoothness of the underlying measure.…

We show that, given a family of discs centered at a nice curve, the analytic capacities of arbitrary subsets of these discs add up. However we need that the discs in question would be slightly separated, and it is not clear whether the…

If $f$ is a real-valued function on $[-\pi,\pi]$ that is Henstock--Kurzweil integrable, let $u_r(\theta)$ be its Poisson integral. It is shown that $\|u_r\|_p=o(1/(1-r))$ as $r\to 1$ and this estimate is sharp for $1\leq p\leq\infty$. If…

Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups…

For a function $f\colon [0,1]\to\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\colon [0,1]\to\mathbb R$ and a Cantor set $D\subset [0,1]$ with $\{0,1\}\subset D$, we obtain conditions equivalent…

We prove a Julia-Wolff-Carath\'edory theorem on angular derivatives of infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit ball $B^n\subset\mathbb{C}^n$, starting from results recently obtained by…

Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the…

There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane $\Bbb C$ has the…

We study the entanglement entropy of a scalar filed in 2+1 spacetime where space is modeled by a fuzzy sphere and a fuzzy disc. In both models we evaluate numerically the resulting entropies and find that they are proportional to the number…

The main goal of this article is to bring together the theories of holomorphic iteration in the unit disc and semigroups of holomorphic functions. We develop a technique that allows us to partially embed the orbit of a holomorphic self-map…

The entanglement entropy for smooth regions $\cal A$ has a logarithmic divergent contribution with a shape dependent coefficient and that for regions with conical singularities an additional $\log ^2$ term. Comparing the coefficient of this…

The embeddability problem is a very old and hard problem in discrete holomorphic iteration which deals with determining general conditions on a given univalent self-map $\varphi$ of the unit disc $\mathbb D$ in order to be contained in a…

Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction…

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show…

We derive general formula for the fourth coefficient of the functions belonging to the Carath\'{e}odory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

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 given class whose number of asymptotic…

If $\mathcal{G}$ is the group (under composition) of diffeomorphisms $f : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ of the closed unit disc ${\bar{D}}(0;1)$ which are the identity map $id : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ on the…

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…