Related papers: The Julia-Wolff-Caratheodory theorem in polydisks

We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and…

Polya-Carlson theorem asserts that if a power series with integer coefficients and convergence radius 1 can be extended holomorphically out of the unit disc, it must represent a rational function. In this note, we give a generalization of…

We use Bowen's definition of topological entropy and Ahlfors five islands theorem, as well as the theory of polynomial-like mappings, to show that the topological entropy of any entire transcendental function is infinity. In addition the…

Let $F=(\phi, \psi):\mathbb{D}^2\to\mathbb{D}^2$ denote a holomorphic self-map of the bidisk without interior fixed points. It is well-known that, unlike the case with self-maps of the disk, the sequence of iterates $$\{F^n:=F\circ F\circ…

A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type results for compositions of self-maps of the unit disc. Here, we tackle the particular case of inner functions: if $f_n:\mathbb{D}\to\mathbb{D}$ are inner…

We investigate the set of uniform limits of polynomials on any closed Jordan domain with respect to the chordal metric $\chi$ on $\mathbb{C}\cup\{\infty \}$. We conclude that Mergelyan's Theorem may be extended to the case of uniform…

Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable…

We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the…

Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose…

We obtain upper bounds for the norm of the Schwarzian derivative of convex holomorphic mappings defined on the polydisk and the unit ball in $\mathbb{C}^n$. For coordinate-wise convex mappings on the polydisk, we derive a sharp estimate…

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia…

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb…

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…

A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on…

We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth…

We show that if $f$ is a nonzero, noninvertible function on a smooth complex variety $X$ and $J_f$ is the Jacobian ideal of $f$, then ${\rm lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities.…

Let $\Delta\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbb D \to \Delta$ be a Riemann map and let $\{z_k\}\subset \Delta$ be a compactly divergent sequence. Using Gromov's hyperbolicity theory, we show that…

The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…

The purpose of this article is threefold. The first is to construct a Nevanlina theory for meromorphic mappings from a polydisc to a compact complex manifold. In particular, we give a simple proof of Lemma on logarithmic derivative for…

We study in detail the one-variable local theory of functions holomorphic over a finite-dimensional commutative associative unital $\mathbb{C}$-algebra $\mathcal{A}$, showing that it shares a multitude of features with the classical…