Related papers: Rotation domains and Stable Baker omitted value

We consider the structure of substantially dissipative complex H\'enon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points…

Given a reproducing kernel $k$ on a nonempty set $X$, we define the reproductive boundary of $X$ with respect to $k$. Furthermore, we generalize the well known nontangential and horocyclic approach regions of the unit circle to this new…

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen…

We construct a transcendental entire $f:\mathbb{C}\rightarrow\mathbb{C}$ such that (1) $f$ has bounded singular set, (2) $f$ has a wandering domain, and (3) each singular value of $f$ escapes to infinity under iteration by $f$.

Let X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere or a torus. We construct a variety of examples of analytic functions g:W->X, where W is an arbitrary subdomain of X, that satisfy Epstein's "Ahlfors islands…

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most…

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…

We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible…

Let $\Omega\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary…

In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape…

We show that, if $f\colon \mathbb{B}^q\to \mathbb{B}^q$ is a holomorphic self-map of the unit ball in $\mathbb{C}^q$ and $\zeta\in \partial \mathbb{B}^q$ is a boundary repelling fixed point with dilation $\lambda>1$, then there exists a…

We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$ in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha \in(0,1)$. We establish…

We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit…

We constructed Yoccoz puzzle for cosine functions $f(z)=ae^z+be^{-z}$ with bounded post-critical set, and proved that a Fatou component is a Jordan domains if it is bounded and is not eventually a Siegal disk. We proved that $f$ is…

We study the dynamics of polynomials with coefficients in a non-Archimedean field $K,$ where $K$ is a field containing a dense subset of algebraic elements over a discrete valued field $k.$ We prove that every wandering Fatou component is…

In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…

We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex…

We will discuss the model of the overdamped Josephson junction in superconductivity, which is given by a family of first order non-linear ordinary differential equations on two-torus depending on three parameters: a fixed parameter $\omega$…

We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville's theorem…

The presence of a boundary enriches the nature of quantum phase transitions. However, the boundary critical phenomena in topological superconductors remain underexplored so far. Here, we investigate the boundary criticality in a…