Related papers: Boundaries of escaping Fatou components

We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $\varphi$ be a holomorphic…

In this article, the dynamics of a one-parameter family of functions $f_{\lambda}(z) = \frac{\sin{z}}{z^2 + \lambda},$ $\lambda>0$, are studied. It shows the existence of parameters $0< \lambda_{1}< \lambda_{2}$ such that bifurcations occur…

A Baker omitted value, in short \textit{bov} of a transcendental meromorphic function $f$ is an omitted value such that there is a disk $D$ centered at the bov for which each component of the boundary of $f^{-1}(D)$ is bounded. Assuming all…

A major open question in transcendental dynamics asks if it is possible for points in a wandering domain to have bounded orbits, and more strongly, for a wandering domain to iterate only in a bounded domain. In this paper we give a partial…

We provide a complete system of analytic invariants for unfoldings of non-linearizable resonant complex analytic diffeomorphisms as well as its geometrical interpretation. In order to fulfill this goal we develop an extension of the Fatou…

We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental…

Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let $X$…

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation \[ \frac{\partial^2 u}{\partial t^2} + \Delta u = 0, \] where $K$ is a p.c.f. set and…

We consider overdetermined problems related to the fractional capacity. In particular we study $s$-harmonic functions defined in unbounded exterior sets or in bounded annular sets, and having a level set parallel to the boundary. We first…

This paper consists of tow parts. One is to study the existence of a point $a$ in the intersection of Julia set and escaping set such that $\arg z=\theta$ is a singular direction if $\theta$ is a limit point of $\{\arg f^n(a)\}$ under some…

This paper works on the structure of infinitely connected Fatou damains of rational maps in terms of Koebe uniformization. Due to the complicated boundary behavior, the existing uniformization results are failed to apply in general. We…

Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,\alpha}$…

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic…

We show that for a transcendental entire function the set of points whose orbit under iteration is bounded can have arbitrarily small positive Hausdorff dimension.

In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we…

Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming…

Boundary analysis is developed for a rich class of generally infinite weighted graphs with compact metric completions. These graph completions have totally disconnected boundaries. The classical notion of $\epsilon$-components and the…

We show that an escape from the potential minimum of Fabry-Perot interferometers can be detected measuring the associated sudden change of reflectivity. We demonstrate that the loss of information that occurs retaining only the sequence of…

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and…

Let $f$ be a transcendental entire function. By a result of Rippon and Stallard, there exist points whose orbit escapes arbitrarily slowly. By using a range of techniques to prove new covering results, we extend their theorem to prove the…