Related papers: Hyperbolic Components in Exponential Parameter Spa…

This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker…

We give a complete combinatorial description of the bifurcation structure in the space of exponential maps $z\mapsto\exp(z)+\kappa$. This combinatorial structure is the basis for a number of important results about exponential parameter…

We establish certain uniform a priori bounds for hyperbolic components of disjoint type. As an application, we will prove that Sierpinski carpet hyperbolic components of disjoint type are bounded. Furthermore, we show that for each map $f$…

Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called…

We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself which have degree two or more on each copy. In any space $\p^{S}$ of suitably normalized maps of…

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques…

In complex dynamics, the boundaries of higher dimensional hyperbolic components in holomorphic families of polynomials or rational maps are mysterious objects, whose topological and analytic properties are fundamental problems. In this…

In one-dimensional real and complex dynamics, a map whose post-singular (or post-critical) set is bounded and uniformly repelling is often called a Misiurewicz map. In results hitherto, perturbing a Misiurewicz map is likely to give a…

We give a complete classification of the set of parameters $\kappa$ for which the singular value of $E_{\kappa}:z\mapsto \exp(z)+\kappa$ escapes to infinity under iteration. In particular, we show that every path-connected component of this…

We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we…

For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity…

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $\zeta$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the…

The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be the finite quotients of the Euclidean…

In this paper we provide new characterizations of the Gehring-Hayman theorem from the point of view of Gromov boundary and uniformity. We also determine the critical exponents for the uniformized space to be a uniform space in the case of…

We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+\kappa$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a…

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ \mu_x \}_{x \in X}$ on $\partial X$, we describe a continuous…

Let $\mathbb X$ and $\mathbb Y$ be $\ell$-connected Jordan domains, $\ell \in \mathbb N$, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism $\varphi \colon \partial \mathbb X \to \partial \mathbb Y$…

Let $f$ be a transcendental entire function of finite order which has an attracting periodic point $z_0$ of period at least $2$. Suppose that the set of singularities of the inverse of $f$ is finite and contained in the component $U$ of the…

Suppose $H$ is a finite dimensional reproducing kernel Hilbert space of functions on $X.$ If $H$ has the complete Pick property then there is an isometric map, $\Phi,$ from $X,$ with the metric induced by $H,$ into complex hyperbolic space,…

In this article, we study the hyperbolic components of McMullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the…