Related papers: Criterion for rays landing together

We give a combinatorial definition of "core entropy" for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the $\alpha$ fixed point from its negative). This notion…

Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of…

A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for exactly two external rays with angle which are periodic under doubling. This note will…

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present…

We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain…

A topological mating is a map defined by gluing together the filled Julia sets of two quadratic polynomials. The identifications are visualized and understood by pinching ray-equivalence classes of the formal mating. For postcritically…

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one,…

Let $P: {\mathbb C} \to {\mathbb C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is…

We study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree $d \geq 2$ with connected Julia sets.

We answer a question of Schleicher by showing that, for an exponential map with nonescaping singular value, every periodic ray lands. This is an analog of a theorem of Douady and Hubbard concerning polynomials. We also prove a partial…

In this paper, we use the Carath\'eodory Convergence Theory to prove a landing theorem of rays in hyperbolic components with rational arguments. Although the proof is done in the setting of a family of entire transcendental maps with two…

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a…

For complex quadratic polynomials, the topology of the Julia set and the dynamics are understood from another perspective by considering the Hausdorff dimension of biaccessing angles and the core entropy: the topological entropy on the…

We present a criterion of local Normal Embedding of a semialgebraic (or definable in an o-minimal structure) contained in $R^n$ in terms orders of contact of arcs. Namely, we prove that a semialgebraic set is normally embedded at a point x…

We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the…

Let P be a non-linear polynomial, K_P the filled Julia set of P, f a renormalization of P and K_f the filled Julia set of f. We show, loosely speaking, that there is a finite-to-one function \lambda from the set of P-external rays having…

In this article, we present a landing theorem for periodic dynamic rays for transcendental entire maps which have bounded post-singular sets, by using standard hyperbolic geometry results.

We show that if $P$ is a quadratic polynomial with a fixed Cremer point and Julia set $J$, then for any monotone map $\ph:J\to A$ from $J$ onto a locally connected continuum $A$, $A$ is a single point.

In this article we provide a combinatorial sufficient (and conjecturally, necessary) condition (called $\alpha$-symmetry) for the mating of two postcritically finite polynomials in $\mathcal{S}_1$ to be obstructed. To do this, we study the…

Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The \emph{red dwarf} $J_P$ are nowhere connected im…