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, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set. Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.
In this paper, we prove that any parameter ray at a non-recurrent angle $θ$ lands at a non-recurrent parameter $c$ with $θ$ a characteristic angle of $f_c$; and conversely, every non-recurrent parameter $c$ is the landing point of one or two parameter rays at non-recurrent angles, and these angles are exactly the characteristic angles of $f_c$.
This paper investigates the set of angles of the parameter rays which land on the real slice $[-2,1/4]$ of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for the tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.
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 try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard…
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 the set of rays landing on the real section of the Mandelbrot set, to the right of the given parameter c. We then generalize the result by looking at the entropy of Hubbard trees: namely, we relate the Hausdorff dimension of the set of external angles which land on a certain slice of a principal vein in the Mandelbrot set to the topological entropy of the q…
We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misi…
We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set.
We prove that given any $θ_1,\ldots,θ_{2d-2}\in \R\setminus\Z$, the support of the bifurcation measure of the moduli space of degree $d$ rational maps coincides with the closure of classes of maps having $2d-2$ neutral cycles of respective multipliers $e^{2iπθ_1},\ldots,e^{2iπθ_{2d-2}}$. To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of $(\partial \Mand)^{k}$ are dense in the support of the $k^{th}$-bifurcation current $T^k_\bif$ in general families of rational maps, where $\Mand$ is the Mandelbrot set. As a consequence, we also get sharp dimension est…
On subsets E of the Mandelbrot set M, homeomorphisms are constructed by quasi-conformal surgery. When the dynamics of quadratic polynomials is changed piecewise by a combinatorial construction, a general theorem yields the corresponding homeomorphism h: E to E in the parameter plane. Each h has two fixed points in E, and a countable family of mutually homeomorphic fundamental domains. Possible generalizations to other families of polynomials or rational mappings are discussed. The homeomorphisms on subsets E of M constructed by surgery are extended to homeomorphisms of M, and employed to study…
For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\hat{c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt{|c-\hat{c}|})$ when $c$ belongs to a parameter ray that lands on $\hat{c}$. We also characterize the degeneration of the dynamics along the parameter ray.
We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components $H$ of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on $H$ to the set of angles of external rays landing on the real axis.
A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we prove {\it a priori} bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.
It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn $\mathcal{M}_d^*$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existenc…
A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called biaccessible if two parameter rays land at $z$. In both cases, we say that the external angles of these two rays are biaccessible as well. In this paper we give upper and lower bounds for the Hausdorff dimension of biaccessible external angles of quadratic polynomials, both in the dynamical and parameter space. In particular, explicitly describe those quadratic polynomials where this dimension equals 1 (if and only if the Julia set is an interval)…
In this paper, we use the Carathéodory 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 singular values, the method can be generalized to many other one parameter families, with some modifications.
The aim of this work is to describe the equivalence relations in $\Q/\Z$ that arise as the rational lamination of polynomials with all cycles repelling. We also describe where in parameter space one can find a polynomial with all cycles repelling and a given rational lamination. At the same time we derive some consequences that this study has regarding the topology of Julia sets.
One of the fundamental properties of the Mandelbrot set is that the set of postcritically finite parameters is structured like a tree. We extend this result to the set of quadratic kneading sequences and show that this space contains no irrational decorations. Along the way, we prove a combinatorial analogue to the correspondence principle of dynamic and parameter rays. Our key tool is to work simultaneously with the two equivalent combinatorial concepts of Hubbard trees and kneading sequences.
This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of piecewise-linear maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^2 that map rays from the origin into rays from the origin. The action on rays defines a map S_{ab} of the circle, which has a well-defined rotation number. This paper characterizes …
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 Hubbard tree. These quantities are related according to Thurston. Tiozzo [arXiv:1305.3542] has shown continuity on principal veins of the Mandelbrot set M . This result is extended to all veins here, and it is shown that continuity with respect to the external angle theta will imply continuity in the parameter c . Level sets of the biaccessibility dimension are …
