Connectedness Locus Research Articles

Holomorphic correspondences mating Chebyshev-like maps with Hecke groups

We prove structure theorems for holomorphic correspondences realizing matings between pinched polynomial-like maps with connected Julia sets and Hecke groups (Fuchsian representations of free products C2 * Cn of cyclic groups of orders 2 and n). We show that such matings are generated by two-to-two subcorrespondences if and only if the maps are Chebyshev-like. We describe the dynamical behaviour of these matings in detail in the case n = 4 and we present a conjectured combinatorial description of the connectedness locus in parameter space in this case.

On the ‘Mandelbrot set’ for pairs of linear maps: asymptotic self-similarity

We continue the investigation of iterated function systems (IFS) {λz, λz + 1} in the complex plane, depending on a parameter λ in the open unit disc. Let Aλ be the attractor, and let denote the connectedness locus, that is, the set of λ for which Aλ is connected. We show that the set is locally asymptotically self-similar and asymptotically similar to the attractor of the IFS {λz − 1, λz, λz + 1}, for certain ‘landmark’ points on the boundary of . We also study the parameters for which the attractor Aλ is tree-like, and prove that there are uncountable many such λ, answering a question of Bandt.

A STUDY OF THE DYNAMICS OF λsinz

This paper studies the dynamics of the family λ sin z for some values of λ. First we give a description of the Fatou set for values of λ inside the unit disc. Then for values of λ on the unit circle of parabolic type (λ = exp (i2πθ), θ = p/q, (p, q) = 1), we prove that if q is even, there is one q-cycle of Fatou components, if q is odd, there are two q cycles of Fatou components. Moreover the Fatou components of such cycles are bounded. For λ as above there exists a component Dqtangent to the unit disc at λ of a hyperbolic component.There are examples for λ such that the Julia set is the whole complex plane. Finally, we discuss the connectedness locus and the existence of buried components for the Julia set.

JULIA AND MANDELBROT SETS OF CHEBYSHEV FAMILIES

We present one-parameter families of rational functions of arbitrary degree d which are globally generalized polynomial-like of degree d and roughly speaking locally quadratic-like everywhere, where the parameter appears not only as a purely multiplicative factor but also in a more complicated nonlinear way. The connectedness locus of these families contains homeomorphic copies of the Mandelbrot set. Main emphasis is put on the explicit construction (and not as usual on the existence only) of the sets on which generalized polynomial-likeness and quadratic-likeness are given as well as on the explicit description of the regions where the homeomorphic copies of the Mandelbrot set are located.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Given $d \ge 2$ consider the family of polynomials $P_c(z) = z^d + c$ for $c \in {\mathbb C}$. Denote by $J_c$ the Julia set of $P_c$ and let ${\cal M}_d = \{ c \mid J_c \hbox{ is connected}\}$ be the connectedness locus; for $d = 2$ it is called the Ma

Harmonic measure and expansion on the boundary of the connectedness locus

The paper develops a technique for proving properties that are typical in the boundary of the connectedness locus with respect to the harmonic measure. A typical expansion condition along the critical orbit is proved. This condition implies a number of properties, including the Collet-Eckmann condition, Hausdorff dimension less than 2 for the Julia set, and the radial continuity in the parameter space of the Hausdorff dimensions of totally disconnected Julia sets.

Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture

We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and hairiness of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N > 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.

Hausdorff dimension of subsets of the parameter space for families of rational maps. (A generalization of Shishikura's result)

In this paper we extend Shishikura's result on the Hausdorff dimension of the boundary of the Mandelbrot set to higher-dimensional parameter spaces, for example the space of degree d polynomials, and show that some parameter subsets, including the boundary of the connectedness locus, have Hausdorff dimensions equal to the real dimension of the parameter space (which is four for cubic polynomials). PACS number: 0545

The Fibonacci partition on the connectedness locus and the Farey sequence

As in the case of the Mandelbrot set, the index introduces a Fibonacci partition on the connectedness locus of polynomials of higher degree f c ( z) = z n + c, n ⩾ 3. It is shown that the appearance of the Fibonacci partition is explained in terms of the Farey sequence.

Mating quadratic maps with the modular group

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences $$\mathcal {F}_a$$: $$\begin{aligned} \left( \frac{aw-1}{w-1}\right) ^2+\left( \frac{aw-1}{w-1}\right) \left( \frac{az+1}{z+1}\right) +\left( \frac{az+1}{z+1}\right) ^2=3 \end{aligned}$$and proved that for every value of $$a \in [4,7] \subset \mathbb {R}$$ the correspondence $$\mathcal {F}_a$$ is a mating between a quadratic polynomial $$Q_c(z)=z^2+c,\,\,c \in \mathbb {R}$$, and the modular group $$\varGamma =PSL(2,\mathbb {Z})$$. They conjectured that this is the case for every member of the family $$\mathcal {F}_a$$ which has a in the connectedness locus. We show here that matings between the modular group and rational maps in the parabolic quadratic family $$Per_1(1)$$ provide a better model: we prove that every member of the family $$\mathcal {F}_a$$ which has a in the connectedness locus is such a mating.

Connectedness of the tricorn

AbstractIn this note, we show the connectedness of the tricorn, the connectedness locus for the family of antiquadratic maps: fc(z) = + c, c ∈ C.

Dynamics of Certain Nonconformal Degree-Two Maps of the Plane

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates): $$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$ When $\alpha=1$, these maps are quadratic ($z \maps z^2 + c$), and their dynamics and bifurcation theory are to some degree understood. When $\alpha$ is different from one, the dynamics is no longer conformal. In particular, the dynamics is not completely determined by the orbit of the critical point. Nevertheless, for many values of the parameter c, the dynamics has strong similarities to that of the quadratic family. For other parameter values the dynamics is dominated by 2 dimensional behavior: saddles and the like. The objects of study are Julia sets, filled-in Julia sets and the connectedness locus. These are defined in analogy to the conformal case. The main drive in this study is to see to what extent the results in the conformal case generalize to that of maps which are topologically like quadratic maps (and when $\alpha$ is close to one, close to being quadratic).