Quadratic Julia Sets Research Articles

On the area rate of perturbed Siegel disks

Abstract In the famous work by Buff and Chéritat constructing quadratic Julia sets with positive area, the control of the shape of perturbed Siegel disks is a key technique. To do it, Buff and Chéritat added a restrictive condition on rotation numbers. In this paper, we show that this restrictive condition is not necessary. As a consequence, more quadratic Julia sets with positive area can be constructed.

Hausdorff Dimension of Julia Sets in the Logistic Family

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set \({{\mathcal {M}}}\), the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on c and find explicit bounds at the tip of \({{\mathcal {M}}}\) for most real parameters in the the sense of 1-dimensional Lebesgue measure.

Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

We show that there exist real parameters $$c\in (-2,0)$$ for which the Julia set $$J_{c}$$ of the quadratic map $$z^{2} + c$$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold T(n), there exist a real parameter c such that the computational complexity of computing $$J_{c}$$ with n bits of precision is higher than T(n). This is the first known class of real parameters with a non-poly-time computable Julia set.

Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets

We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.

Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

We prove that Collet–Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.

Entire functions whose Julia sets include any finitely many copies of quadratic Julia sets

We prove that for any finite collection of quadratic Julia sets, a polynomial and a transcendental entire function exist whose Julia sets include copies of the given quadratic Julia sets. In order to prove the result, we construct quasiregular maps with required dynamics and employ the quasiconformal surgery to obtain the desired functions.

Quartic Julia sets including any two copies of quadratic Julia sets

If the Julia set of a quartic polynomial with certain conditions isneither connected nor totally disconnected, there exists a homeomorphismbetween the set of all components of the filled-in Julia setand some subset of the corresponding symbol space.The question is to determine the quartic polynomialsexhibiting such a dynamicsand describe the topology of the connected componentsof their filled-in Julia sets.In this paper, we answer the question, namelywe show that for any two quadratic Julia sets,there exists a quartic polynomial whose Julia set includescopies of the two quadratic Julia sets.

Convergence properties of the Gronwall area formula for quadratic Julia sets

Using parabolic enrichment, it is shown that Gronwall area formula for the filled Julia set along the boundary of the main cardioid of the Mandelbrot set cannot be well approximated by replacing it by a finite sum. A revised version of this article will appear in the Journal of the London Mathematical Society.

Distributional chaos in dendritic and circular Julia sets

If x and y belong to a metric space X, we call (x,y) a DC1 scrambled pair for f:X→X if the following conditions hold:1)for all t>0, limsupn→∞1n|{0≤i<n:d(fi(x),fi(y))<t}|=1, and2)for some t>0, liminfn→∞1n|{0≤i<n:d(fi(x),fi(y))<t}|=0. If D⊂X is an uncountable set such that every x,y∈D form a DC1 scrambled pair for f, we say f exhibits distributional chaos of type 1. If there exists t>0 such that condition 2) holds for any distinct points x,y∈D, then the chaos is said to be uniform. A dendrite is a locally connected, uniquely arcwise connected, compact metric space. In this paper we show that a certain family of quadratic Julia sets (one that contains all the quadratic Julia sets which are dendrites and many others which contain circles) has uniform DC1 chaos.

Non-Computable Impressions of Computable External Rays of Quadratic Polynomials

We discuss computability of impressions of prime ends of compact sets. In particular, we construct quadratic Julia sets which possess explicitly described non-computable impressions.

The -limit sets of quadratic Julia sets

AbstractIn this paper we characterize$\omega $-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin’s symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an$\omega $-limit set of a point if, and only if, it is internally chain transitive.

Hausdorff dimension of quadratic rational Julia sets

Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.

Graphical exploration of the connectivity sets of alternated Julia sets

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness) which characterizes the standard Julia sets. Via computer graphics, we unveil these connectivity domains which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.

Shadowing and internal chain transitivity

The main result of this paper is that a map f : X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with shadowing have the property that every internally chain transitive set is an ω-limit set of a point, and we also show that topologically hyperbolic maps and certain quadratic Julia sets have a closed space of ω-limit sets

Quadratic Julia sets with positive area

We recently proved the existence of quadratic polynomials having a Julia set with positive Lebesgue measure. We present the ideas of the proof and the techniques involved.

Laplacians on a family of quadratic Julia sets II

This paper continues the work started in [4] to construct $P$-invariant Laplacians on the Julia sets of $P(z) = z^2 + c$ for $c$ in the interior of the Mandelbrot set, and to study the spectra of these Laplacians numerically. We are able to deal with a larger class of Julia sets and give a systematic method that reduces the construction of a $P$-invariant energy to the solution of nonlinear finite dimensional eigenvalue problem. We give the complete details for three examples, a dendrite, the airplane, and the Basilica-in-Rabbit. We also study the spectra of Laplacians on covering spaces and infinite blowups of the Julia sets. In particular, for a generic infinite blowups there is pure point spectrum, while for periodic covering spaces the spectrum is a mixture of discrete and continuous parts.

Julia sets converging to filled quadratic Julia sets

AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

Laplacians on a family of quadratic Julia sets I

Laplacians on a family of quadratic Julia sets I

On the derivative of the Hausdorff dimension of the quadratic Julia sets

Let d ( c ) d(c) denote the Hausdorff dimension of the Julia set J c J_c of the polynomial f c ( z ) = z 2 + c f_c(z)=z^2+c . The function c ↦ d ( c ) c\mapsto d(c) is real-analytic on the interval ( − 3 / 4 , 1 / 4 ) (-3/4,1/4) , which is included in the main cardioid of the Mandelbrot set. It was shown by G. Havard and M. Zinsmeister that the derivative d ′ ( c ) d’(c) tends to + ∞ +\infty as fast as ( 1 / 4 − c ) d ( 1 / 4 ) − 3 / 2 (1/4-c)^{d(1/4)-3/2} when c ↗ 1 / 4 c\nearrow 1/4 . Under numerically verified assumption d ( − 3 / 4 ) &gt; 4 / 3 d(-3/4)&gt;4/3 , we prove that d ′ ( c ) d’(c) tends to − ∞ -\infty as − ( c + 3 / 4 ) 3 d ( − 3 / 4 ) / 2 − 2 -(c+3/4)^{3d(-3/4)/2-2} when c ↘ − 3 / 4 c\searrow -3/4 .

On local connectivity for the Julia set of rational maps: Newton’s famous example

We show that Newton’s cubic methods (famous rational maps) have a locally connected Julia set except in some very specific cases. In particular, when those maps are infinitely renormalizable their Julia set is locally connected and contains small copies of nonlocally connected quadratic Julia sets. This also holds when Newton’s method is renormalizable and has Cremer points, unlike the polynomial case. After a dynamical description we show the necessity of the Brjuno condition within this family.