Main Cardioid Research Articles

Fractals: Exploring Mandelbrot Coordinates and qualitative characteristics of the corresponding Julia Set

The Mandelbrot Set and Julia Sets are fractals that are generated employing the cyclic equation zn+1 = zn2 + C an infinite number of times, where zn and C are complex numbers. For the Mandelbrot set, the value of zn is static and the equation projects points that do not diverge to infinity for all possible values of C derived from coordinates on the complex plane. For Julia Sets, every value of C remains static while every value for zn that does not diverge to infinity is projected. Because the standard, 2D Mandelbrot set assumes the value of zn = 0, there is only one primary depiction of the Mandelbrot Set. For Julia Sets, the value of C is not assumed, and therefore, there are an infinite number of Julia Sets. In this study, the effect of changing the coordinate from which the value of C is derived in the complex plane for the equation zn+1 = zn2 + C on the qualitative characteristics of the coordinate’s corresponding, standard 2D Julia Set in relation to the standard, 2D Mandelbrot Set is observed —­ ­specifically, Julia Sets derived from points outside the Mandelbrot Set, in the main cardioid of the Mandelbrot Set, in the primary bulbs of the Mandelbrot Set, along the edges of the Mandelbrot set, along the real and imaginary axes.

Generalizations of Douady’s magic formula

AbstractWe 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.

Total disconnectedness of Julia sets of random quadratic polynomials

AbstractFor a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

Global graph of metric entropy on expanding Blaschke products

We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

The size of Mandelbrot bulbs

We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map z→z2−μ. For the (p, q) orbit with winding number p/q, the associated stability bulb is an approximate circle with radius 1q2sinπpq.

Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy

We study the dynamics of a family K_alpha of discontinuous interval maps whose (infinitely many) branches are Moebius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus E_KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K_alpha is analytic outside the bifurcation set but not differentiable at points of E_KU, and that the entropy is monotone as a function of the parameter. Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.

Conformal trace theorem for Julia sets of quadratic polynomials

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

Heat kernel estimates on Julia sets

We give heat kernel estimates on Julia sets J(fc) for quadratic polynomials fc(z)= z2 + c for c in the main cardioid or the ±1k-bulbs where k ≥2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.

The Hausdorff Dimension is Convex on the Left Side of 1/4

AbstractLet d(c) denote the Hausdorff dimension of the Julia set Jc of the polynomial fc(z) = z2 +c. The function c ↦ d(c) is real-analytic on the interval (–3/4, 1/4), which is in the domain bounded by the main cardioid of the Mandelbrot set. We prove that the function d is convex close to 1/4 on the left side of it.

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.

The main cubioid

The connectedness locus in the parameter space of quadratic polynomials is called the Mandelbrot set. A good combinatorial model of this set is due to Thurston. By definition, the principal hyperbolic domain of the Mandelbrot set consists of parameter values, for which the corresponding quadratic polynomials have an attracting fixed point. The closure of the principal hyperbolic domain of the Mandelbrot set is called the main cardioid. Its topology is completely described by Thurston's model. Less is known about the connectedness locus in the parameter space of cubic polynomials. In this paper, we discuss cubic analogues of the main cardioid and establish relationships between them.

CHECKERBOARD JULIA SETS FOR RATIONAL MAPS

In this paper, we consider the family of rational maps [Formula: see text] where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or [Formula: see text] where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

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 ) > 4 / 3 d(-3/4)>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 .

Boundaries of bounded Fatou components of quadratic maps

In this paper, we characterize those external rays that land on the bounded Fatou components of hyperbolic and parabolic quadratic maps. For those maps not in the main cardioid of the Mandelbrot set, we prove that the arguments of these rays form a Cantor subset of the circle at infinity. Our techniques involve both the orbit portraits of Goldberg and Milnor that relate the dynamic and parameter planes and the Thurston theory of laminations for quadratic maps.

Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form ∑ k = 1 ∞ c k exp ( − z g ( k ) ) , Re ( z ) > 0 as z approaches the boundary i R , under the assumption g ′ → ∞ and further assumptions on c k . These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with c k = 1 , the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z = 0 . When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If g ( k ) = k b , c k = 1 , b = n or b = ( n + 1 ) / n , n ∈ N , behavior near the boundary is roughly of the standard form Re ( z ) − b ′ Q ( x ) where Q ( x ) = 1 / q if x = p / q ∈ Q and zero otherwise. The Bötcher map at infinity of polynomial iterations of the form x n + 1 = λ P ( x n ) , | λ | < λ 0 ( P ) , turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P ( x ) = x − x 2 , λ 0 = 1 , and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

Tessellation and Lyubich–Minsky laminations associated with quadratic maps, II: Topological structures of 3-laminations

According to an analogy to quasi-Fuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky’s affine and hyperbolic33-laminations associated with hyperbolic and parabolic quadratic maps.We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same33-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient33-laminations of Douady’s rabbit, the Cauliflower, andz↦z2z \mapsto z^2are described.The descriptions employ a new method oftessellationinside the filled Julia set introduced in Part I [Ergodic Theory Dynam. Systems29(2009), no. 2] that works like external rays outside the Julia set.

External arguments of Douady cauliflowers in the Mandelbrot set

Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated structure, and how the external rays land without intersecting in its Misiurewicz points and minute Mandelbrot sets is not really well known. In this work we study the Douady cauliflowers, giving a binary tree model to calculate the binary expansions of the external arguments of both the main cardioid of each minute Mandelbrot set and Misiurewicz points, and creating a horsetail model that explains the arrangement of its corresponding external rays.

GEOMETRY OF THE ANTENNAS IN THE MANDELBROT SET

In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p/q-bulbs. The reason for this is that the largest component of the interior of these bulbs consists of c-values for which the quadratic function Qc(z) = z2 + c admits an attracting cycle with rotation number p/q. In this paper, we give a geometric method to read off p/q from the geometry of the antenna attached to the bulb.

REAL CONTINUATION FROM THE COMPLEX QUADRATIC FAMILY: FIXED-POINT BIFURCATION SETS

This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family: [Formula: see text] where z is a complex dynamic (phase) variable, [Formula: see text] its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varied. For A=0, the resulting two-real-parameter family is a familiar one-complex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A ≠ 0, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of ℛ2. Saddle-node and period-doubling loci evolve from points on the main cardioid for A=0 into closed curves for A ≠ 0. As A is varied further from 0 in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.

Describing quadratic Cremer point polynomials by parabolic perturbations

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.