Connectedness Locus Research Articles

Embedding unicritical connectedness loci

In this article, for degree d ≥ 1 d\geq 1 , we construct an embedding Φ d \Phi _d of the connectedness locus M d + 1 \mathcal {M}_{d+1} of the polynomials z d + 1 + c z^{d+1}+c into the connectedness locus of degree 2 d + 1 2d+1 bicritical odd polynomials.

Symmetric cubic laminations

To investigate the degree d d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

Analytic structures and harmonic measure at bifurcation locus

We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci Md for unicritical polynomials fc(z)=zd+c. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove C1+α2d+α−ϵ-conformality, α=2−HD(Jc0), of the parameter-phase space similarity maps ϒc0(z):C↦C at typical c0∈∂Md and establish that globally quasiconformal similarity maps ϒc0(z), c0∈∂Md, are C1-conformal along external rays landing at c0 in C∖Jc0 mapping onto the corresponding rays of Md. This conformal equivalence leads to the proof that the z-derivative of the similarity map ϒc0(z) at typical c0∈∂Md is equal to 1/T(c0), where T(c0)=∑n=0∞(D(fc0n)(c0))−1 is the transversality function.The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on ∂Md, [27]. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in ∂Md effectively blocking a good access of ∂Md from the outside.

Matings of cubic polynomials with a fixed critical point. Part II: α-symmetry of limbs

Following Milnor, define to be the space of monic, centred cubic polynomials with a marked fixed critical point. In this article, a sequel to [Thomas Sharland, Matings of cubic polynomials with a fixed critical point, part I: Thurston obstructions, Conform. Geom. Dyn. 23 (12) (2019), pp. 205–220], we provide a combinatorial sufficient (and conjecturally, necessary) condition (called α-symmetry) for the mating of two postcritically finite polynomials in to be obstructed. To do this, we study the rotation sets associated to the parameter limbs in the connectedness locus of , which allows us to determine when there exist ray classes in the formal mating which contain a closed loop. We give a proof of the necessity of α-symmetry for a particular subset of postcritically finite maps in . Many examples are given to illustrate the results of the paper.

Dynamics of modular matings

We develop a dynamical theory for the family of holomorphic correspondences Fa proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice Per1(1) ([2]). Such a mating endows the complement of the limit set of Fa with the geometry of the hyperbolic plane, equipped with the action of the modular group. We introduce bi-infinite coding sequences for geodesics in this complement, utilising continued fraction expressions of end points; we prove landing theorems for periodic and preperiodic geodesics, and we establish a stronger Yoccoz inequality for repelling fixed points of these correspondences than Yoccoz's classical inequality for quadratic polynomials. We deduce that the connectedness locus of the family Fa is contained in a particular lune in parameter space.

Accessibility and porosity of harmonic measure at bifurcation locus

We study hyperbolic geodesics running from ∞ $\infty$ to a generic point, by the harmonic measure with the pole at ∞ $\infty$ , on the boundary of the connectedness locus M d ${\cal M}_d$ for unicritical polynomials f c ( z ) = z d + c $f_c(z)=z^d+c$ . It is known that a generic parameter c ∈ ∂ M d $c\in \partial {\cal M}_d$ is not accessible within a John angle and ∂ M d $\partial {\cal M}_d$ spirals round them infinitely many times in both directions. We prove that almost every point from ∂ M d $\partial {\cal M}_d$ is asymptotically accessible by a flat angle with apperture decreasing slower than ( log ∘ ⋯ ∘ log dist ( c , ∂ M d ) ) − 1 $(\log \circ \dots \circ \log {\rm dist}\,(c,\partial {\cal M}_d))^{-1}$ for any iterate of log $\log$ . This is a consequence of an iterated large deviation estimate for exponential distribution. Additionally, for an arbitrary β > 0 $\beta >0$ , the bifurcation locus is not β $\beta$ -porous on a set of scales of positive density along almost every external ray with respect to the harmonic measure.

Relations Between Escape Regions in the Parameter Space of Cubic Polynomials

We describe a topological relationship between slices of the parameter space of cubic maps. In the paper [9], Milnor defined the curves \(\mathcal {S}_{p}\) as the set of all cubic polynomials with a marked critical point of period p. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves \(\mathcal {S}_{1}\) and \(\mathcal {S}_{2}\).

Discontinuity of Straightening in Anti-holomorphic Dynamics: I

It is well known that baby Mandelbrot sets are homeomorphic to the original one. We study baby Tricorns appearing in the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials, and show that the dynamically natural straightening map from a baby Tricorn to the original Tricorn is discontinuous at infinitely many explicit parameters. This is the first known example of discontinuity of straightening maps on a real two-dimensional slice of an analytic family of holomorphic polynomials. The proof of discontinuity is carried out by showing that all non-real umbilical cords of the Tricorn wiggle, which settles a conjecture made by various people including Hubbard, Milnor, and Schleicher.

On a quartic polynomials family of two parameters

ABSTRACT We consider a family of quartic polynomials generated by the composition of two quadratic polynomials. The elements of this family have two complex parameters, however they have at most two dynamic behaviors, since every map in this family have two critical points with the same forward orbits. In this paper, we study this quartic family in the complex parameter space, and we describe the dynamical plane for some special parameters. Moreover, we analyze the parameter space for these quartic polynomials with a super attracting fixed point. We describe the connectedness locus for this family, and we prove the locally connectedness of the boundary of hyperbolic components in the parameter space.

Mating quadratic maps with the modular group II

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences mathcal {F}_a: aw-1w-12+aw-1w-1az+1z+1+az+1z+12=3\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( \\frac{aw-1}{w-1}\\right) ^2+\\left( \\frac{aw-1}{w-1}\\right) \\left( \\frac{az+1}{z+1}\\right) +\\left( \\frac{az+1}{z+1}\\right) ^2=3 \\end{aligned}$$\\end{document}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.

Geometric limits of Julia sets and connectedness locus of the family of polynomials Pc(z) = zn + czk

ABSTRACTLet n and k be positive integers. For , we consider the connectedness locus of the family of polynomials , where c is a complex parameter. We show that the geometric limit of the connectedness locus sets , when n tends to infinity, exists and is the closed unit disk. In addition, we give an upper bound for the geometric size of .When parameter c belongs to the open unit disk, we show that the geometric limit of the Julia sets , when n tends to infinity, exists and is the unit circle. Finally, we have established some properties on the hyperbolic components of this family.

Models for spaces of dendritic polynomials

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the pinched disk model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.

Distribution of postcritically finite polynomials III: Combinatorial continuity

In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped by Kiwi and Dujardin-Favre. In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the boundary of the Tricorn. We also establish an approximation property by Misiurewicz parameters in the spirit of the previous one. Finally, we answer a question of Kiwi, exhibiting in the moduli space of degree 4 polynomials, non-trivial Impression of specific combinatorics.

Self-Similarity for the Tricorn

ABSTRACTWe discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author [IK 12], we show that there are many homeomorphic copies of the Mandelbrot sets. With the help of rigorous numerical computation, we also prove that the straightening map is not continuous for the candidate of a “baby tricorn” centered at the airplane, hence not a homeomorphism.

Fine structure of connectedness loci

We study distributions of the harmonic and Lebesgue measures on the boundary of the connectedness locus \(\mathcal{M}_{d}\) for uni-critical polynomials \(z^{d}+c\) and dynamical properties which are typical with respect to the harmonic measure. One result is a conformal similarity between \(\mathcal{M}_{d}\) and the corresponding Julia set for almost every point of \(\mathcal{M}_{d}\) with respect to the harmonic measure. It is also shown that the harmonic measure is supported on Lebesgue density points of the complement of \(\mathcal{M}_{d}\) which are not accessible from outside within John angles and at which the boundary of \(\mathcal{M}_{d}\) ‘spirals’ infinitely often in both directions. In the case of quadratic polynomials, we prove that every parameter c from the boundary of the Mandelbrot set \(\mathcal{M}\) that is recurrent under iterates of \(z^{2}+c\) but not infinitely tunable is a Lebesgue weak density point of the complement of \(\mathcal{M}\). A direct consequence of our theory is the Yoccoz local connectivity theorem. A new technical ingredient in the paper is the method of “amplification” which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding Julia set. The pointwise conformality of the similarity map is derived through inducing techniques and a classical TWB-theory.

Laminations from the main cubioid

Polynomials from the closure of the principal hyperbolic domain of the cubic connectedness locus have some specific properties, which were studied in a recent paper by the authors. The family of (affine conjugacy classes of) all polynomials with these properties is called the Main Cubioid. In this paper, we describe a combinatorial counterpart of the Main Cubioid --- the set of invariant laminations that can be associated to polynomials from the Main Cubioid.

The space of 2-generator postcritically bounded polynomial semigroups and random complex dynamics

We investigate the dynamics of 2-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space B of such semigroups. We show that for a parameter h in the intersection of B, the hyperbolicity locus H and the closure of the disconnectedness locus (the space of parameters for which the Julia set is disconnected), the corresponding semigroup satisfies either the open set condition (and Bowen's formula) or that the Julia sets of the two generators coincide. Also, we show that for such a parameter h, if the Julia sets of the two generators do not coincide, then there exists a neighborhood U of h such that for each parameter in U, the Hausdorff dimension of the Julia set of the corresponding semigroup is strictly less than 2. Moreover, we show that the intersection of the connectedness locus and B∩H has dense interior. By using the results on the semigroups corresponding to these parameters, we investigate the associated functions which give the probability of tending to ∞ (complex analogues of the devil's staircase or Lebesgue's singular functions) and complex analogues of the Takagi function.

On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

Themulticornsare the connectedness loci of unicritical antiholomorphic polynomials$\bar{z}^{d}+c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

Let$\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$and$T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$. Let$A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$be the unique compact set satisfying$A=T_{1}(A)\cup T_{-1}(A)$. In this paper, we give a detailed analysis of$A$and the parameters$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$where$A$satisfies various topological properties. In particular, we show that if$\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$, then$A$has a non-empty interior, thus significantly improving the bound from Dajaniet al[Self-affine sets with positive Lebesgue measure.Indag. Math. (N.S.)25(2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series.Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of$A$which have a unique address has positive Hausdorff dimension for all$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$. Finally, we investigate simultaneous$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$-expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings.Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference2008, pp. 743–748].

Non-landing parameter rays of the multicorns

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 existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of $\mathcal{M}_d^*$, and the Misiurewicz parameters are not dense on the boundary of $\mathcal{M}_d^*$.