Quasiconformal Surgery Research Articles

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

Julia Sets, Jordan Curves and Quasi-circles

In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in (Carleson and Gamelin in Complex dynamics, Springer, Berlin, 1993; Steinmetz in Math Ann 307:531–541, 1997), will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critical points in simply connected (super-)attracting and parabolic basins into a single critical point of highest possible multiplicity.

On the correspondence of external angles under renormalization

Let P $P$ be a monic polynomial of degree D ⩾ 3 $D \geqslant 3$ whose filled Julia set K P $K_P$ has a non-degenerate periodic component K $K$ of period k ⩾ 1 $k \geqslant 1$ and renormalization degree 2 ⩽ d < D $2 \leqslant d<D$ . Let I = I K $I=I_K$ denote the set of angles θ $\theta$ on the circle T = R / Z ${\mathbb {T}}={\mathbb {R}}/{\mathbb {Z}}$ for which the (smooth or broken) external ray R θ P $R^P_\theta$ for P $P$ accumulates on ∂ K $\partial K$ . We prove the following. I $I$ is a compact set of Hausdorff dimension ⩽ log d / ( k log D ) < 1 $\leqslant \log d/(k \log D)<1$ and there is a continuous surjection Π : I → T $\Pi : I \rightarrow {\mathbb {T}}$ which semiconjugates θ ↦ D k θ ( mod Z ) $\theta \mapsto D^k \theta \ (\operatorname{mod} {\mathbb {Z}})$ on I $I$ to θ ↦ d θ ( mod Z ) $\theta \mapsto d \theta \ (\operatorname{mod} {\mathbb {Z}})$ on T ${\mathbb {T}}$ . Moreover, Π $\Pi$ is unique up to postcomposition with a power of the rotation θ ↦ θ + 1 / ( d − 1 ) ( mod Z ) $\theta \mapsto \theta +1/(d-1) \ (\operatorname{mod} {\mathbb {Z}})$ . Any hybrid conjugacy φ $\varphi$ between a renormalization of P ∘ k $P^{\circ k}$ on a neighborhood of K $K$ and a degree d $d$ monic polynomial Q $Q$ induces a semiconjugacy Π : I → T $\Pi : I \rightarrow {\mathbb {T}}$ as above with the property that for every θ ∈ I $\theta \in I$ the external ray R θ P $R^P_\theta$ has the same accumulation set as the curve φ − 1 ( R Π ( θ ) Q ) $\varphi ^{-1}(R^Q_{\Pi (\theta )})$ . In particular, R θ P $R^P_\theta$ lands at z ∈ ∂ K $z \in \partial K$ if and only if R Π ( θ ) Q $R^Q_{\Pi (\theta )}$ lands at φ ( z ) ∈ ∂ K Q $\varphi (z) \in \partial K_Q$ . The projection Π $\Pi$ is uniformly finite-to-one. In fact, the cardinality of each fiber of Π $\Pi$ is ⩽ D − d + 2 $\leqslant D-d+2$ , and the inequality is strict when the component K $K$ has period k = 1 $k=1$ . The upper bound in the above result is sharp. Using a new type of quasiconformal surgery, we construct polynomials P $P$ with a prescribed hybrid class near K = P ( K ) $K=P(K)$ and a prescribed set of D − d + 1 $D-d+1$ consecutive fixed points of θ ↦ D θ ( mod Z ) $\theta \mapsto D \theta \ (\operatorname{mod} {\mathbb {Z}})$ in the same fiber of Π $\Pi$ .

On hyperbolic rational maps with finitely connected Fatou sets

In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show that any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence of rational maps of a fixed degree with infinitely many non-monomial rescaling limits.

Primitive tuning via quasiconformal surgery

Using quasiconformal surgery, we prove that any primitive, postcritically-finite hyperbolic polynomial can be tuned with an arbitrary generalized polynomial with non-escaping critical points, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves affirmatively a conjecture by Inou and Kiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable.

Siegel disks of the tangent family

We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a Siegel disk is unbounded if and only if the boundary of its image contains at least one asymptotic value. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.

Schwarz reflections and anti-holomorphic correspondences

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in [14,15]. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic anti-rational maps with the abstract modular group. We further illustrate our mating framework by studying the correspondence associated with the Schwarz reflection map of a deltoid.

Bank-Laine Functions with Real Zeros

Suppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that E'(z) = pm 1 whenever E(z) = 0. Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

Bank–Laine functions, the Liouville transformation and the Eremenko–Lyubich class

The Bank–Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if the set of finite critical and asymptotic values of the coefficient function is bounded. It is also shown that if E is a Bank–Laine function of finite order with infinitely many zeros, all real and positive, then its zeros must have exponent of convergence at least 3/2, and an example is constructed via quasiconformal surgery to demonstrate that this result is sharp.

Quasiconformal surgery and linear differential equations

We describe a new method of constructing transcendental entire functions A such that the differential equation w″ + Aw = 0 has two linearly independent solutions with relatively few zeros. In particular, we solve a problem of Bank and Laine by showing that there exist entire functions A of any prescribed order greater than 1/2 such that the differential equation has two linearly independent solutions whose zeros have finite exponent of convergence. We show that partial results by Bank, Laine, Langley, Rossi and Shen related to this problem are in fact best possible. We also improve a result of Toda and show that the estimate obtained is best possible. Our method is based on gluing solutions of the Schwarzian differential equation S(F) = 2A for infinitely many coefficients A.

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.

Bounded type Siegel disks of finite type maps with few singular values

Let $U$ be an open subset of the Riemann sphere $\S$.We give sufficient conditions for which a finite type map $f:U\to~\S$ with at most three singular values has a Siegel disk compactly contained in $U$ and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery ala Douady-Ghys-Herman-§w. We also give sufficient conditions for which, instead, $\Delta$ has not compact closure in $U$. The main tool is the Schwarzian derivative and area inequalities ala Graczyk-§w.

Constructing entire functions with non-locally connected Julia set by quasiconformal surgery

We give an alternative way to construct an entire function with quasiconformal surgery so that all its Fatou components are quasi-disks but the Julia set is non-locally connected.

Surgery on Herman rings of the standard Blaschke family

Let $B_{\alpha ,a}$ be the Blaschke product of the following form:\begin{document}${B_{\alpha ,a}}(z) = {e^{2\pi {\rm{\mathbf{i}}}\alpha }}{z^{d + 1}}{(\frac{{z - a}}{{1 - az}})^d}.$\end{document}If $B_{\alpha ,a}|_{S^1}$ is analytically linearizable, then there is a Herman ring admitting the unit circle as an invariant curve in the dynamical plane of $B_{\alpha ,a}$. Given an irrational number $θ$, the parameters $(\alpha ,a)$ such that $B_{\alpha ,a}|_{S^1}$ has rotation number $θ$ form a curve $T_d(θ)$ in the parameter plane. Using quasiconformal surgery, we prove that if $θ$ is of Brjuno type, the curve can be parameterized real analytically by the modulus of the Herman ring, from $a=M(θ)$ up to $∞$ with $M(θ)≥q 2d+1$, for which the Herman ring vanishes.Moreover, we can show that for a certain set of irrational numbers $θ ∈ \mathcal {B}\setminus\mathcal {H}$, the number $M(θ)$ is strictly greater than $2d+1$ and the boundary of the Herman rings consist of two quasicircles not containing any critical point.

The Fine Structure of Herman Rings

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting.

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.

On a family of rational perturbations of the doubling map

The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products . First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter a. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials . In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.

On Connectivity of Fatou Components concerning a Family of Rational Maps

I. N. Baker established the existence of Fatou component with any given finite connectivity by the method of quasi-conformal surgery. M. Shishikura suggested giving an explicit rational map which has a Fatou component with finite connectivity greater than 2. In this paper, considering a family of rational mapsRz,tthat A. F. Beardon proposed, we prove thatRz,thas Fatou components with connectivities 3 and 5 for anyt∈0,1/12. Furthermore, there existst∈0,1/12such thatRz,thas Fatou components with connectivity nine.

Connectedness properties of the set where the iterates of an entire function are bounded

AbstractWe investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K(f) is totally disconnected if and only if each component of K(f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is totally disconnected and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.

Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with a bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation that are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set Jγ to satisfy that Jγ is a Jordan curve but not a quasicircle, the unbounded component of C ^ ∖ J γ is a John domain and the bounded component of ℂ \ Jγ is not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated.