Workshop on Complex Dynamics

RIMS, Kyoto  –  December 11-15, 2017

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Romain DUJARDIN, Degenerations of $SL(2,\mathbb C)$ representations, and Lyapunov exponents I and II
Let $G$ be a finitely generated group endowed with some probability measure $\mu$ and $(\rho_\lambda)$ be a non-compact algebraic family of representations of $G$ into $SL(2,\mathbb C)$. This can be understood as a random product of Möbius transformations depending on a parameter $\lambda$. Using non-Archimedean techniques, we study the asymptotics of this random holomorphic dynamical system as $\lambda$ goes to infinity.
This problem is analogous to that of the description of degenerating families of rational maps by DeMarco-Faber, Favre, and others, and the non-archimedean tools used are similar. This is a report of joint work with Charles Favre.

Núria FAGELLA, Wandering domains and singular values [slides]
We present several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the e successive iterates of a wandering component. As a consequence we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that $\text{dist}(p_n,\partial U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\mathbb{C}$) then any sequence of iterates of wandering domains must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

A wandering domain in class B on which all iterates are univalent [slides]
Inspired by the recent surgery construction of an oscillating wandering domain for a map in class B, using quasiconformal folding, we prove that there exists a domain with the same properties but on which the successive iterates of the map are univalent.

Phil RIPPON, The structure of the escaping set of a transcendental entire function [slides]
Much recent work on the iterates of a transcendental entire function $f$ has been motivated by Eremenko's conjecture that all the components of the escaping set $I(f)$ are unbounded, which is still unsolved. Many partial results on this conjecture are proved using an important subset of $I(f)$ called the fast escaping set $A(f)$. For example, $A(f)$ can be used to show that $I(f)$ must contain at least one unbounded component.
Also, there are many examples of entire functions for which $I(f)$ consists of uncountably many disjoint unbounded curves, most points of which are fast escaping, and also many examples where $I(f)$ is connected and has the structure of an infinite spider's web. For the set $A_R(f)$, which is the `core' of the fast escaping set, we prove that in a certain sense exactly one of these two structures must happen. This is joint work with Gwyneth Stallard.

Gwyneth STALLARD, Wandering domains and commuting transcendental entire functions [slides]
The classical classification of periodic Fatou components is one of the foundations of complex dynamics. For transcendental functions, Fatou components may be wandering domains and these are much less well understood. For multiply connected wandering domains, however, we do have a complete description of the dynamics within the wandering domain and of the geometry of the wandering domain. We will describe some of the key properties of multiply connected wandering domains from joint work with Walter Bergweiler and Phil Rippon, and more recent work with Phil Rippon on the boundary components of such domains. We also discuss an application of our results in joint work with Phil Rippon and Anna Benini on the question of whether the Julia sets of commuting functions must be equal.

Michel ZINSMEISTER, Variations of the Hausdorff dimension of real quadratic Julia sets [slides]
(Joint work with Ludwik Jaksztas)
If $c\in\mathbb{C}$, let $f_c(z)=z^2+c$, $J_c$ its Julia set and $d(c)$ the Hausdorff dimension of $J_c$.
If $c$ is real and $c\in (c_{\text{Feig}},1/4)$ then either $f_c$ is hyperbolic or else $f_c$ has a parabolic cycle whose Leau-Fatou flower has two petals.
Let $c_0$ be a parameter of the latter type. McMullen proved that $d$ is continuous at $c_0$ along the real line. We prove that:
(1) if $d(c_0)>4/3$ then $d$ is differentiable at $c_0$.
(2) if $d(c)\leqslant 4/3$ then $d'(c_0)=-\infty$. More precisely we show that there exist two constants $K_+>K_->0$ such that $d'(c)\sim -K_\pm |c-c_0|^{\tfrac{3}{2}d(c_0)-2}$ (if $d(c_0)\neq 4/3$).


Matthieu ASTORG, Collet, Eckmann and the bifurcation measure
Joint work with T. Gauthier, N. Mihalache and G. Vigny.
The bifurcation measure (in the moduli space of degree d rational maps) is defined as the Monge-Ampère of the Lyapunov exponent. Its support describes parameters which bifurcate maximally in some sense. We prove that the support of this measure has positive Lebesgue measure in moduli space.

Fabrizio BIANCHI, Holomorphic motions of Julia sets
For a family of rational maps, results by Lyubich, Mané-Sad-Sullivan and DeMarco provide a fairly complete understanding of dynamical stability. I will review this one-dimensional theory and present a recent generalisation to several complex variables. I will focus on the arguments that do not readily generalise to this setting, and introduce the tools and ideas that allow one to overcome these problems.

Vasiliki EVDORIDOU, Non-escaping endpoints of disjoint-type functions
Let f be a transcendental entire function of disjoint type and finite order. The Julia set of f consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Following recent results on the topology of the set of non-escaping endpoints for functions in the exponential family, we show that the union of non-escaping endpoints of f with infinity is a totally separated set. Combined with a result of Alhabib and Rempe-Gillen this gives a strong dichotomy on the topological properties of the set of endpoints which escape and those which do not escape for disjoint-type functions. This is joint work with D. Sixsmith.

Hiroyuki INOU, On perturbation of a polynomial with parabolic fixed point
We present a new example of polynomials having a parabolic fixed point, whose perturbation is either in the shift locus or has an attracting fixed point. In particular, such a polynomial cannot be approximated by Misiurewicz polynomials. This is a joint work in progress with Sabyasachi Mukherjee.

Yutaka ISHII, $\mathcal{M}_4$ is regular-closed [pictures]
For each $n\geq 2$, we investigate a family of iterated function systems which is parameterized by the unique contraction ratio $s\in\mathbb{D}^{\times}\equiv\{s\in\mathbb{C} : 0<|s|<1\}$ and possesses a rotational symmetry of order $n$. Let $\mathcal{M}_n$ be the locus of the contraction ratios $s$ where the corresponding self-similar sets are connected. The purpose of my talk is to show that $\mathcal{M}_n$ is regular-closed, i.e. $\overline{\mathrm{int}\,\mathcal{M}_n}=\mathcal{M}_n$ for $n\geq 4$. This gives a new result for $n=4$ and a simple geometric proof of the previously known result by Bandt and Hung for $n\geq 5$.

Tomoki KAWAHIRA, Dynamical and parametric Zalcman functions: Similarity between the Julia sets, the Mandelbrot set, and the tricorn [slides]
We apply Zalcman's lemma to: (1) dynamics of rational maps on the Riemann sphere of degree two or more; and (2) the bifurcation loci of families of rational maps. Then we have families of non-constant meromorphic functions that we call the dynamical and parametric Zalcman functions. In this talk, we present some basic properties of these families following the ideas of Steinmetz. We also give a simple proof of Tan Lei's theorem about the local similarities between the Julia sets and the Mandelbrot set, (and the tricorn) by using the intersection of the families of dynamical and parametric Zalcman functions.

David MARTÍ-PETE, Fingers in the parameter space of the complex standard family [slides]
We study the parameter space of the complex standard family $F_{\alpha,\beta}(z)=z+\alpha+\beta \sin z$ where the parameter $0<\beta\ll 1$ is considered to be fixed and the bifurcation is studied with respect to the parameter $\alpha\in\mathbb{C}$. In the real axis of that parameter plane one can observe the so-called Arnold tongues, and from them arise some finger-like structures which were observed for the first time by Fagella in her PhD thesis. Similar structures can also be observed in the parameter spaces of families of Blaschke products or Henon maps in higher dimension. We study the qualitative and quantitative aspects of the fingers via parabolic bifurcation. This is a work in progress joint with Mitsuhiro Shishikura (Kyoto University).

Shizuo NAKANE, On formal normal forms of holomorphic germs at super-saddle fixed points [slides]
Fiber Julia sets for polynomial skew products is discontiunuous if there exists connection between two sadlles. In a joint work with Inou, we explained this discontinuity by an analogous argument as parabolic implosion. Here we used linearizing coordinates instead of Fatou coordinates. For this purpose, we had to assume local invertiblility at saddles. In case of super-saddles, we need another normal forms. Although, for rigid germs, formal normal forms are obtained by Ruggiero, formal conjugacies might diverge. We consider a certain class of rigid germs and show that, for most maps, formal conjugacies diverge. In case they converge, the argument in Inou and the author works.

Yusuke OKUYAMA, Discontinuity of the escape rate of a degenerating meromorphic family of rational maps [slides]
For a holomorphic family $f_t$ of rational maps of degree $d>1$ on the projective line parametrized by the punctured unit disk ${0<|t|<1}$ whose coefficients extends to meromorphic functions on the whole unit disk ${|t|<1}$, the function which associates each parameter $0<|t|<1$ with the Lyapunov exponent $L(f_t)$ of $f_t$ (with respect to its maximal entropy measure) is continuous and subharmonic on ${0<|t|<1}$, and there is a constant $\eta$ such that the function $L(f_t)-\eta\log|t|$ has the order $o(\log|t|^{-1})$ around the puncture $t=0$. In this talk, we provide several examples of families $f_t$ where the function $L(f_t)-\eta\log|t|$ on ${0<|t|<1}$ fails to be bounded around the puncture $t=0$. This is in contrast to the recent result of Favre-Gauthier that the continuity of $L(f_t)-\eta\log|t|$ on the whole ${|t|<1}$ is always the case for any polynomials family $f_t$, and also succeeds in providing a counterexample to a conjecture posed by Favre in 2016. This is a joint work with Professor Laura DeMarco (Northwestern University).

Hiroki SUMI, Weak mean stability in random holomorphic dynamical systems [long abstract]
We consider random holomorphic dynamical systems generated by holomorphic families of rational maps on the Riemann sphere. We introduce the notion of \weak mean stability" and show several properties of weakly mean stable systems. Also, we show the following (1) (2). (1) Generic random dynamical systems of polynomials of degree two or more are weakly mean stable. (2) Generic random relaxed Newton's method systems are weakly mean stable. For the preprint, see H. Sumi, Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods,

Takato UEHARA, On a construction of transcendental K3 surfaces: application of Arnol’d’s theorem
K3 surfaces have recently been intensively investigated from a dynamical point of view, since some of them admit automorphisms with positive topological entropy. In this talk, we construct K3 surfaces by gluing two 9-point blowups of the complex projective plane, where the existence of overlaps is guaranteed by Arnol’d’s theorem. A calculation of their period maps shows that such K3 surfaces constitute a large family, including transcendental K3 surfaces. Furthermore, we comment on a relation with automorphisms of K3 surfaces having positive entropy.

Shigehiro USHIKI, Reversible complex dynamical systems and exotic rotation domains
Complex dynamical systems of 2-dimensional complex manifolds may have exotic rotation domains. Exotic rotation domain is a rotation domain conjugate to the product of annulus and disk. Computer generated pictures strongly suggest the existence of such domains.


Shunsuke MOROSAWA, Dynamics of semigroups of transcendental entire functions [slides]
We consider Dynamics of semigroups of transcendental entire functions. We see some examples of semigroups of transcendental entire functions whose Fatou sets have wandering domains or Baker domains.

Leticia PARDO SIMÓN, Escaping singular orbits in the class B [slides]
In 1989, Eremenko conjectured for transcendental maps that every point in their escaping set can be connected to infinity by a curve in the escaping set. After this was proven to hold for functions in Class B of finite order, the question of when those curves, called “hairs” or “rays”, land has been an active topic of research. Even if this might not always be the case, it has been shown for some functions with bounded postsingular set that their Julia set is structured as a (pinched) Cantor Bouquet, that is, an embedding in the plane of a straight brush (with possibly identified endpoints).
In this talk I will consider certain functions with unbounded postsingular set whose singular orbits escape at some minimum speed. In this setting, certain hairs will split when they hit critical points. I will present a new structure for their Julia set as a modified Cantor Bouquet that will allow me to conclude that their hairs, if maybe now with split ends, still land.

Tomoko SHINOHARA, Local invariant set for a rational map of two variables at a fixed indeterminate point
For some fixed indeterminate point of a rational map of two variables, it is known that there exist a Cantor bouquet or a local stable set. In this talk we discuss properties of a rational mapping to have a local invariant set at a fixed indeterminate point.

Anand Prakash SINGH, On escaping sets of composition of transcendental entire functions
If $ f $ and $ g $ are transcendental entire functions, then so are $ f \circ g $ and $ g \circ f $. We discuss here the dynamics of various types of escaping sets and also non-escaping sets of composition of transcendental entire functions $ f \circ g $ and $ g \circ f $ and also its relations with regard to its factors $ f $ and $ g $.If $ f $ and $ g $ are transcendental entire functions, then so are $ f \circ g $ and $ g \circ f $. We discuss here the dynamics of various types of escaping sets and also non-escaping sets of composition of transcendental entire functions $ f \circ g $ and $ g \circ f $ and also its relations with regard to its factors $ f $ and $ g $.

Kohei UENO, Bottcher coordinates for holomorphic skew products [slides]
Let $f$ be a holomorphic skew product of the form $f(z,w)=(p(z),q(z,w))$ with a superattracting fixed point at the origin. Under one or two assumptions, we construct a Bottcher coordinate on an invariant open set whose closure contains the fixed point, which conjugates $f$ to a monomial map. The monomial map and the open set are determined by the order of $p$ at the origin and the Newton polygon of $q$.