On geometric complexity of Julia sets - II

24.08.2020 - 27.08.2020 | Online

Abstracts + Slides + Videos

Adam Abrams (IMPAN)
Dynamics of complex continued fractions via partitions
For a large class of real-valued continued fraction algorithms, the natural extension of the Gauss map has a global attractor with a simple structure. Using finite and countable partitions, we show a similar phenomenon for a class of complex continued fraction algorithms. For some algorithms with the "finite building property", the domain in ℂ × ℂ of the natural extension of complex Gauss map can be described as a finite union of Cartesian products. In one complex coordinate, the sets come from explicit manipulation of the continued fraction algorithm, while in the other coordinate the sets are determined experimentally.
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Krzysztof Barański (University of Warsaw)
On the dimension of points which escape to infinity at given rate under exponential iteration
We determine the Hausdorff and packing dimension of sets of points which escape to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps. This is a joint work with Bogusława Karpińska.
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Fabrizio Bianchi (CNRS & Université de Lille)
A spectral gap for the transfer operator on complex projective spaces
We study the transfer (Perron-Frobenius) operator on Pk(C) induced by a generic holomorphic endomorphism and a suitable continuous weight. We prove the existence of a unique equilibrium state and we introduce various new invariant functional spaces, including a dynamical Sobolev space, on which the action of f admits a spectral gap. This is one of the most desired properties in dynamics. It allows us to obtain a list of statistical properties for the equilibrium states such as the equidistribution of points, speed of convergence, K-mixing, mixing of all orders, exponential mixing, central limit theorem, Berry-Esseen theorem, local central limit theorem, almost sure invariant principle, law of iterated logarithms, almost sure central limit theorem and the large deviation principle. Most of the results are new even in dimension 1 and in the case of constant weight function, i.e., for the operator f_*. Our construction of the invariant functional spaces uses ideas from pluripotential theory and interpolation between Banach spaces. This is a joint work with Tien-Cuong Dinh.
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Jordi Canela (Centre de Recerca Matemàtica)
Achievable connectivities of Fatou components in singular perturbations
A periodic Fatou component can only have connectivity 1, 2, or infinity. Despite that, preperiodic Fatou components can have arbitrarily large finite connectivity. There exist explicit examples of rational maps with Fatou components of any prescribed connectivity. However, the degree of these maps grows as the required connectivity increases.
We study a family of singular perturbations of rational maps with a single free critical point. Under certain conditions, the dynamical planes of these singular perturbations contain Fatou components of arbitrarily large finite connectivity. In this talk we will analyze the dynamical conditions under which these Fatou components of arbitrarily large connectivity appear. We will also study which are the achievable connectivities depending on the parameters of the singular perturbation.
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Neil Dobbs (University College Dublin)
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. On the boundary of the Mandelbrot set M, Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we find the local behaviour of the Hausdorff dimension of quadratic Julia sets at the tip -2 of the Mandelbrot set for most real parameters in the sense of 1-dimensional Lebesgue measure. This is joint work with J. Graczyk and N. Mihalache.
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Vasiliki Evdoridou (The Open University)
Constructing examples of oscillating wandering domains
Let U be a Fatou component of a transcendental entire function. If U is not eventually periodic then it is called a wandering domain. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. The first wandering domain of oscillating type was constructed by Eremenko and Lyubich in 1987. Motivated by their construction and the recent classification of simply connected wandering domains obtained by Benini, E., Fagella, Rippon and Stallard, we give a general technique, based on Approximation Theory, for the construction of bounded oscillating wandering domains. We show that this technique can be used to produce examples of oscillating wandering domains of all six different types that arise by the classification. This is joint work with P. Rippon and G. Stallard.
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Gustavo Rodrigues Ferreira (The Open University)
The Permutable Mystery Tour for transcendental meromorphic functions
Which pairs of analytic functions commute? This question has been studied for a century now, and we have several reasonable answers for polynomials and rational functions. For transcendental meromorphic functions (entire and non-entire), however, little is known. In this talk, I will focus on a dynamical approach to this question: do permutable transcendental meromorphic functions have the same Julia set? As we will see, in many cases the answer is "yes", but the most general case remains open.
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Igors Gorbovickis (Jacobs University Bremen)
Critical points of the multipliers in the quadratic family: equidistribution and accumulation
A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a \textit{critical point of a period $n$ multiplier}, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. Information about the location of critical points and critical values of the multipliers might play a role in the study of the geometry of the Mandelbrot set.
We will discuss asymptotic behavior of critical points of the period $n$ multipliers as $n\to\infty$. We will show that while the critical points equidistribute on the boundary of the Mandelbrot set $\mathbb M$, their accumulation set $\mathcal X$ is strictly larger than $\partial\mathbb M$. We will further show that the accumulation set $\mathcal X$ is bounded, path connected and contains the Mandelbrot set as a proper subset. This is joint work with Tanya Firsova.
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Bogusława Karpińska (Warsaw University of Technology)
Fatou components and singularities of meromorphic functions
In this talk we discuss the relation between the postsingular set and the boundary of Fatou components, which are specific for transcendental meromorphic maps: Baker domains and wandering domains. In particular we answer a question of Mihaljevic-Brandt and Rempe-Gillen for Baker domains. For wandering domains 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 the distance from p_n to the boundary of U_n tends to 0. This allows to exclude the existence of wandering domains for some meromorphic maps. The talk is based on a joint work with Krzysztof Baranski, Nuria Fagella and Xavier Jarque.
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Krzysztof Lech (University of Warsaw)
Julia sets in non-autonomus quadratic iteration
We shall discuss typical connectedness of Julia sets of non-autonomous compositions of quadratic functions. Some examples of phenomena that do not occur in autonomous iteration will be provided. We also answer a question about whether picking the parameter randomly from a large enough disk typically yields a totally disconnected Julia set. This is based on joint work with Anna Zdunik.
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Luna Lomonaco (IMPA)
Mating quadratic maps with the modular group
Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere (implicit maps sending z to w). The iteration of such multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalise rational maps and finitely generated Kleinian groups). We consider a specific 1-(complex)parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every a in the connectedness locus M_{\Gamma}, this family is a mating between the modular group and rational maps in the family Per_1(1), and we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials.
This is joint work with S. Bullett.
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Carsten Lunde Petersen (Roskilde Universitet)
Geography of the cubic connectedness locus: Lemon limbs
The geographical atlas of the cubic connectedness locus still have very many blank pages. The aim of this project is to fill-in some of these pages. We define and explore a family of subsets of the cubic connectedness locus, which we for now are calling lemon limbs or Jack’s limbs because their intersection with Per_1(0), the complex line of cubic polynomials with a super attracting fixed point, are precisely the limbs of the lemon, the central hyperbolic component of Per_1(0). Part of our work extends in a certain sense the intertwining surgery of Epstein and Yampolsky as the title suggest. The lemon limbs extends quite far in the cubic connectedness locus and the corresponding Julia sets are amenable to puzzles.
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Lasse Rempe (University of Liverpool)
Classes of criniferous entire functions
The escaping set of a transcendental entire function f is the set of points whose orbits under f tend to infinity. Fatou observed in 1926 that the escaping set of certain transcendental entire functions contains curves to infinity. In the 1980s, Devaney (with various co-authors), showed that in some of the simplest cases, the escaping set consists entirely of such curves, now often called “Devaney hairs”. When this is the case, we say that the function is “criniferous”.
The Eremenko-Lyubich class B consists of those transcendental entire functions whose set of critical and asymptotic values is bounded. If f is in B and has finite order of growth, it is known that f is criniferous. However, it is also known that there are functions in B for which the escaping set contains no nontrivial curves. In this talk, we shall describe two natural classes of criniferous entire functions, one the subject of joint work with Albrecht and Benini and the other of joint work with Pardo Simón, and their properties.
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David Sixsmith (The Open University)
Fatou's associates
We use an inner function to study the dynamics of a transcendental entire function on a simply connected Fatou component.
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Giulio Tiozzo (University of Toronto)
Metrics on trees, laminations, and core entropy
The notion of core entropy, defined as the entropy of the restriction to the Hubbard tree, was formulated by W. Thurston to produce a combinatorial invariant which captures the topological complexity of polynomial Julia sets and varies in a rich fractal way over parameter space.
Core entropy has been so far defined by looking at a Markov partition on the tree, or by a combinatorial construction involving infinite graphs. We will introduce a new interpretation of core entropy based on metrics on trees and, dually, on transverse measures on laminations defining the Julia set.
On the one hand, this will define a new notion of transverse measures on quadratic laminations, completing the analogy with laminations on surfaces on the “other side” of Sullivan’s dictionary. Moreover, this is also related to a question of Milnor on a piecewise-linear analogue of Thurston iteration on Teichmueller space.
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Jonguk Yang (Stony Brook University)
Polynomials with Bounded Type Siegel Disks
Consider a polynomial with a Siegel disc of bounded type rotation number. It is known that the Siegel boundary is a quasi-circle that contains at least one critical point. In the quadratic case, this means that the entire post-critical set is trapped within the Siegel boundary, where the theory of real analytic circle maps provides us with excellent control. However, in the higher degree case, there exist multiple critical points. A priori, these “free” critical points may accumulate on the Siegel boundary in a complicated way, causing extreme distortions in the geometry nearby. In my talk, I show that in fact, this does not happen, and that the Julia set is locally connected at the Siegel boundary.
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Anna Zdunik (University of Warsaw)
Thermodynamic formalism for coarse expanding dynamical systems
We consider a class of dynamical systems which we call weakly coarse expanding, which generalize to the postcritically infinite case expanding Thurston maps as discussed by Bonk–Meyer and are closely related to coarse expanding conformal systems as defined by Hassinsky–Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all such results even in presence of periodic (repelling) branch points. This is a joint work with Tushar Das, Feliks Przytycki, Giulio Tiozzo and Mariusz Urbański.
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Alexandre Eremenko (Purdue University)
Open Problems Session on geometric complexity of Julia sets II
Problems Session by Alexander Eremenko
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