Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe
Complex dynamics, in the sense of iteration theory of holomorphic mappings, is a very active and deep field with many connections all across mathematics, with profound recent developments and important goals. Its connections to other branches of mathematics include geometry of 3-manifolds (through Sullivan’s celebrated “dictionary” between holomorphic dynamics and Kleinian groups, as well as Thurston’s geometrization theory), algebra (in particular group theory, through Nekrashevych’ theory of Iterated Monodromy Groups), renormalization (that plays an important role in the study of holomorphic dynamics as well as in physics), all the way to numerical analysis (many equation solvers and root finders are iterated holomorphic maps). Our focus will be on selected topics of holomorphic dynamics that have seen substantial developments in the past years that open up new perspectives and vistas. We will put particular emphasis on their interconnections, helping an actively developing field to maintain coherence and a view towards “the whole picture”. One substantial ingredient of the conference is the Advanced Grant of the European Research Council “Hologram” by one of the organizers; the intended time period for the conference (September 2021) marks the conclusion of this grant, so it is a good opportunity to “pass on the torch” to the community-at-large in terms of accomplishments and questions achieved. We hope that the results of this grant will inspire our colleagues to apply for other major grants, ERC and other wise.
27
2021
75
22 hours 20 minutes
27 results
1:02:49
Mukherjee, SabyasachiCombination theorems play an important role in several areas of dynamics, geometry, and group theory. In this talk, we will expound a framework to conformally combine Kleinian (reflection) groups and (anti-)holomorphic rational maps in a single dynamical plane. In the anti-holomorphic setting, such hybrid dynamical systems are generated by Schwarz reflection maps arising from univalent rational maps. A crucial technical ingredient of this study is a recently developed David surgery technique that turns hyperbolic conformal dynamical systems to parabolic ones. We will also mention numerous consequences of this theory, including 1. an explicit dynamical connection between various rational Julia and Kleinian limit sets,2. existence of new classes of welding homeomorphisms and conformally removable Julia/limit sets, and3. failure of topological orbit equivalence rigidity for Fuchsian groups acting on the circle.
2021Centre International de Rencontres Mathématiques (CIRM)
33:11
20Lyubich, MikhailThe Feigenbaum point is a remarkable parameter in the quadratic family focusing some of the deepest phenomena of non-linear dynamics (regular vs chaotic behavior, rigidity and MLC, universal self-similarity and renormalization). A theory of this point has been under construction for about 45 years. I will outline this story, completing it with one missing piece, MLC at this point (based on a joint work with Dima Dudko).
2021Centre International de Rencontres Mathématiques (CIRM)
47:52
1Drach, KostiantynThe concept of a complex box mapping (or puzzle mapping) is a generalization of the classical notion of polynomial-like map to the case when one allows for countably many components in the domain and finitely many components in the range of the mapping. In one-dimensional dynamics, box mappings appear naturally as first return maps to certain nice sets, and hence one arrives at a notion of box renormalization. We say that a rational map is box renormalizable if the first return map to a well-chosen neighborhood of the set of critical points (intersecting the Julia set) has a structure of a box mapping. In our talk, we will discuss various features of general box mappings, as well as so-called dynamically natural box mappings, focusing on their rigidity properties. We will then show how these results can be used almost as 'black boxes' to conclude similar rigidity properties for box renormalizable rational maps. We will give several examples to illustrate this procedure, these examples include, most prominently, complex polynomials of arbitrary degree and their Newton maps. (The talk is based on joint work with Trevor Clark, Oleg Kozlovski, Dierk Schleicher and Sebastian van Strien.)
2021Centre International de Rencontres Mathématiques (CIRM)
1:02:26
4Cheraghi, DavoudIn this talk we present a toy model for the (sector) renormalisation of holomorphic maps with an irrationally indifferent fixed points. The model depends on the arithmetic of the rotation number at the fixed point, and exhibits the geometry of a non-degenerate holomorphic map with an irrationally indifferent fixed point. We present some sufficient conditions on a give (sector) renormalisation which guarantees the underlying map has the same dynamics as the one of the toy model.
2021Centre International de Rencontres Mathématiques (CIRM)
1:06:58
5Bonk, MarioEvery expanding Thurston map gives rise to a fractal geometry on its underlying 2-sphere. Many dynamical properties of the map are encoded in this fractal, called the 'visual sphere' of the map. An interesting question is how to determine the (Ahlfors regular) conformal dimension of the visual sphere if the map is obstructed. In my talk I will give an introduction to this subject and discuss some recent progress.
2021Centre International de Rencontres Mathématiques (CIRM)
33:58
2Bogdanov, KonstantinIn complex dynamics it is usually important to understand the dynamical behavior of critical (or singular) orbits. For quadratic polynomials, this leads to the study of the Mandelbrot set and of its complement. In our talk we present a classification of some explicit families of the transcendental entire functions for which all singular values escape, i.e. functions belonging to the complement of the 'transcendental analogue' of the Mandelbrot set. This classification allows us to introduce higher dimensional analogues of parameter rays and to explore their properties. A key ingredient is a generalization of the famous Thurston's Topological Characterization of Rational Functions, but for the case of infinite rather than finite postsingular set. Analogously to Thurston's theorem, we consider the sigma-iteration on the Teichmüller space and investigate its convergence. Unlike the classical case, the underlying Teichmüller space is infinite-dimensional which leads to a completely different theory.
2021Centre International de Rencontres Mathématiques (CIRM)
58:41
Petersen, Carsten LundeIn a recently completed paper Pascale Roesch and I have given a complete proof that the connectedness locus M_{1} in the space moduli space of quadratic rational maps with a parabolic fixed point of multiplier 1 is homeomorphic to the Mandelbrot set. In this talk I will outline and discus the proof, which in an essential way involves puzzles and a theorem on local connectivity of M_{1} at any parameter which is neither renormalizable nor has all fixed points non-repelling similar to Yoccoz celebrated theorem for local connectivity of M at corresponding parameters.
2021Centre International de Rencontres Mathématiques (CIRM)
59:41
Fagella, Nuria2021Centre International de Rencontres Mathématiques (CIRM)
1:03:26
3Dudko, DzmitryA fundamental fact about Riemann surfaces is that they degenerate in a specific pattern — along thin annuli or wide rectangles. As it was demonstrated by W. Thurston in his realization theorem, we can often understand the dynamical system by establishing a priori bounds (a non-escaping property) in the near-degenerate regime. Similar ideas were proven to be successful in the Renormalization Theory of the Mandelbrot set. We will start the talk by discussing a dictionary between Thurston and Renormalization theories. Then we will proceed to neutral renormalization associated with the main cardioid of the Mandelbrot set. We will show how the Transcendental Dynamics naturally appear on the renormalization unstable manifolds. In conclusion, we will describe uniform a priori bounds for neutral renormalization — joint work with Misha Lyubich.
2021Centre International de Rencontres Mathématiques (CIRM)
1:11:08
1Thurston, DylanWe can conveniently represent post-critically finite topological branched selfcovers of the sphere to itself using maps of graphs. With this representation, there is also a positive characterization of hyperbolic rational maps among these topological branched self-covers, using energies that control elastic 'stretchiness'. In broad terms, a map is rational iff a network of elastic bands gets looser and looser as you pull it back. This complements the older negative characterization of W. Thurston.
2021Centre International de Rencontres Mathématiques (CIRM)
1:06:44
5Shishikura, MitsuhiroWe discuss the idea of renormalization for complex dynamical systems. There various types of renormalizations defined via a first return map, appear in complex dynamics, for unimodal maps, homeomorphisms of circle, and germs of irrationally indifferent fixed points of holomorphic maps. The target of renormalization is usually tame and fragile dynamics and the connecting maps are often expanding maps and the exding property helps us to understand the rigid nature of the target maps. We propose the idea of dynamical charts for irrationally indifferent fixed points, in order to reconstruct the original map from the sequence of renormalizations.
2021Centre International de Rencontres Mathématiques (CIRM)
28:13
5Reinke, BernhardIterated monodromy groups are self-similar groups associated to partial selfcoverings. In my talk I will give an overview of iterated monodromy groups of post-singularly finite entire transcendental functions. These groups act self-similarly on a regular rooted tree, but in contrast to IMGs of rational functions, every vertex of the tree has countably infinite degree. I will discuss the similarities and differences of IMGs of entire transcendental functions and of polynomials, in particular in the direction of amenability.
2021Centre International de Rencontres Mathématiques (CIRM)
35:40
1Lazebnik, KirillWe show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorffdimension 1 Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon+Stallard. Our proof relies on a quasiconformal-surgery approach. This is joint work with Jack Burkart.
2021Centre International de Rencontres Mathématiques (CIRM)
32:47
Estevez Jacinto, Gabriela AlexandraIn this talk we study real analytic bi-cubic circle maps with bounded type rotation number. We define a suitable functional space where the renormalization operator is analytic, and we construct a hyperbolic attractor of renormalization with codimension-two stable foliation. This is joint work with Michael Yampolsky.
2021Centre International de Rencontres Mathématiques (CIRM)
1:01:26
3Bartholdi, LaurentQuadratic polynomials have been investigated since the beginnings of complex dynamics, and are often approached through combinatorial theories such as laminations or Hubbard trees. I will explain how both of these approaches fit in a more algebraic framework: that of iterated monodromy groups. The invariant associated with a quadratic polynomial is a group acting on the infinite binary tree, these groups are interesting in their own right, and provide insight and structure to complex dynamics: I will explain in particular how the conversion between Hubbard trees and external angles amounts to a change of basis, how the limbs and wakes may be defined in the language of group theory, and present a model of the Mandelbrot set consisting of groups. This is joint work with Dzmitry Dudko and Volodymyr Nekrashevych.
2021Centre International de Rencontres Mathématiques (CIRM)
29:23
5Waterman, JamesSeveral important problems in complex dynamics are centered around the local connectivity of Julia sets of polynomials and of the Mandelbrot set. Importantly, when the Julia set of a polynomial is locally connected, the topological dynamics ofthe map can be completely described as a quotient of a power map on the circle.Local connectivity of the Julia set is less significant for transcendental entire functions. Nevertheless, by restricting to a class of transcendental entire functions, known as docile functions, we obtain a similar concept by describing the topological dynamics as a quotient of a simpler disjoint-type map. We will discuss the notion ofdocile functions, as well as some of their properties. This is joint work with Lasse Rempe.
2021Centre International de Rencontres Mathématiques (CIRM)
59:08
2Nekrashevych, VolodymyrOne can associate with every finitely generated contracting self-similar group (for example, with the iterated monodromy group of a sub-hyperbolic rational function) and every positive p the associated \ell_{p}-contraction coefficient. The critical exponent of the group is the infimum of the set of values of p for which the \ell_{p}-contraction coefficient is less than 1. Another number associated with a contracting self-similar group is the Ahlfors-regular conformal dimension of its limit space. One can show that the critical exponent is not greater than the conformal dimension. However, the inequality may be strict. For example, the critical exponent is less than 1 for many groups of intermediate growth (while the corresponding conformal dimension is equal to 1 ). We will also discuss a related notion of the degree of complexity of an action of a group on a set.
2021Centre International de Rencontres Mathématiques (CIRM)
27:20
1Pardo-Simon, Leticia2021Centre International de Rencontres Mathématiques (CIRM)
26:01
2Martí-Pete, DavidWe construct a transcendental entire function for which infinitely many Fatou components share the same boundary. This solves the long-standing open problem whether Lakes of Wada continua can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. Our theorem also provides the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a recent question of Boc Thaler. Using the same techniques, we give new counterexamples to a conjecture of Eremenko concerning curves in the escaping set of an entire function. This is joint work with Lasse Rempe and James Waterman.
2021Centre International de Rencontres Mathématiques (CIRM)
59:25
2Hubbard, John H.W. Thurston's theorems almost all aim to give a purely topological problem an appropriate geometry, or to identify an appropriate obstruction.. We will illustrate this in two examples: --The Thurston pullback map to make a rational map from a post-critically finite branched cover of the sphere, and --The skinning lemma, to find a hyperbolic structure for a Haken 3-manifold. In both cases, either the relevant map on Teichmüller space has a fixed point, solving the geometrization problem, or there is an obstruction consisting a multicurve.
2021Centre International de Rencontres Mathématiques (CIRM)
21:59
1Brown, AndrewFatou noticed in 1926 that certain transcendental entire functions have Julia sets in which there are curves of points that escape to infinity under iteration and he wondered whether this might hold for a more general class of functions. In 1989, Eremenko carried out an investigation of the escaping set of a transcendental entire function f, $I(f)=\left \{ z\in\mathbb{C}:\left | f^{n}\left ( z \right ) \right | \rightarrow \infty \right \}$ and produced a conjecture with a weak and a strong form. The strong form asks if every point in the escaping set of an arbitrary transcendental entire function can be joined to infinity by a curve in the escaping set. This was answered in the negative by the 2011 paper of Rottenfusser, Rückert, Rempe, and Schleicher (RRRS) by constructing a tract that produces a function that cannot contain such a curve. In the same paper, it was also shown that if the function was of finite order, that is, log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )$ as $\left | z \right |\rightarrow \infty$, then every point in the escaping set can indeed be connected to infinity by a curve in the escaping set. The counterexample $f$ used in the RRRS paper has growth such that log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )^{k}$ where $K > 12$ is an arbitrary constant. The question is, can this exponent, K, be decreased and can explicit calculations and counterexamples be performed and constructed that improve on this?
2021Centre International de Rencontres Mathématiques (CIRM)
1:05:21
1Bishop, ChristopherI will introduce the Speiser and Eremenko-Lyubich classes of transcendental entire functions and give a brief review of quasiconformal maps and the measurable Riemann mapping theorem. I will then discuss tracts and models for the Eremenko-Lyubich class and state the theorem that all topological tracts can occur in this class. A more limited result for the Speiser class will also be given. I will then discuss some applications of these ideas, focusing on recent work with Kirill Lazebnik (prescribing postsingular orbits of meromorphic functions) and Lasse Rempe (equilateral triangulations of Riemann surfaces).
2021Centre International de Rencontres Mathématiques (CIRM)
54:47
1Benini, Anna MiriamIn this first lecture we will introduce some the main differences between the dynamics of polynomials and the dynamics of transcendental entire functions: Baker and wandering domains, the new features of the escaping set, new features in the Julia set and some information about parameter spaces for some specific classes of entire functions.
2021Centre International de Rencontres Mathématiques (CIRM)
32:19
Shemyakov, SergeyThurston's topological characterization theory asks whether there is a holomorphic dynamical system that realizes topological (even combinatorial) data, this often allows to describe all possible dynamical systems in a certain parameter space. I work on extending Thurston topological characterization theory to different classes of transcendental functions. In my talk I will start with some explicit families of functions for which we have established an extension of Thurston's theory, and then describe further extensions by compositions of such functions. The presented ideas are part of my PhD thesis under the supervision of Dierk Schleicher. This is work in progress.
2021Centre International de Rencontres Mathématiques (CIRM)
59:24
6Winarski, RebeccaThurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques — adapting tools used tostudy mapping class groups — to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.
2021Centre International de Rencontres Mathématiques (CIRM)
1:05:00
Buff, Xavier2021Centre International de Rencontres Mathématiques (CIRM)
54:58
4Hlushchanka, MikhailThere are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). We will discuss several natural decompositions that arise in the study of rational maps, such as Pilgrim's canonical decomposition and Levy decomposition (by Bartholdi and Dudko). I will also introduce a new decomposition of rational maps based on the topology of their Julia sets (obtained jointly with Dima Dudko and Dierk Schleicher). At the end of the talk, we will briefly consider connections of this novel decomposition to geometric group theory and self-similar groups.
2021Centre International de Rencontres Mathématiques (CIRM)