HOLODYN: A research team in holomorphic dynamics
Seminar
We organize a weekly seminar in holomorphic dynamics and related areas in the lecture room of IMUB (Institute of Mathematics at Universitat de Barcelona). If you wish to be added to the mailing list, or have any question about the seminar please contact Jordi Canela, the organizer of the seminar since 2023. (We thank Toni Garijo who did this for many years.)
Upcoming talks (2024/25, Room IA (ground floor of Fac. Matemàtiques i Informàtica UB))
Working seminar series: What is the Teichmüller space of a holomorphic map? Organized by Gustavo R. Ferreira with lectures shared among voluntary participants. Tentative program is HERE.
| Date | Time | Speaker | Title | Abstract |
|---|---|---|---|---|
| 21.11.2024 | 12:00 | Gustavo R. Ferreira CRM / UB | What is the Teichmüller space of a holomorphic map? (V) | After an incursion into algebraic topology, we’re ready to introduce the hyperbolic geometry of the unit disc. Then, we will use what we have learned to define the hyperbolic metric of hyperbolic Riemann surfaces and its main properties. |
Past talks (recent)
| Date | Time | Speaker | Title | Abstract |
|---|---|---|---|---|
| 14.11.2024 | 12:00 | Beno Učakar (University of Ljubljana) | A beginner’s guide to quasiconformal folding | In 2014, Christopher J. Bishop pioneered a technique called quasiconformal folding, which, on a given domain, allows us to construct families of quasiconformal maps with uniformly controlled dilatations. This is done by adding tree-like structures to the domain’s boundary. The inverses of these maps will take the boundary of the codomain and fold it onto itself to create these tree-like structures, hence the name quasiconformal folding. This technique turned out to be very powerful and was used to prove many results in complex dynamics, approximation theory, and the study of maps of bounded and finite type. Sadly, the original proof of the main tool in quasiconformal folding, the folding lemma, is rather daunting and hard to wrap your head around. My goal for this talk is to give some intuition behind quasiconformal folding, show why it might be useful, and go over the details of the proof of the folding lemma in a slightly simpler setting, where our domain is the outside of the unit disc. |
| 7.11.2024 | 12:00 | Gustavo R. Ferreira CRM / UB | What is the Teichmüller space of a holomorphic map? (IV) | After an incursion into algebraic topology, we’re ready to introduce the hyperbolic geometry of the unit disc. Then, we will use what we have learned to define the hyperbolic metric of hyperbolic Riemann surfaces and its main properties. |
| 31.10.2024 | 12:00 | Gustavo R. Ferreira CRM / UB | What is the Teichmüller space of a holomorphic map? (III) | Using some basic facts from algebraic topology, we will discuss covering maps between Riemann surfaces and their properties. Then, in order to further develop the theory of surfaces covered by the unit disc, we will introduce the hyperbolic metric and its relation to Möbius maps of the disc. |
| 24.10.2024 | 12:00 | Nikolai Prochorov Aix Marseille Université | Thurston’s theory in transcendnetal dynamics. | In the 1980s, William Thurston demonstrated a famous characterization of postcritically finite branched coverings on the sphere. Simply put, this theorem provides a criterion for determining whether a continuous map is equivalent (in a certain combinatorial and dynamical sense) to a rational function defined on the Riemann sphere. This result forms the basis of a field within holomorphic dynamics, called Thurston theory, which analyzes the dynamical behavior of postcritically finite branched coverings and uses them to study the dynamics of rational functions. At the same time, in complex dynamics, there is also interest in families of transcendental holomorphic functions, which take values in the Riemann sphere but are not defined everywhere on it. The goal of this presentation is to show how Thurston’s theory can be extended to this context. Specifically, I will explain how Thurston’s characterization theorem can be generalized to certain fairly broad families of functions that are not defined on the entire sphere. Time permitting I try to explain how this subject is related to the dynamics of holomorphic self-maps of the unit disk, and how one can try to extend this subject to the setting of finite type maps. |
| 17.10.2024 | 12:00 | Arnaud Chéritat CNRS/Institut de Mathématiques de Toulouse | Parabolic implosion and pearl necklaces in the cubic family | In one complex dimensional discrete time dynamical systems, parabolic implosion is the name given to the phenomena appearing with specific perturbations of holomorphic maps having a parabolic cycle. For instance Julia sets get enriched and this was explained by the appearance of geometric limits by Douady and Lavaurs. It has proved to be of great importance in the understanding of holomorphic dynamics. Enrichment also occurs at the level of parameter spaces: the space of cubic maps with a fixed point of given multiplier that is a root of unity can be parameterized by one complex number. In this complex plane one can draw a bifurcation locus. When replacing the root of unity by nearby values, the bifurcation locus gets enriched, and this can be also explained using parabolic implosion. In this talk I will define the objects of study, make observations, state proven results and conjectures, and our work in progress on these questions. |
| 10.10.2024 | 12:00 | Gustavo R. Ferreira CRM / UB | What is the Teichmüller space of a holomorphic map? (II) | Using some basic facts from algebraic topology, we will discuss covering maps between Riemann surfaces and their properties. Then, in order to further develop the theory of surfaces covered by the unit disc, we will introduce the hyperbolic metric and its relation to Möbius maps of the disc. |
| 03.10.2024 | 12:00 | Gustavo R. Ferreira CRM / UB | What is the Teichmüller space of a holomorphic map? (I) | This is the initial session of a seminar series with this title. We shall present an overview of the program and start speaking about Universal coverings, Fuchsian groups and the hyperbolic metric. |
| 11.07.2024 | 15:00 | Xavier Jarque (Universitat de Barcelona) | Old and new results in dynamical systems | I will talk about old results on interval dynamics and new results on holomorphic dynamics. |
| 14.05.2024 | 15:00 | Lasse Rempe (The University of Liverpool) | Wandering dynamics of entire functions | Let f be a transcendental entire or meromorphic function. A compact set K is called wandering if any two forward iterates of K under f are disjoint from each other. In this talk, we discuss which wandering dynamics can be realised entire and meromorphic functions. I will begin by motivating the problem, and then present our main result, which states effectively that, for any sequence of compact sets and holomorphic maps between them, the behaviour of this sequence can be realised by the orbit of a wandering compactum of a transcendental meromorphic function. If each of the compact sets is full (that is, its complement in the plane is connected), then the function can be chosen to be entire. The results are proved using approximation theory. This is joint work with Vasiliki Evdoridou and David Martí-Pete |
| 07.05.2024 | 15:00 | Jordi Canela (Universitat Jaume I) | Numerical analysis of parameter planes with several critical points | When studying a family of rational maps depending on one parameter it is convenient to draw a parameter plane to better understand the dynamics of the maps. However, it may happen that such maps have several free critical points. In this talk we will discuss different options for drawing parameter planes in this situation and we will explain the advantatges and inconvenients of each option. We will focus on the drawing of parameter planes of polynomials and families of rational maps arising from root-finding algorithms. |
| 30.04.2024 | 15:00 | Xavier Jarque (Universitat de Barcelona) | Chaotic dynamics at the boundary of an attracting basin via non transversal intersections for a non global smooth diffeomorphism | In this talk we study the existence of transversal homoclinic points for a family of smooth maps in $\mathbb R^2$ with no global smooth inverse which come out as a truncated map governing the local dynamics of a critical period three cycle associated to the Secant map. Using Moser-Birkoff-Smale’s Theorem, we show that the boundary of the basin of attraction of the origin of $T_d$ contains a Cantor-like invariant subset whose restricted dynamics is conjugated to the full shift of 2-symbols. |
| 19.04.2024 | 15:30 | Luka Boc Thaler (University of Ljubljana) | Dynamics of skew-products tangent to the identity | We will look at the family maps P_b(z,w) = (z + z^2, w + w^2 + bz^2) with b\in\mathbb{C}. We will see that already such simple family of maps exabits a rich variety of dynamical behaviors, in particular for certain values b>1/4 we get wandering domains. Also we will see that there infinitely many parameters b_n converging to ¼ , for which maps P_{b_n} are not topologically conjugate to each other. In particular this implies that the family P_b is not structuraly stable at b=1/4. |
| 16.04.2024 | 15:00 | Leticia Pardo Simón (Universitat de Barcelona) | An approximation technique based on Hörmander’s solution to the dbar-equation (III) | In this third talk, we will cover two applications of the technique. First, we will show how to use it to construct maps with fast escaping wandering domains. Second, we will discuss an approximate answer to Erdős’s question about maximum modulus points. |
| 9.04.2024 | 15:00 | Leticia Pardo Simón (Universitat de Barcelona) | An approximation technique based on Hörmander’s solution to the dbar-equation (II) | In this series of talks, we aim to explore a technique, based on a theorem of Hörmander, to construct entire functions with prescribed features. In this first talk, we shall review some background material, present the main theorem, and provide an outline of the method. |
| 2.04.2024 | 15:00 | Leticia Pardo Simón (Universitat de Barcelona) | An approximation technique based on Hörmander’s solution to the dbar-equation (I) | In this series of talks, we aim to explore a technique, based on a theorem of Hörmander, to construct entire functions with prescribed features. In this first talk, we shall review some background material, present the main theorem, and provide an outline of the method. |
| 19.03.2024 | 15:00 | Gustavo Rodrigues-Ferreira (CRM) | Mixing and ergodicity for compositions of inner functions | Thanks to the work of Aaronson, Doering and Mañé, Crazier, and many others, we know that the properties of an inner function and its dynamics in the unit disc are closely related to the dynamics of its boundary extension. If, however, we consider compositions of inner functions, i.e. non-autonomous dynamics in the unit disc, much less is known about its relation to the corresponding non-autonomous dynamical system on the unit circle given by composing the boundary extensions. In this talk, we will tackle this problem from the point of view of ergodicity. We will derive necessary and sufficient conditions for a composition of inner functions fixing the origin to be ergodic on the unit circle, construct examples and counterexamples, and discuss some consequences of ergodicity. Time allowing, we will also explore mixing of compositions of inner functions, strengthening a result of Pommerenke and discussing what the correct definition of mixing is in the non-autonomous context. This is joint work with Artur Nicolau. |
| 12.03.2024 | 15:00 | Núria Fagella (Universitat de Barcelona) | Grand orbit relations in wandering domains (II) | We consider dynamical systems generated by the iteration of a complex entire map with an essential singularity at infinity. We say that two points are in the same Grand Orbit, if they are eventually iterated to the same point, even if the number of iterates needed are different for each of them. Grand orbits induce and equivalent relation in the complex plane, where two points belong to the same class if they belong to the same Grand Orbit. Motivated by the work of McMullen and Sullivan in 98 to to study the Teichmuller space of a rational map, we study the nature of Grand Orbit relations in stable components, specially in those specific for entire transcendental maps (Baker and wandering domains). We will show that all grand orbit relations are discrete or all are indiscrete, when considering connected components of the stable set minus the closure of marked points. We will also give an example of a wandering component, on which discrete and indiscrete grand orbits coexist, something that never occurs for stable components of rational maps. This talk is based in joint work with Christian Henriksen in ’06 and ’09, and in work in progress with Vasiliki Evdoridou, Lukas Geyer and Leticia Pardo. |
| 5.03.2024 | 15:30 | Núria Fagella (Universitat de Barcelona) | Grand orbit relations in wandering domains (I) | We consider dynamical systems generated by the iteration of a complex entire map with an essential singularity at infinity. We say that two points are in the same Grand Orbit, if they are eventually iterated to the same point, even if the number of iterates needed are different for each of them. Grand orbits induce and equivalent relation in the complex plane, where two points belong to the same class if they belong to the same Grand Orbit. Motivated by the work of McMullen and Sullivan in 98 to to study the Teichmuller space of a rational map, we study the nature of Grand Orbit relations in stable components, specially in those specific for entire transcendental maps (Baker and wandering domains). We will show that all grand orbit relations are discrete or all are indiscrete, when considering connected components of the stable set minus the closure of marked points. We will also give an example of a wandering component, on which discrete and indiscrete grand orbits coexist, something that never occurs for stable components of rational maps. This talk is based in joint work with Christian Henriksen in ’06 and ’09, and in work in progress with Vasiliki Evdoridou, Lukas Geyer and Leticia Pardo. |
| 27.02.2024 | 15:30 | Leticia Pardo-Simón (Universitat de Barcelona) | Wandering domains with nearly bounded orbits | In this talk, I will explain how to construct a bounded wandering domain with the property that, in a sense, we will make precise, nearly all of its forward iterates are contained within a bounded domain. This is based on joint work with D. Sixsmith. |
| 8.02.2024 | 15:30 | Anna Miriam Benini (University of Parma) | Entire maps with measures of maximal entropy whose support is the Julia set | We will see how to obtain measures with infinite entropy for several classes of entire maps, whose support is all of the Julia set. This is joint work with Leandro Arosio, Han Peters, and John Erik Fornaess. |
| 20.12.2023 | 9:30 | Kostiantyn Drach (Universitat de Barcelona) | Local connectivity of boundaries of Fatou components (V) | In the talk, I will re-prove the famous result of Roesch and Yin about local connectivity of boundaries of bounded Fatou components of polynomials (except Siegel). For this, I will present techniques of puzzles and generalized renormalizations (box mappings) and will show how to use these techniques to get the result. All necessary background will be provided. |
| 14.12.2023 | 12:15 | Neil Dobbs (University College Dublin) | Optimal constants for bounds on the Hausdorff dimension of some quadratic Julia sets | In recently published work, joint with J. Graczyk and N. Mihalache, we proved some asymptotic estimates for the Hausdorff dimension of quadratic Julia sets as one approaches the parameter -2 (the leftmost tip of the Mandelbrot set), inside a large subset of the real line: the dimension behaved ≈1+const.√c+2. In the meantime, L. Jaksztas conjectured the value of the optimal constant. Combining techniques from thermodynamic formalism with a more refined inducing argument, we obtain a proof for the optimal constant. Moreover, we extend our lower bound estimates to a fat cusp of complex parameters. |
| 1.12.2023 | 14:00 | Kostiantyn Drach (Universitat de Barcelona) | Local connectivity of boundaries of Fatou components (IV) | In the talk, I will re-prove the famous result of Roesch and Yin about local connectivity of boundaries of bounded Fatou components of polynomials (except Siegel). For this, I will present techniques of puzzles and generalized renormalizations (box mappings) and will show how to use these techniques to get the result. All necessary background will be provided. |
| 24.11.2023 | 14:00 | Kostiantyn Drach (Universitat de Barcelona) | Local connectivity of boundaries of Fatou components (III) | In the talk, I will re-prove the famous result of Roesch and Yin about local connectivity of boundaries of bounded Fatou components of polynomials (except Siegel). For this, I will present techniques of puzzles and generalized renormalizations (box mappings) and will show how to use these techniques to get the result. All necessary background will be provided. |
| 16.11.2023 | 12:15 | Kostiantyn Drach (Universitat de Barcelona) | Local connectivity of boundaries of Fatou components (II) | In the talk, I will re-prove the famous result of Roesch and Yin about local connectivity of boundaries of bounded Fatou components of polynomials (except Siegel). For this, I will present techniques of puzzles and generalized renormalizations (box mappings) and will show how to use these techniques to get the result. All necessary background will be provided. |
| 10.11.2023 | 14:00 | Kostiantyn Drach (Universitat de Barcelona) | Local connectivity of boundaries of Fatou components | In the talk, I will re-prove the famous result of Roesch and Yin about local connectivity of boundaries of bounded Fatou components of polynomials (except Siegel). For this, I will present techniques of puzzles and generalized renormalizations (box mappings) and will show how to use these techniques to get the result. All necessary background will be provided. |
| 2.11.2023 | 12:15 | Joanna Horbaczewska (University of Warsaw) | Local connectivity of the boundary of a Fatou component | In this talk, I will present my current work on local connectivity of the boundary of the Baker Domain of the function f(x)=z-(1-e^z)/(1-2e^z). |
| 26.10.2023 | 12:15 | Gustavo Rodrigues Ferreira (Centre de Recerca Matemàtica) | Transcendental dynamics and wandering domains III | In this three-part talk, we will discuss the internal dynamics of wandering domains of meromorphic functions. Focusing on the hyperbolic distance between pairs of orbits, we will show that every wandering of a meromorphic function locally resembles a wandering of an entire function. |
| 19.10.2023 | 12:15 | Xavier Jarque (Universitat de Barcelona) | Connectivity of the basins of attraction of fixed points for some root finding algorithms | We consider the connectivity of the immediate basins of attraction for different dynamical systems, basically inspired by root-finding algorithms, as the title already explains. |
| 16.10.2023 | 12:00 | Kevin Pilgrim (Indiana University Bloomington) | Real bimodal quadratic rational maps: moduli space and entropy | Bruin-van Strien and Kozlovski showed that for multimodal self-maps f of the unit interval, the function f↦h(f) sending f to its topological entropy is monotone. K. Filom and I showed that for interval maps arising from real bimodal quadratic rational maps, this monotonicity fails. A key ingredient in our proof is an analysis of a family f_{p/q}, p/q \in Q/Z of critically finite maps on which the dynamics on the postcritical set is conjugate to the rotation x↦ x+p mod q on Z/qZ, where x=0 and x=1 correspond to the two critical points. The recent PhD thesis of S. Kang constructs a piecewise-linear (PL) copy of the well-known Farey tree whose vertices are expanding PL quotients of the f_{p/q}’s. This PL model, conjecturally, sheds light on the moduli space of the real quadratic bimodal family, and on the variation of entropy among such maps. (Joint work with K. Filom and S. Kang) |
| 5.10.2023 | 12:15 | Oleg Ivrii (Tel Aviv University) | Pesin theory and inner functions | An inner function is a holomorphic self-map of the unit disk which extends to a measure-theoretic dynamical system of the unit circle. Even though the forward dynamics of an inner function can be very wild, if its derivative belongs to the Nevanlinna class, then backward iteration is asymptotically linear along almost every inverse orbit. We give several applications. (This is joint work with Mariusz Urbański.) |
| 28.09.2023 | 12:20 | Gustavo Rodrigues Ferreira (Centre de Recerca Matemàtica) | Transcendental dynamics and wandering domains II | In this three-part talk, we will discuss the internal dynamics of wandering domains of meromorphic functions. Focusing on the hyperbolic distance between pairs of orbits, we will show that every wandering of a meromorphic function locally resembles a wandering of an entire function. |
| 21.09.2023 | 12:30 | Gustavo Rodrigues Ferreira (Centre de Recerca Matemàtica) | Transcendental dynamics and wandering domains | In this three-part talk, we will discuss the internal dynamics of wandering domains of meromorphic functions. Focusing on the hyperbolic distance between pairs of orbits, we will show that every wandering of a meromorphic function locally resembles a wandering of an entire function. |
For the archive of all talks, see here.