Conformal Dynamics and Groups


Organizers: Junyi Xie, Wenyuan Yang, Zhiqiang Li



(Talks with * are co-hosted in the School Colloquium series by the School of Mathematical Sciences of Peking University; talks with ** are co-hosted in the Distinguished Lecture series by the School of Mathematical Sciences of Peking University; talks with *** are co-hosted in the Colloquium series by the Beijing International Center for Mathematical Research of Peking University; talks with **** are co-hosted in the Progress in Mathematics series by the Beijing International Center for Mathematical Research of Peking University; talks with ***** are co-hosted in the Distinguished Lecture series by the Beijing International Center for Mathematical Research of Peking University)


Nov. 13 2020 - Weixiao Shen (沈维孝) (Fudan University)

Time: 3:00-4:00pm (Beijing Time)    Location: Room 1114, Science Building 1, Peking University

Title: Primitive tuning via quasiconformal surgery****

Abstract: In 1980s, Douady-Hubbard developed a complex counterpart of the Feigenbaum renormalization theory for quadratic-like maps and used this theory to prove existence of small copies in the Mandelbrot set. Inou and Kiwi have extended most of Douady-Hubbard's theory to higher degree polynomial-like maps, but a key surjectivity property was left as a conjecture. We will show how to use quasiconformal surgery to prove this surjectivity conjecture by Inou-Kiwi, under a primitive assumption.  This is a joint work with Wang Yimin.


Nov. 20 2020 - Guizhen Cui (崔贵珍) (Chinese Academy of Sciences)

Time: 3:00-4:00pm (Beijing Time)    Location: Room 1114, Science Building 1, Peking University

Title: Rational maps with constant Thurston maps*

Abstract: A rational map with marked points induces a holomorphic map between Teichmüller spaces of punctured spheres, which is called the Thurston pullback map. One natural problem is to classify the marked rational maps with constant Thurston maps. In this talk, I will introduce the background about this problem at first. Then present a new class of rational maps with constant Thurston pullback maps. This result can be stated in elementary geometry. In the last I will present some relations about this problem with monodromy groups of rational maps.


Nov. 27 2020 - Robert Tang (Xi’an Jiaotong-Liverpool University)

Time: 10:00-11:00am (Beijing Time)    Location: Zoom (ID: 647 1604 0205, Password: 112487)

Title: Affine diffeomorphism groups are undistorted

Abstract: Studying the distortion properties of naturally occurring subgroups of the mapping class group plays an important part in understanding its large-scale geometry. In this talk, I will focus on the affine diffeomorphism group Aff(S,q) of a half-translation surface (S,q). This is the group of self-diffeomorphisms with constant differential away from the singularities. This group also coincides with the stabilizer of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space. Our main result is that any finitely generated subgroup of Aff(S,q) is undistorted in the mapping class group.


Dec. 4 2020 - Mario Bonk (UCLA)

Time: 11:00am-12:00pm (Beijing Time)    Location: Zoom (ID: 649 4519 4579, Password: 143386)

Title: Sullivan's dictionary and beyond*****

Abstract: The classical theory of holomorphic dynamics includes the theory of Kleinian groups and the iteration theory of rational maps. In the 1980s Sullivan pointed out several analogies between these two subjects. For example, the concept of a Julia set of a rational map corresponds to the concept of a limit set of a Kleinian group. Even if one goes beyond holomorphic dynamics, an “extended” Sullivan dictionary can be very helpful as a guide for investigations. In my talk I will give an introduction to this subject and will discuss some recent developments. 


Dec. 10 2020 - Huiping Pan (潘会平) (Jinan University)

Time: 4:00-5:00 pm (Beijing Time)    Location: Zoom (ID: 653 2910 3639, Password: 175707)

Title: On the geometry of the saddle connection graph

Abstract: Every half-translation surface induces a singular flat metric. A saddle connection on a half-translation surface is an open geodesic segment which connects singular points and which contains no singular points in the interior. The saddle connection graph is a graph whose vertices are saddle connections and edges are pairs of disjoint saddle connections. In this talk, we will discuss the geometry of saddle connection graphs. We will show that on the one hand saddle connection grpahs are isometrically rigid, namely two saddle connection graphs are isometric if and only if the underlying half-translation surfaces are affine equivalent. While on the other hand all saddle connections graphs are uniformly quasi-isometric to the regular countably infinite-valent tree. This talk is partially based on a joint work with Valentina Disarlo, Anja Rendecker and Robert Tang.


Dec. 25 2020 - Yan Gao (高延) (Sichuan University)

Time: 10:00-11:00 am (Beijing Time)    Location: Zoom (ID: 690 5359 1120, Password: 294214)

Title: Boundedness and compactification of hyperbolic components for Newton maps

Abstract: In complex dynamics, an interesting question is to determine which kinds of hyperbolic components are bounded in the moduli space of rational maps. In this talk, we study this problem in a well-known slice called Newton family. We prove that, in the moduli space of quartic Newton maps, a hyperbolic component is bounded if and only if it is not type-IE(immediately escaping); furthermore, the GIT-compactification of each type-IE hyperbolic component at infinity boundary is either an analytic closed disk or one point. The proof is based on a convergence theorem of internal rays we establish for degenerate Newton sequences. This is a joint work with Hongming Nie.


Jan. 1 2021 - Break for New Year


Jan. 8 2021 - Wenbo Li (李文博) (University of Toronto)

Time: 11:00am-12:00pm (Beijing Time)    Location: Zoom (ID:  649 1586 5872, Password: 416250)

Title: Conformal dimension and minimality of stochastic objects

Abstract: In this talk, we discuss the conformal dimension of some stochastic objects. The conformal dimension of a metric space is the infimum of the Hausdorff dimension of all its quasisymmetric images. We call a metric space minimal if its conformal dimension equals its Hausdorff dimension. We begin with a construction of a graph of a random function which is a.s. minimal. Inspired by this, we apply the same techniques to the study of 1-dimensional Brownian graphs. The main tool is the Fuglede modulus. This is a joint work with Ilia Binder and Hrant Hakobyan.


Jan. 13 2021 - Tim Mesikepp (University of Washington)

Time: 11:00am-12:00pm (Beijing Time)    Location: Zoom (ID:  649 1586 5872, Password: 416250)

Title:How to weld. Or, towards a unified approach to weldings, driving functions, and energies.

Abstract: How do you conformally weld? That is, given a welding homeomorphism $\varphi:\mathbb{S}^1 \rightarrow \mathbb{S}^1$, how do you compute the associated conformal maps? The measurable Riemann mapping theorem provides a non-constructive answer for quasisymmetric weldings, but sometimes an explicit approximation is helpful. A beautiful algorithm developed by Don Marshall and Lennart Carleson in the early 1980's, called the zipper, provides a numerically robust approach by composing large numbers of "slit" mappings. We give the first convergence results for the zipper by shifting the problem from weldings to driving functions. In the process, we also give some results on minimal-energy curves and, if time permits, draw connections to universal Teichmuller space. There will be many pictures and the talk should be widely accessible.


Mar. 24 2021 – Zhe Sun (IHES)

Time: 16:00-17:00pm (Beijing Time)    Location: Zoom (ID:  674 1443 0868, Password: 904596)

Title: Surface group representations, tropical points and webs.

Abstract: Kuperberg introduced oriented 3-valent graphs on the surface, called 3-webs, to study the SL_3-invariant tensor products T of irreducible representations of SL_3. Then Kuntson-Tao found a family of linear inequalities to characterize when T contains an invariant vector. Let A be a variation of the SL_3 character variety which generalizes the Penner's decorated Teichmuller space. Actually, Goncharov--Shen related Kuntson-Tao inequalities to the positivity of A. On the surface, we identify the space of 3-webs up to homotopy with certain lattice of A mapping class group equivariantly. As a consequence, as predicted by Fock--Goncharov duality conjecture, these tropical points parameterize a linear basis of the regular function ring of the dual space explicitly. This is a joint work with Daniel Douglas.


Mar. 30 2021 - Suzhen Han (韩素珍) (Peking University)

Time: 10:00am-12:00pm (Beijing Time)    Location: Room 1114, Science Building 1, Peking University

Title: The proper actions of relatively hyperbolic groups on finite products of quasi-trees

Abstract: A finitely generated group has property (QT) if it can act isometrically on a finite product of quasi-trees so that the orbit map is a quasi-isometric embedding. This notion is introduced by M. Bestvina, K. Bromberg and K. Fujiwara, who also established such property for residually finite hyperbolic groups and mapping class groups. In a joint work with H.T. Nguyen and Wenyuan Yang, we generalize their result for hyperbolic groups, and showed that residually finite relatively hyperbolic groups have property (QT) if their peripheral subgroups satisfy some conditions.


April 7 2021 – Hao Liang (梁灏) (Sun Yat-sen University)

Time: 3:00-4:00pm (Beijing Time)    Location: Zoom (ID:  611 5027 7004, Password: 329424)

Title: A factorization theorem for homomorphisms into Fuchsian groups

Abstract: I will present a factorization theorem for homomorphisms into Fuchsian groups (discrete subgroups of the isometry group of the hyperbolic plane). It is a useful tool for understanding the structure of the collection of all discrete representations of any fixed finitely generated group G into PSL(2, R). It will also have some applications in discrete limit groups of PSL(2, R). I will discuss the background and application of the theorem and the main idea of the proof.


April 12 2021 – Yusheng Luo (骆宇盛) (University of Michigan – Ann Arbor / Stony Brook University)

Time: 9:00-10:00am (Beijing Time)    Location: Zoom (ID:  677 2262 6461, Password: 304728)

Title: Circle packings, kissing reflection group and critically fixed anti-rational maps

Abstract: Circle packings appear frequently in the studies of dynamics, geometry. One can naturally associate a reflection group to a finite circle packing, generated by reflections along the corresponding circles. In this talk, we will establish an explicit correspondence between such reflection groups with anti-holomorphic maps of the Riemann sphere where all the critical points are fixed. We will explore the correspondence both in the dynamical plane and the parameter spaces. In particular, we will explain how the analogue of Thurston’s compactness theorem for acylindrical hyperbolic 3-manifold holds for critically fixed anti-rational maps.

We will also brief discuss some open questions motivated by the correspondence.


April 21 2021 – Ma Biao (马彪) (Université de Nice)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom (ID:  821 5134 9008, Password: 370777)

Title: Boundary representations of mapping class groups

Abstract: Let S = S_g be a closed orientable surface of genus g > 1 and Mod(S) be the mapping class group of $S$. In this talk, we show that the boundary representation of Mod(S) is ergodic, which generalizes the classical result of Masur on ergodicity of the action of Mod(S) on the projective measured foliation space PMF(S). As a corollary, we show that the boundary representation of Mod(S) is irreducible.


May 12 2021 – Lizhi Chen (陈立志) (Lanzhou University)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom (ID:  895 8085 2996, Password: 339132)

Title:  Homology and homotopy complexity of manifolds via systolic geometry 

Abstract: We discuss homology and homotopy complexity of manifolds in terms of Gromov’s systolic inequality. The optimal constant in systolic inequality is usually called systolic volume. A central theorem in systolic geometry relates systolic volume to simplicial volume. Since for hyperbolic manifolds there exist proportionality principle, this theorem builds a bridge between systolic geometry and hyperbolic geometry. In the talk, we will present some applications of this theorem to the problem of homology and homotopy complexity of manifolds.


May 17 2021 – Roland Roeder (Indiana University – Purdue University Indianapolis)

Time: 9:00-10:00am (Beijing Time)    Location: Zoom (ID:  813 3007 9112, Password: 828437)

Title:  Dynamics of groups of birational automorphisms of cubic surfaces and Fatou/Julia decomposition for Painlevé 6

Abstract: We study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces

S_{A,B,C,D} = {(x,y,z)  \in C^3   :    x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D},

where A,B,C, and D are complex parameters.  It arises naturally in the dynamics on character varieties and it also describes the monodromy of the famous Painlevé 6 differential equation.  We explore the Fatou/Julia dichotomy for this group action (defined analogously to the usual definition for iteration of a single rational map) and also the Locally Discrete / Non-Discrete dichotomy (a non-linear version from the classical discrete/non-discrete dichotomy for Lie groups). The interplay between these two dichotomies allow us to prove several results about the topological dynamics of this group.  Moreover, we show the coexistence of non-empty Fatou sets and Julia sets with non-trivial interior for a large set of parameters.

This is joint work with Julio Rebelo.


May 26 2021 – Fei Yang (杨飞) (Nanjing University)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom (ID:  884 5933 7010, Password: 684103)

Title:  Local connectivity of the Julia sets with bounded type Siegel disks

Abstract: Let f be a holomorphic map containing an irrationally indifferent fixed point z0. If f is locally linearizable at z0, then the maximal linearizable domain containing z0 is called the Siegel disk of f centered at z0. The topology of the boundaries of Siegel disks has been studied extensively in past 3 decades. This was motivated by the prediction of Douady and Sullivan that the Siegel disk of every non-linear rational map is a Jordan domain.

For the topology of whole Julia sets of holomorphic maps with Siegel disks, the results appear less. Petersen proved that the quadratic Julia sets with bounded type Siegel disks are locally connected. Later Yampolsky proved the same result by an alternative method based on the existence of complex a prior bound of unicritical circle maps. A big progress was made by Petersen and Zakeri in 2004. They proved that for almost all rotation numbers, the quadratic Julia sets with Siegel disks are locally connected. Recently J. Yang proved a striking result that the Julia set of any polynomial (assumed to be connected) is locally connected at the boundary points of their bounded type Siegel disks.

As a generalization of Petersen's result, we prove that the Julia sets of a number of rational maps and transcendental entire functions with bounded type Siegel disks are locally connected. This is based on establishing an expanding property of a long iteration of a class of quasi-Blaschke models near the unit circle.


June 9 2021 – Yulan Qing (卿于兰) (Fudan University)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom (ID:  857 3683 8743, Password: 566197)

Title:  The Large scale geometry of big mapping class groups

Abstract: In this talk, we introduce the framework of the coarse geometry of non-locally compact groups in the setting of big mapping class groups, as studied by Rosendal. We will discuss the characterization results of Mann-Rafi and Horbez-Qing-Rafi that illustrate big mapping groups' rich geometric and algebraic structures. We will outline the proofs in these results and their implications. If time permits, we will discuss some open problems in this area. 


June 18 2021 – Xiaoguang Wang (王晓光) (Zhejiang University)

Time: 10:00-11:00am (Beijing Time)    Location: Room 1560, Science Building 1, Peking University

Title: Boundaries of capture hyperbolic components

Abstract: In the field of complex dynamics, the boundaries of higher dimensional hyperbolic components in meaningful families of polynomals or rational maps are mysterious topological objects.  In this talk, we discuss some typical families of polynomials, and show that the boundary of a (high-dimensional) capture hyperbolic component is homeomorphic to the sphere. More strikingly, the Hausdorff dimension of this boundary can be given explicitly. This is a joint work with Jie Cao and Yongcheng Yin.   


July 2 2021 – Gaofei Zhang (张高飞) (Nanjing University)

Time: 3:00-4:00pm (Beijing Time)    Location: Zoom (ID: 822 0642 3584, Password: 822747)

Title:  On the local connectivity of the Julia sets of rational maps with bounded type Siegel disks***

Abstract: In the talk, I will sketch the idea of the proof of the following result.  Suppose f is a rational map with bounded type Siegel disks such that any infinite critical orbit intersects either the closure of some bounded type Siegel disk or the basin of some attracting periodic point.  Suppose additionally the Julia set J(f) is connected. Then J(f) is locally connected.  This was previously proved under the assumption that the boundaries of attracting bastions do not intersect the boundary of any Siegel disk.  


September 24 2021 – Jinsong Liu (刘劲松) (Chinese Academy of Sciences)

Time: 2:00-3:00pm (Beijing Time)    Location: Room 77201, Jingchunyuan 78, BICMR, Peking University

Title:  Extensions of quasi-isometries between complex domains***

Abstract: In this talk, by using the Gehring-Hayman-type theorem on some complex domains. We will give some results on bi-Holder extensions not only for biholomorphisms, but also for more general Kobayashi metric quasiisometries between these domains.


October 28 2021 – Jianyu Chen (陈剑宇) (Soochow University)

Time: 3:00-4:00pm (Beijing Time)    Location: Zoom (ID: 886 6229 2968, Password: 733229)

Title:  Inducing schemes with finite weighted complexity

Abstract: We consider a measurable map of a compact metric space which admits an inducing scheme. Under the finite weighted complexity condition, we establish a thermodynamic formalism for a parameter family of potentials $\varphi+t\psi$ in an interval containing $t=0$.  Furthermore, if there is a generating partition compatible to the inducing scheme, we show that all ergodic invariant measures with sufficiently large pressure are liftable. Our results are applicable to the Sinai dispersing billiards with finite horizon, that is, we establish the equilibrium measures for the family of geometric potentials in a slightly restricted class. This is a joint work with Fang Wang and Hong-Kun Zhang.


November 9 2021 – Renxing Wan (万仁星) (Peking University)

Time: 3:10-5:00pm (Beijing Time)    Location: Room 1114, Science Building 1, Peking University

Title:  Uniform growth in groups of exponential growth

Abstract: This talk is based on two surveys about the growth of groups by Grigorchuk and de la Harpe. At first, I will briefly introduce some notions and list known results and open problems. Later, we will focus on some basic examples including free groups, non-elementary hyperbolic groups, some solvable groups , mapping class group and so on.


November 26 2021 – Bin Yu (余斌) (Tongji University)

Time: 3:00-5:00pm (Beijing Time)    Location: Zoom (ID: 879 8424 0017, Password: 928052)

Title:  R-covered Anosov flows and freely homotopic periodic orbits

Abstract: In the last 30 years, by using foliation theory and group action, Barbot and Fenley developed a powerful tool to qualitatively understand 3-dimensioanl Anosov flows.  In this seminar talk,  we will introduce some basic ideas due to Fenley in this direction.  In particular, we will explain a significant property discovered by Fenley: every free homotopy class of an  R-covered Anosov flow on a hyperbolic 3-manifold contains infinitely many periodic orbits.


December 3 2021 – Inhyeok Choi (Seoul National University)

Time: 3:00-4:00pm (Beijing Time)    Location: Zoom (ID: 871 9276 2553, Password: 897719)

Title:  Limit laws for random walks on mapping class groups

Abstract: Random walks on groups acting on non-positively-curved spaces have been studied in depth. Various limiting behaviors of these random walks, including the positive escape rate, laws of large numbers and central limit theorems, were known under some geometric assumptions on the space and the group action. However, the actions of mapping class groups on Teichmüller spaces and curve complexes do not satisfy these assumptions, the former one being not Gromov hyperbolic and the latter one being not locally compact. Nonetheless, recent developments imply that many limiting behaviors of random walks on mapping class groups follow directly from the (partial) hyperbolicity of Teichmüller spaces and curve complexes. In this talk, I will focus on the principle behind these results and two possible applications: the regularity of the harmonic measure and the counting problem in mapping class groups. Partially joint with Hyungryul Baik and Dongryul M. Kim.


December 24 2021 – Jinsong Zeng (曾劲松(Guangzhou University)

Time: 3:00-4:00pm (Beijing Time)    Location: Zoom (ID: 875 7470 1894, Password: 101624)

Title:  Decomposition of rational maps

Abstract: This talk is mainly about the dynamics of rational maps on the Riemann sphere. We will show that every postcritically finite rational map with non-empty Fatou set can be decomposed into bubble rational maps and Sierpinski rational maps. Based on this theory, an invariant and finite connected graph can be constructed in the Julia set. This is a joint work with Guizhen Cui and Yan Gao.


March 12 2022 – Zhuchao Ji (冀诸超) (Shanghai Center for Mathematical Sciences)

Time: 2:00-3:00pm (Beijing Time)    Location: Quanzhai 9, Jingchunyuan 78, BICMR, Peking University

Title:  On wandering domains in higher dimensions

Abstract:  See here.


May 26 2022 – Tim Mesikepp (University of Washington)

Time: 10:00-11:00am (Beijing Time)    Location: Zoom (ID: 826 4953 9587, Password: 083683)

Title:  A deterministic approach to Loewner-energy minimizers

Abstract: The Loewner energy is a conformally-invariant functional of curves in the upper half plane H that has connections to hyperbolic geometry, SLE theory, Teichmuller theory and geometric function theory.  In this talk we study "one-point" minimizers of Loewner energy, asking what curves minimize the energy among all which pass through a given point in H, and what curves minimize the energy among all which weld given points x<0<y.  The former question was partially studied by Yilin Wang, who used SLE techniques to calculate the minimal energy and show it is uniquely attained.  We revisit the question using a purely deterministic methodology, however, and re-derive the energy formula and also obtain further results, such as an explicit computation of the driving function. Our approach also yields existence and uniqueness of minimizers for the welding question, as well as an explicit energy formula and explicit driving function.  We discuss parallel properties of both families, such as "universality" (all curves in either family can be generated by a single driving function), energy usage, and connections to SLE with forcing. 


May 26 2022 – Jingyin Huang (Ohio State University)

Time: 8:00-9:00pm (Beijing Time)    Location: Zoom (ID: 860 9860 6310, Password: 399654)

Title: The non-positive curvature geometry of some fundamental groups of complex hyperplane arrangement complements

Abstract: A complex hyperplane complement is a topological space obtained by removing a collection of complex codimension one affine hyperplanes from C^n (or a convex cone of C^n). Despite the simple definition, these spaces have highly non-trivial topology. They naturally emerge from the study of real and complex reflection groups, braid groups and configuration spaces, and Artin groups. More recently, the fundamental groups of some of these spaces start to play important roles in geometric group theory, though most of these groups remain rather mysterious. We introduce a geometric way to understand classes of fundamental groups of some of these spaces, by equivariantly “thickening” these groups to metric spaces which satisfy a specific geometric property that is closely related to convexity and non-positive curvature. We also discuss several algorithmic, geometric and topological consequences of such a non-positive curvature condition. This is joint work with D. Osajda.


June 21 2022 – Fabrizio Bianchi (Université de Lille)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom (ID: 823 4284 1727, Password: 366599)

Title: A spectral gap for the transfer operator on complex projective spaces

Abstract: 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.


September 20 2022 – Carlangelo Liverani (University of Rome Tor Vergata) 

Time: 4:00-5:00pm (Beijing Time)    Location: Room 77201, Jingchunyuan 78, BICMR, Peking University; Zoom (ID: 812 0977 4828, Password: 813004)

Title: Statistical properties of hyperbolic billiards*****

Abstract: I will discuss some recent and less recent results concerning the statistical properties of hyperbolic billiards, with particular emphasis on properties motivated by fundamental questions in non-equilibrium statistical mechanics.


October 7 2022 – Wenbo Li (李文博) (Peking University)

Time: 10:30-11:30am (Beijing Time)    Location: Room 77201, Jingchunyuan 78, BICMR, Peking University

Title: Quasiconformal geometry and the boundary of hyperbolic groups

Abstract: A Hyperbolic space is a metric space whose geodesic triangles are "thin" and a hyperbolic group is a group whose Cayley graph is hyperbolic. We focus on the boundary of hyperbolic groups in this talk and go through topics around two rigidity conjectures: The Cannon Conjecture and the Kapovich-Kleiner Conjecture. Roughly speaking, these two conjectures ask about whether special topological restrictions on the boundary of hyperbolic groups will uniquely determine them up to quasisymmetries. In an effort to answer these questions, many people have studied them from different approaches. We will go through the work by Bonk and Kleiner. In the end, we provide our trying on these conjectures. In particular, we have constructed a special case of metric Sierpinski carpet, dyadic slit carpets, and completely characterize its planar quasisymmetric embeddability.


October 14 2022 – Xiaolei Wu (伍晓磊) (Fudan University)

Time: 3:00-4:00pm (Beijing Time)    Location: Tencent (ID: 522-848-241, Password: 221014)

Title: Finiteness properties of asymptotic mapping class groups

Abstract: Asymptotic mapping class groups was introduced by Funar and Kapoudjian. They can be treated as a generalization of the Thompson groups in the mapping class group setting. We will discuss how these groups are constructed and show that they are in fact of type F_infinity. The proof boils down to prove certain subsurface complexes are highly connected. This is based on joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Nansen Petrosyan.


October 21 2022 – Ilia Binder (University of Toronto)

Time: 8:00-9:00pm (Beijing Time)    Location: Zoom (ID: 870 3163 5732, Password: 955080)

Title: Computing Julia sets, Conformal maps, and Harmonic measure

Abstract: In this talk, I will discuss recent advances in the computability of various objects arising in Complex Dynamics and the Theory of Univalent functions. After a brief introduction to the general Computability Theory, I will talk about the computation properties of polynomial Julia sets, conformal maps, and Harmonic measure. Based on joint work with M. Braverman (Princeton), A. Glucksam (Northwestern), C. Rojas ( Pontificia Universidad Catolica de Chile), and M. Yampolsky (University of Toronto).


October 27 2022 – Bobo Hua (华波波) (Fudan University)

Time: 3:00-4:00pm (Beijing Time)    Location: Tencent (ID: 464-146-395, Password: 221028)

Title: Ollivier curvature on graphs and discrete harmonic functions

Abstract: Ollivier introduced a curvature notion on graphs via the optimal transport, which is a discrete analog of the Ricci curvature on manifolds. Analytic properties of discrete harmonic functions are closely related to the Ollivier curvature. We prove that the number of ends is at most two for an infinite graph with nonnegative Olliver curvature. This is joint work with Florentin Muench.


November 4 2022 – Mariusz Urbański (University of North Texas)

Time: 9:00-10:00am (Beijing Time)    Location: Zoom (ID: 827 6082 3142, Password: 057934)

Title: The exact value of Hausdorff dimension of escaping sets for class $\mathcal B$ of meromorphic functions

Abstract: We consider the subclass of class $\mathcal B$ consisting of meromorphic functions $f\C\to\C$ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in [BwKo2012] and the Hausdorff dimension HD(I(f)) of the set I(f) of all points escaping to infinity under forward iteration of $f$ was estimated therein. In this lecture, based on joint paper with Volker Mayer, we provide a closed formula for the exact value of HD(I(f)) identifying it with the critical exponent of the natural series introduced in [BwKo2012]. This exponent is very easy to calculate for many concrete functions. In particular, we construct a function from this class which is of infinite order and for which HD(I(f))=0.


November 25 2022 – Daniel Meyer (University of Liverpool)

Time: 5:00-6:00pm (Beijing Time)    Location: Zoom (ID: 838 0851 2867, Password: 686499)

Title: Fractal spheres, visual metrics, and rational maps

Abstract: Quasisymmetric maps map ratios of distances in a controlled way. They generalize conformal maps. The quasisymmetric uniformization theorem asks if a certain metric space is quasisymmetrically equivalent to some model space. Of particular interest in this context is the question to characterize quasispheres, i.e., metric spaces that are quasisymmetrically equivalent to the standard $2$-sphere. A simple class of fractal spheres are ``snowballs'', which are topologically $2$-dimensional analogs of the van Koch snowflake curve.

A Thurston map is a topological analog of a rational map (i.e., a holomorphic self-map of the Riemann sphere). Thurston gave a criterion when such a map ``is'' rational. Given such a map $f$ that is expanding, we can equip the sphere with a ``visual metric''. With respect to this metric, the sphere is a quasisphere if and only if $f$ ``is'' rational.

This is joint work with Mario Bonk (UCLA).


December 2 2022 – Sang-hyun Kim (KIAS)

Time: 3:00-5:00pm (Beijing Time)    Location: Zoom (ID: 565 147 0713, Password: 944743)

Title: First order rigidity of manifold homeomorphism groups

Abstract: We prove that two compact connected manifolds are homeomorphic only if their homeomorphism groups are elementarily equivalent, i.e. have exactly the same set of the true first-order group-theoretic sentences . This is a work in progress. (Joint with Thomas Koberda and Javier de la Nuez--Gonzalez).


March 17 2023 – Sylvester Eriksson-Bique (University of Oulu)

Time: 4:30-5:30pm (Beijing Time)    Location: Zoom  (ID: 892 5622 7347, Password: 094591)

Title: A new lemma on Hausdorff content

Abstract: What is in common between random or dynamically generated limit superior sets and boundaries of sets of finite perimeter. In both settings problems can be reduced to estimating the Hausdorff content of some set which has a type of self-similarity. We introduce a new lemma that yields a Hausdorff content lower bound in both of these settings, and which seems useful in further applications. We illustrate its usefulness by deriving three well known conclusions from it: the Federer characterization of sets of finite perimeter, the Hausdorff dimension of random limit superior sets and the Beresnevitch-Velani Mass Transference Principle.


April 14 2023 – Insung Park (Brown University)

Time: 10:00-11:00am (Beijing Time)    Location: Zoom  (ID: 856 2139 9573, Password: 111111)

Title: Julia sets having Ahlfors-regular conformal dimension one

Abstract: The Ahlfors-regular conformal dimension, ARCdim(X), of a compact metric space X is the infimal Hausdorff dimension in the Ahlfors-regularly quasi-symmetric class of X. As a fractal embedded in the Riemann sphere, the Julia set J_f of a hyperbolic rational map f has Ahlfors-regular conformal dimension between 1 and 2. We have ARCdim(J_f)=2 iff J_f is the entire Riemann sphere. The other extreme case ARCdim(J_f)=1, however, contains a variety of Julia sets, including Julia sets of critically finite polynomials and Newton maps. In this talk, we show that for a critically finite hyperbolic rational map f, ARCdim(J_f)=1 if and only if there exists an f-invariant graph G containing all the critical points such that the topological entropy of the induced dynamics on G is zero. We also show that for a (possibly non-hyperbolic) critically finite rational map f, ARCdim(X)=1 is attained as the minimal Hausdorff dimension if and only if f is conjugate to the monomial map z^{\pm} or the Chebyshev polynomial. This talk is partially based on joint work with Angela Wu.


April 21 2023 – John Mackay (University of Bristol)

Time: 5:00-6:00pm (Beijing Time)    Location: Zoom  (ID: 859 6470 2961, Password: 111111)

Title: Maps between relatively hyperbolic groups and their boundaries

Abstract: Isometries of real hyperbolic spaces correspond to Mobius transformations of their spheres at infinity.  Thanks to Paulin and Bonk-Schramm, quasi-isometries of Gromov hyperbolic spaces correspond to quasisymmetries between their boundaries at infinity.  After discussing this background, I'll describe how certain maps between relative hyperbolic groups correspond to maps between their Bowditch boundaries, and an application.  Joint work with Alessandro Sisto.


May 19 2023 – Kevin M. Pilgrim (Indiana University Bloomington)

Time: 8:00-9:00pm (Beijing Time)    Location: Zoom  (ID: 847 5703 2480, Password: 111111)

Title: On amenable iterated monodromy groups (joint with V Nekrashevych and D Thurston)

Abstract: We show that a contracting recurrent self-similar group of polynomial activity growth is amenable.  The proof proceeds by first showing that the associated limit space has conformal dimension equal to 1. We then show this implies the orbital graphs of its standard action are recurrent. Finally, we deduce amenability by applying results of Juschenko, Nekrashevych, and de la Salle.  Examples of such groups include the iterated monodromy groups of critically finite  hyperbolic “crochet” rational functions; these are maps for which any two points in the Fatou set are joined by a curve that meets the Julia set in an at most countable set.


May 24 2023 – Yulan Qing (卿于兰) (Fudan University)

Time: 1:00-3:00pm (Beijing Time)    Location: Room 1114, Sciences Building No. 1

Title: Boundary of groups

Abstract: Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it is an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce a generalization of the Gromov boundary for all finitely generated groups. We construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space and it is metrizable. We show the geometric genericity of points in this boundary using Patterson Sullivan measure on the visual boundary of CAT(0) spaces. As an application we discuss the connection between the sublinearly Morse boundary and random walk on groups. We answer open problems regarding QI-invariant models of random walk on CAT(0) groups and on mapping class groups. If time permits, we also will also look at how this boundary behaves under sublinear bilipschitz equivalences.


May 26 2023 – Araceli M. Bonifant (University of Rhode Island)

Time: 8:00-9:00pm (Beijing Time)    Location: Zoom  (ID: 889 4828 5549, Password: 111111)

Title: Understanding the dynamics of cubic polynomials


Abstract: For each p>0 there is a family S_p of complex cubic maps with a marked critical orbit of period p. For each q>0, I will describe a dynamically defined tessellation of S_p. Each face of these tessellations is associated with one particular behavior for periodic orbits of period q. (Joint work with John Milnor.) 


Jun 5 2023 – Tim Mesikepp (Peking University)

Time: 11:00am-12:00pm (Beijing Time)    Location: Room 1114, Sciences Building No. 1

Title: SLE large deviations and the regularity of Loewner transform

Abstract: The Loewner energy is a functional on Jordan curves that is the large-deviations rate function for Schramm-Loewner Evolutions (SLE's).  Recent work has shown it also has surprising connections to other areas of probability, complex analysis, geometric measure theory, hyperbolic geometry, and even Teichmuller theory.  In the first part of this talk we survey some of these connections, providing an "appetizer" for Yilin Wang's upcoming summer school.

In the second part of the talk we show that Loewner-energy minimizers give insight into the regularity of the inverse Loewner transform (the map taking a driving function to its associated curve).  In particular, we use energy minimizers to exhibit curves of differing regularity that have driving functions of the same regularity, thus showing a theorem of Carto Wong is sharp.  This portion of the talk will be deterministic in nature.


Jun 5 2023 to Jun 12 2023 – Mario Bonk (UCLA)

Time & Location: 

Jun 5 2023 9:00-11:00am (Beijing Time)   Room 1114, Sciences Building No. 1

Jun 10 2023 10:00am-12:00pm (Beijing Time)   Room 1556, Sciences Building No. 1

Jun 12 2023 3:00-5:00pm (Beijing Time)   Room 1556, Sciences Building No. 1


Title: Teichmüller spaces and the Thurston pull-back map Mario Bonk

Abstract: A postcritically-finite branched covering map on a topological 2-sphere is known as a Thurston map. It induces a holomorphic map on an associated Teichmüller space. This map is known as the Thurston pull-back map. In this mini course, I will give a quick introduction to necessary background (Teichmüller spaces and Thurston maps), outline a proof of the holomorphicity of the pull-back map, and discuss a landmark theorem by Thurston on the characterization of postcritically-finite rational maps.

I’ll discuss concrete examples for the pull-back maps for Thurston maps with four postcritical points. As will be seen in the course, this leads to a certain type of polymorphic functions related to modular functions and forms. If time permits, I will also discuss ongoing research about the existence of a global finite curve attractor in this framework.


June 7 2023 – Mario Bonk (UCLA)

Time: 4:00-5:00pm (Beijing Time)    Location: Room 1114, Sciences Building No. 1

Title: Fractals and the dynamics of Thurston maps**

Abstract: A Thurston map is a branched covering map on a topological 2-sphere for which the forward orbit of each critical point under iteration is finite. Each such map gives rise to a fractal geometry on its underlying 2-sphere. The study of these maps and their associated fractal structures links diverse areas of mathematics such as dynamical systems, ergodic theory, classical conformal analysis, hyperbolic geometry, Teichmüller theory, and analysis on metric spaces. In my talk I will give an introduction to this subject and report on ongoing research.


June 16 2023 – Sabyasachi Mukherjee (Tata Institute of Fundamental Research)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom  (ID: 852 0183 6977, Password: 111111)

Title: Matings, holomorphic correspondences, and a Bers slice

Abstract: There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: an algebraic correspondence framework due to Bullett-Penrose-Lomonaco and an orbit equivalence mating framework using Bowen-Series maps. The latter is analogous to the Douady-Hubbard theory for polynomial mating. We will discuss how these two frameworks can be unified and generalized. As a consequence, we will construct holomorphic correspondences that are matings of hyperbolic orbifold groups (including Hecke groups) with Blaschke products. Time permitting, we will introduce an analog of a Bers slice of the above orbifolds in the algebraic parameter space of correspondences. Based on joint work with Mahan Mj. 


June 30 2023 – Scott Sutherland (Stony Brook University)

Time: 9:00-10:00pm (Beijing Time)    Location: Zoom  (ID: 871 7088 4305, Password: 111111)

Title: Real quadratic rational maps

Abstract: We consider the family or degree two rational maps of the Riemann sphere to itself that preserve the real line, with a goal of understanding the parameter space of all such mappings.  As in the case of polynomial maps,  the postcritically finite case (which are Thurston maps) play an important role; we also discuss the associated combinatorics of such maps.  [Joint work with Araceli Bonifant and John Milnor.]


July 14 2023 – Dylan Thurston (Indiana University, Bloomington)

Time: 2:30-3:30pm (Beijing Time)    Location: BICMR 77201

Title: Rational maps via spines

Abstract: How does a (post-critically finite) rational map act on the Riemann sphere, topologically and as a dynamical system? We give a concrete way to understand this using spines, appropriate graphs embedded in the sphere. The entire dynamics of the map can be faithfully captured combinatorially by a pair of graphs and a pair of maps between them.


July 14 2023 – Dylan Thurston (Indiana University, Bloomington)

Time: 4:00-5:00pm (Beijing Time)    Location: BICMR JiaYiBing Lecture Hall

Title: Expansivity

Abstract: We start with a detour to recall part of the Nielsen-Thurston classification of surface self-maps, and in particular how curves on the surface are stretched under iteration. The analogous notion for a general dynamical system, which we call the expansivity, appears to be captured for rational maps by a stretch factor naturally arising from the graphical description, and the minimal Lipschitz stretch of maps between graphs. 


July 15 2023 – Dylan Thurston (Indiana University, Bloomington)

Time: 10:00-11:00am (Beijing Time)    Location: BICMR 77201

Title: Positive Characterization of Rational Maps

Abstract: When does a topological self-map of the sphere come from an actual rational map? Can we give a positive combinatorial certificate that this happens? Expansivity from the previous lecture gives one constraint, but to give the full answer we need another "energy" associated to graph maps, the stretch factor related to rubber bands and, with some caveats, to electrical networks. 


September 8 2023 – Oliver Jenkinson (Queen Mary, University of London)

Time: 4:00-5:00pm (Beijing Time)    Location: Room 1114, Sciences Building No. 1

Title: When simple is best: Ergodic optimization, Sturmian orbits, and ergodic dominance*

Abstract: For a given dynamical system, and a given real-valued function, the field of Ergodic Optimization seeks to understand those orbits (or invariant measures) that realise the largest possible ergodic average. Rather often, these orbits turn out to be in some sense `simple', for example periodic, or non-periodic but of low complexity (e.g. Sturmian). Ergodic dominance is one strategy, appealing to ideas from stochastic dominance, for understanding the optimizing orbits and measures for certain classes of functions. One application is to constrained optimization problems for digit expansions: e.g. if we fix the mean value of the decimal digits of a number, or equivalently fix the arithmetic mean of an orbit under the map x-> 10x (mod 1), how can we minimize the variance around the mean, and how can we maximize the geometric mean? 


October 10 2023 – Ilia Binder (University of Toronto)

Time: 4:00-5:00pm (Beijing Time)    Location: BICMR 77201

Title: Computability in Complex Dynamics and Complex Analysis***

Abstract: In this talk, I will discuss computability questions arising in Complex Dynamics and Complex Analysis. More specifically, I will discuss the existence of non-computable polynomial Julia sets and of polynomial Julia sets with arbitrarily high computational complexity. I will also discuss the conditions necessary and sufficient for the computability of conformal maps and their Caratheodory extensions. The computability of another central object of Complex Analysis, the Harmonic measure, will also be discussed. A crash course in general Computability Theory will be given. The talk is based on joint projects with M. Braverman (Princeton), A. Glucksam (Northwestern), C. Rojas (Universidad Catolica de Chile), and M. Yampolsky (University of Toronto). 


October 27 2023 – Guy C. David (Ball State University)

Time: 9:00-10:00am (Beijing Time)    Location: Zoom  (ID: 861 9382 6449, Password: 647898)

Title: Embeddings, carpets, curves, and conformal dimension

Abstract: We discuss two results on bi-Lipschitz embeddings of metric spaces. The first is a (negative) answer to a 1997 question of Heinonen and Semmes asking whether spaces that can be “folded” into Euclidean space can be embedded bi-Lipschitzly. The second is a more general theorem showing that spaces with “thick” curve families cannot have bi-Lipschitz embeddings in Euclidean space, unless they admit some “infinitesimal splitting”. This latter result has some applications to the study of conformal dimension. This is joint work with Sylvester Eriksson-Bique (University of Jyvaskyla). 


November 27 2023 – Zachary Smith (UCLA)

Time: 10:00-11:00am (Beijing Time)    Location: Zoom  (ID: 824 3911 1432, Password: 745324)

Title: The Moduli Space Correspondence of Thurston Theory

Abstract: A Thurston map f: (S^2, A) → (S^2, A) with marking set A induces a map σ_f on Teichmüller space by pulling back complex structures. One way to study this pullback map is via a correspondence on moduli space. In this talk I will discuss both the general correspondence and some special cases. Specifically, I will give a careful description of the correspondence for the case where A has four points and the postcritical set (which is a subset of A) has three points. The diagram I will present has been previously used by Kelsey and Lodge to study quadratic Thurston maps with few postcritical points, and I also use it in a forthcoming paper proving special cases of the curve attractor problem for rational maps with four marked points. Time permitting I will further discuss the results of this paper. 


December 29 2023 – Shaosong Liu (刘少松) (University of Rochester)

Time: 10:00-11:00am (Beijing Time)    Location: Zoom  (ID: 891 8732 0712, Password: 615723)

Title: Lee-Yang zeros and its critical behavior for the Ising model on Sierpinski gasket

Abstract: It is known that Ising model on Sierpinski gasket is exactly solvable. In 1980s Gefen, Aharony, Shapir and Mandelbrot gave the recursive relation for such model and argued no finite-temperature phase transitions. In this talk I will give a proof for exact formulas of its partition functions. The recursive relation is conjugated to a quadratic map, from which I can give a precise description about zero distributions and thermodynamic limit.


January 9 2024 – Pekka Pankka (University of Helsinki)

Time: 4:00-5:00pm (Beijing Time)    Location: BICMR 77201

Title: Picard theorems in quasiconformal geometry***

Abstract: The classical Picard theorem from complex analysis states that a holomorphic map from the complex plane to itself omits at most one point. Of course, the exponential mapping shows that Picard's theorem is sharp. In higher dimensional quasiconformal geometry (i.e. when we allow bounded distortion of the conformal structure), Picard's theorem appears in the form of Rickman's theorem (1980): A quasiregular mapping from the Euclidean space to itself the cardinality of the omitted set is bounded by a constant depending only on the dimension and distortion of the mapping. Also this result is sharp. In this talk, I will discuss these Picard type results and their cohomological counterparts for quasiregular mappings from Euclidean spaces to closed Riemannian manifolds, especially the following result: If a closed Riemannian n-manifold admits a quasiregular mapping from the n-Euclidean space, then its de Rham cohomology embeds into the exterior algebra of R^n. 


January 11 2024 to January 12 2024 – Pekka Pankka (University of Helsinki)

Time & Location: 

January 11 2024 2:30-3:30pm (Beijing Time)   Hou Zhu Lou 1124, Beijing Normal University

January 11 2024 4:00-5:00pm (Beijing Time)   Hou Zhu Lou 1124, Beijing Normal University

January 12 2024 4:00-5:00pm (Beijing Time)   BICMR 77201, Peking University


Title: Quasiregular mappings -- past, present, future

Abstract: In this minicourse, I will discuss quasiconformal geometry from the point of view of mapping theory, especially of quasiregular mappings. Quasiregular mappings can be viewed as a generalization of planar holomorphic mappings in higher dimensional Riemannian geometry. The prefix 'quasi-' refers to bounded distortion of the conformal structure. The classical Liouville's theorem states that conformal mappings between domains of a Euclidean n-space are restrictions of Möbius transformtion in dimensions n\ge 3. Therefore, in higher Euclidean dimensions, distortion is needed to obtain a non-trivial 'conformal' mapping theory.


On the first lecture, I will discuss how conformal invariants lead to familiar results from complex analysis in this higher dimensional setting. On the second lecture, I will discuss measurable conformal structures and a version of complex dynamics in this context and how these ideas lead to cohomological boundedness of closed manifolds which admit quasiregular mappings from Euclidean spaces. Finally, on third lecture I will discuss non-equidimensional version of the quasiregular theory connecting quasiregular mappings to holomorphic curves and, if time permits, recent developments on Picard theorems for mappings without conformality assuptions.


Lecture I: Quasiregular mappings and conformal invariants

Lecture II: Quasiregular dynamics and cohomological results

Lecture III: Quasiregular curves and Picard theorems beyond conformality


January 19 2024 – Mikhail Hlushchanka (University of Amsterdam)

Time: 5:00-6:00pm (Beijing Time)    Location: Zoom  (ID: 851 2054 7588, Password: 126435)

Title: Invariant graphs for rational maps: construction, application, and open problems

Abstract:  Invariant graphs provide nice combinatorial models for dynamical systems under consideration. As such, they appear naturally in various aspects of complex dynamics and have multiple applications. For instance, "Hubbard trees" were used to classify all postcritically-finite polynomials in the 80’s. I will present the main approaches for the construction of invariant graphs, overview some of their applications, and discuss several open combinatorial problems in this area.  


February 2 2024 – Richard Sharp (University of Warwick)

Time: 4:00-5:00pm (Beijing Time)    Location: Zoom  (ID: 863 6531 9969, Password: 573084)

Title: Random walks on groups, amenability and ratio limit theorems

Abstract: A famous result of Kesten from 1959 relates symmetric random walks on countable groups to amenability. Precisely, provided the support of the walk generates the group, the probability of return to the identity in 2n steps decays exponentially fast if and only if the group is not amenable. This led to many analogous “amenability dichotomies”, for example for the spectrum of the Laplacian of manifolds and critical exponents of discrete groups of isometries. I will present a version of the dichotomy for non-symmetric walks. I will also discuss a new ratio limit theorem for amenable groups. This is joint work with Rhiannon Dougall (Durham University). 


March 8 2024 – Christian Wolf (City University of New York)

Time: 9:00-10:00pm (Beijing Time)    Location: Zoom  (ID: 842 9270 8644, Password: 106244)

Title: Measures of maximal entropy for symbolic systems beyond SFT

Abstract: In this talk we present results about the uniqueness of measures of maximal entropy for symbolic systems beyond subshifts of finite type (SFT). In particular, we consider coded shifts which include several well-known examples of shift spaces. A coded shift space is defined as the closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We also discuss flexibility results for the entropy on the sequential and residual set. Finally, we present a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov. The results presented in this talk are joint work with Tamara Kucherenko and Martin Schmoll.



April 12 2024 – Cristobal Rojas (Pontificia Universidad Catolica de Chile)

Time: 9:00-10:00pm (Beijing Time)    Location: Zoom  (ID: 889 2406 3457, Password: 792934)

Title: Computational tractability in the description of typical asymptotic behaviors in dynamical systems

Abstract: A major goal in dynamical system theory is the classification of systems according to the asymptotic behavior exhibited by a typical trajectory. The quadratic family is an example of a class of systems where the status of this classification in both topological (attractors/repellers) and statistical (physical measures) terms is essentially complete -- there are only a few possible different asymptotic "regimes".  A natural question in this context is the possibility of using computers, for instance, to decide which regime a system belongs to, or to accurately describe the invariant objects relevant to a given regime. In this talk we will review the state of the art regarding this question in the case of the real quadratic family. 











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