Complex Analysis and Dynamics Seminar at CUNY's GC
## Complex Analysis and Dynamics Seminar

### Fall 2022:

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September 9: Hongming Nie (Stony Brook University)

*Polynomials in the boundary of shift locus*

The shift locus consists of polynomials with fixed degree whose all critical points escape to infinity. It is interesting to give a “nice” description of polynomials in the boundary of shift locus. In complex dynamics, this seems extremely hard. In this talk, I will talk the counterpart in nonarchimedean seting, and under a tameness assumption, state a sufficient and necessary condition for a nonarchimedean polynomial in the boundary of shift locus. The main ingredient is a rigid result for polynomials on the basin of infinity. This is a joint work with Jan Kiwi.

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September 16: Tao Chen (Laguardia Community College and the Graduate Center)

*Enumerate “hyperbolic” components *

Hyperbolic maps are the dynamically simplest among all maps in a certain family, and each connected component consisting of all hyperbolic maps are called hyperbolic components. Hyperbolic components of entire maps $\lambda e^z$ and meromorphic maps $\lambda \tan z$ have been well investigated before. In this talk, we will discuss the dynamics and parameter space of the family of meromorphic functions with only two asymptotic values, one of which is a pole, in particular, enumerate all "hyperbolic" components in this family by coding. This is a joint work with Linda Keen.

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September 23: Yusheng Luo (Stony Brook University)

*Hyperbolic rational maps with finitely connected Fatou set*

A hyperbolic component of rational maps with connected Julia set is homeomorphic to a ball. It has a unique center, corresponding to the unique post-critically finite map in the hyperbolic component. This post-critically finite map provides a model to understand the dynamics on the Julia set. When the Julia set is disconnected, there is no such post-critically finite rational maps. In this talk, we will restrict ourselves to the case where every Fatou component is finitely connected. In this case, I will explain how a post-critically finite model, which is some post-critically finite map on a tree of spheres, exists. This post-critically finite model can be interpreted as the `center at infinity’ for a hyperbolic component with finitely connected Fatou set. I will also give an admissible conditions for such model, and discuss some applications in rescaling limits.

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September 30: Anca Rădulescu (SUNY New Paltz)

*Complex dynamics in networks, templates and mutated systems*

We explore three directions extending the traditional theory of complex quadratic iterations to: (1) **complex quadratic networks**; (2) **template iterations** and (3) **mutated systems**. In all three cases, we define Julia and Mandelbrot sets, and describe how their properties are different than those of the traditional sets for single map iterations.

In the case of networks, the dynamics of each node are driven by the same complex quadratic map $f(z) = z^2+c$, with coupling specified by the adjacency matrix $A_{ij}$ and weights $g_{ij}$, so that the system takes the form of an iteration in $\mathbb{C}^n$. We discuss how the topology of the network Mandelbrot changes under perturbation of the network architecture, and how this can be used to create and analyze a fractal profile for a person's brain network.

In the case of templates, we study non-autonomous iterations of a pair of complex quadratic functions, $f_{c_0}$ and $f_{c_1}$, applied according to a general binary symbolic sequence ${\bf s}$. Here, the Mandelbrot set is a subset of the product space $\mathbb{C}^2 \times \{ 0,1 \}^\infty$. We discuss its slices in either subspace, present a Hausdorff convergence theorem and illustrate how the framework can be used in genetics, to understand the how cells differentiate and specialize.

In the case of mutations, we insert an ``erroneous'' function $f_{c_0}$ to replace the intact replication function $f_{c_1}$ within a disk around a mutation *epicenter*. We analyze the perturbations to the Julia set induced by mutations with different parameter $c_0$, different radius and different location. We discuss how this can be used to understand the role of mutations in cancer modeling research.

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October 7: Christian Wolf (City College and the Graduate Center)

*Symbolic dynamics beyond SFT: coded systems, entropy and equilibrium states*

In this talk we discuss recent results concerning the ergodic properties of coded symbolic systems. These systems include S-gap shifts, generalized gap shifts, and Beta-shifts. Specifically we present generalizations of results by V. Climenhaga and R. Pavlov about the topological entropy and equilibrium states for codes systems. We also re-visit the classical example of the Dyck shift to illustrate our results. The topics discussed in this talk are part of an on-going project together with T. Kucherenko, M. Schmoll and Y. Yang.

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October 14: Arkady Etkin (The Graduate Center)

*Julia sets of cubic rational maps with escaping critical points*

It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this talk, we consider any cubic rational map with all critical points converging to a single attracting fixed point under iteration. We show that the dichotomy of Julia sets extends to this type of cubic rational maps. We will also report the progress on the study of a similar problem when the attracting fixed point is replaced by a parabolic fixed point. Intermediate results developed to show the dichotomy or to study the case with a parabolic fixed point have applications in the course of classifying the Julia sets of other two-parameter families in the space of cubic rational maps. If time allows, I will briefly mention them. These are joint works with Prof. Jun Hu.

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October 21: Nick Salter (University of Notre Dame)

*What do we know about the topology of strata?*

Strata of abelian differentials/translation surfaces have long caught the interest of dynamicists and algebraic geometers, but a systematic study of their topology has been comparatively neglected. Following the work of Kontsevich-Zorich around the turn of the millennium which computed their path-components, our understanding of their topology (e.g. fundamental group, (co)homology) has advanced only sporadically. On the other hand, there are indications that this could be a very rich story with deep connections to geometric group theory. I will survey what we know about this topic and perhaps speculate about what we’d like to know.

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October 28: Howard Masur (University of Chicago)

*Counting in the mapping class group*

Let S be a closed surface of genus g at least 2.
Let $Mod(S)=Diff^+(S)/Diff_0(S)$ the mapping class group of S.
The Nielsen-Thurston classification of Mod(S) gives three types of elements: finite order, reducible and pseudo-Anosov. One can try to count mapping class group elements in different contexts. In this talk I will consider the action of Mod(S) on the Teichmuller space of S, with the Teichmuller metric.
The starting point is a theorem of Athreya,Bufetov, Eskin, Mirzakhani. They showed that if one fixes a pair of points x and y in Teichmuller space then the number of orbit points of y in the ball of radius R centered at x grows asymptotically like $c \exp((6g-6)R)$ as R goes to infinity for some c. Joseph Maher then showed that the proportion of mapping classes that are pseudo-Anosov goes to 1 in this lattice counting problem. In this talk I will discuss bounds for the number of finite order elements. This is joint with Spencer Dowdall.

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November 4: Matthew Romney (Stony Brook University)

*Polyhedral approximation and uniformization of metric surfaces*

The classical uniformization theorem states that every simply connected Riemannian 2-manifold is conformally equivalent to the disk, the plane or the 2-sphere. The uniformization theorem has since been extended to various metric space settings. In this talk, we give a solution to the uniformization problem for metric surfaces assuming only locally finite (Hausdorff) 2-measure. This generalizes previous results on the topic due to Bonk–Kleiner, Lytchak–Wenger and Rajala. Our proof is based on a new polyhedral approximation scheme for metric surfaces with locally finite 2-measure: any such surface is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry in a precise way. This result, in turn, is based on a general triangulation theorem for metric surfaces of independent interest. This talk covers joint work with Paul Creutz and Dimitrios Ntalampekos.

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November 11: Yunping Jiang (Queens College and The Graduate Center)

*Rigidity in one-dimensional dynamics and the Furstenberg conjecture*

I will first review an absolutely continuous rigidity result worked out by Mike Shub and Dennis Sullivan in the 1980s for smooth expanding circle endomorphisms. Secondly, I will review a symmetric rigidity result worked out by John Adamski, Yunchun Hu, Zhe Wang, and myself recently for continuous circle endomorphisms with bounded geometry. I will explain why these results are essential in studying Teichmuller spaces of one-dimensional dynamical systems. After that, I will show how to use the geometric method we have developed in studying the Furstenburg conjecture. Finally, I will outline the proof that a non-atomic p- and q-invariant probability measure for two relatively prime positive integers p and q on the circle with balanced geometry must be the Lebesgue measure.

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November 18: Seminar cancelled due to GC MathFest

GC MathFest

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December 2: Amol Aggarwal (Columbia University)

*Large Genus Asymptotics in Flat Surfaces*

In this talk we explain results on the large genus asymptotics for intersection numbers between $\psi$-classes on the moduli space of curves. By combining this result with a combinatorial analysis of formulas of Delecroix-Goujard-Zograf-Zorich, we further describe some features about how random flat surfaces of large genus look. The proof uses a comparison between the recursive relations (Virasoro constraints) that uniquely determine them with the jump probabilities of a certain asymmetric simple random walk.

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December 9: Jun Hu (Brooklyn College and The Graduate Center)

*Totally ramified rational functions - Speiser's graph, dessins d'enfants, existence, classification, and dynamics*

A rational function f (viewed as a map from the Riemann sphere to itself) is said to be totally ramified if for every critical value q, every preimage of q under f is a critical point. We will first give a result on what dynamical pattern of rational map is absent for such rational maps. Then we introduce Speiser's graphs or apply dessins d'enfants to classify all regular ramified rational maps. Thirdly, we use Speiser's graphs to show the existence of totally ramified but not regularly ramified rational maps for any degree bigger than 6. Exact formulas of such maps for degree 7 or 8 will be presented. If time is allowed, I will present some partial results on classifying totally ramified but not regularly ramified rational maps. Some of these are joint works with Xingqing Xiao or Weiwei Cui.