# Complex Analysis and Dynamics Seminar

## Fall 2019 Schedule

#### Sep 13: Sergiy Merenkov (City College and Graduate Center of CUNY) On a Family of (Anti-)Rational Maps with Gasket Julia Sets

In this talk I will describe a construction of a family of (anti-)rational maps whose Julia sets are gasket-like, e.g., the Apollonian gasket. The family is parametrized by topological triangulations of the sphere and the construction relies on the Koebe-Andreev-Thurston theorem on circle packings. Also, for a subclass of certain reduced triangulations, I will discuss the quasisymmetry groups of such gasket-like Julia sets. The talk is based on joint work with Russell Lodge, Misha Lyubich, and Sabyasachi Mukherjee.

#### Sep 27: Ara Basmajian (Hunter College and Graduate Center of CUNY) Extremal Length Calculations and Applications to the Geodesic Flow

A collar neighborhood of a simple closed geodesic in a hyperbolic surface is an open neighborhood of the geodesic which is topologically an annulus. It is well-known that a simple closed geodesic on a hyperbolic surface has a natural (or standard) collar. The outstanding feature of the natural collar is that its size depends on local data, namely its size depends only on the length of the geodesic. Using this collar one can make extremal length calculations of curve families that are transverse to the geodesic.
In this talk, after defining extremal length and discussing its properties, we define a new type of collar which we call a non-standard collar. Using the non-standard collar we are able to improve estimates on the extremal length of curve families that are transverse to the geodesic and give a number of applications to the geodesic flow on an infinite type hyperbolic surface. This is joint work with Hrant Hakobyan and Dragomir Saric.

#### Oct 4: Yunping Jiang (Queens College and Graduate Center of CUNY) Slices of the Parameter Space for Meromorphic Functions with Two Asymptotic Values

In this talk, I will give an introduction to our program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. I will give a description of the bifurcation diagram of the tangent family and the parameter space of meromorphic functions with precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work for the parameter space of quadratic polynomials and the parameter space of degree two rational functions. In particular, for the parameter space of degree two rational maps that have analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity and infinite degree for the covering, our family of meromorphic functions exhibits a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted. This talk is based on a recent joint work with Tao Chen and Linda Keen.

#### Oct 11: Enrique Pujals (Graduate Center of CUNY) Weak Forms of Hyperbolicity

We will try to present an overview on some weak form of hyperbolicity, mainly partial hyperbolicity and dominated splitting and their dynamical consequences. We will focus on examples and some open problems.

#### Oct 18: Samuel Lelièvre (Université Paris-Sud) Periodic Billiard Trajectories in the Regular Pentagon

We present an enumeration of periodic billiard trajectories in the regular pentagon. This enumeration is based on an analogue of the Farey or Stern-Brocot tree, adapted to the $(2, 5, \infty)$ triangle group (also known as Hecke(5) group). The golden ratio $\phi$ is key here, and a gcd algorithm for "golden integers" (elements in the ring ${\mathbb Z}[\phi]$, the ring of integers in the number field ${\mathbb Q}(\phi)$ or ${\mathbb Q}(\sqrt{5})$) appears. Joint work with Diana Davis, with key use of SageMath and CoCalc.

#### Oct 25: Peter Nandori (Yeshiva University) Infinite Measure Mixing for Some Mechanical Systems

Let us consider mechanical systems with some hyperbolicity that preserve a natural physical infinite measure. Some examples are periodic Lorentz gas (with or without an external field), ping-pong models and geodesic flows on surfaces of negative curvature. For such systems, we study two notions of infinite measure mixing: the Hopf-Krickeberg mixing and global mixing. This talk is based on joint work with Dmitry Dolgopyat and in parts with Francoise Pene.

#### Nov 1: Chenxi Wu (Rutgers University) Characterization of Thurston's Teapot

Thurston's "master teapot" is a plot of the entropy of some postcritically finite interval maps together with their Galois conjugate. We gave a description of the closure of this plot in terms of iterated function systems, and relate the shape of the teapot to the limit set of some IFS via a Julia-Mandebrot-like relationship. This is a collaboration with Kathryn Lindsey, Diana Davis and Harrison Bray.

#### Nov 15: Alejandro Cabrera (Universidade Federal do Rio de Janeiro) On the Stack of Orbits of a Dynamical System

In this talk, we will discuss the space of orbits of a dynamical system seen as a differentiable stack. After reviewing the relevant notions, we will present our main result which states that the orbit stack, together with an associated characteristic class, fully characterizes the underlying dynamics up to conjugation. We will also mention some applications, including a geometric analogue of a celebrated C*-algebraic result by M. Rieffel on irrational rotations, open problems and further directions. The talk will be based on a collaboration with M. del Hoyo and E. Pujals (arXiv:1804.00220 [math.DS]).

#### Nov 22: Rafael Potrie (CMAT, Uruguay) Partial Hyperbolicity and Pseudo-Anosov Dynamics

We study some consequences of large scale or coarse dynamics to the topology and ergodic properties of partially hyperbolic diffeomorphisms in dimension 3. For example, we show that every conservative partially hyperbolic diffeomorphism in a hyperbolic 3-manifold is a $K$-system (and therefore ergodic). This is based on joint work with T. Barthelme, S. Fenley and S. Frankel.

#### Dec 6: Yusheng Luo (Stony Brook University) $\mathbb R$-Trees in Complex Dynamics

In this talk, we will give two constructions (one algebraic and one geometric) of dynamics on $\mathbb R$-trees associated to a sequence of rational maps acting on the Riemann sphere. The two constructions turn out to be equivalent and are analogous to Morgan-Shalen’s compactification of spaces of Kleinian groups. We will also talk about an application to the problem of classifying (unbounded) hyperbolic components in the space of all rational maps of a given degree.