Complex Analysis and Dynamics Seminar

Spring 2023 Schedule

February 10: Alex Kapiamba (University of Michigan)
Elephants all the way down: the near-parabolic geometry of the Mandelbrot set

Understanding the geometry of the Mandelbrot set, which records dynamical information about every quadratic polynomial, has been a central task in holomorphic dynamics over the past forty years. Near parabolic parameters, the structure of the Mandelbrot set is asymptotically self-similar and resembles a parade of elephants. Near parabolic parameters on these "elephants'', the Mandelbrot set is again self-similar and resembles another parade of elephants. This phenomenon repeats infinitely, and we see different parades of elephants at each scale. In this talk, we will explore the implications of controlling the geometry of these elephants. In particular, we will partially answer Milnor's conjecture on the optimality of the Yoccoz inequality, and see potential connections to the local connectivity of the Mandelbrot set.

February 17: Pat Hooper (City College and the Graduate Center)
Geodesic representatives on surfaces without metrics

A translation surface is a singular geometric structure on a surface modeled on the plane where transition maps are translations. Some recent research has focused on extending results known for translation surfaces to dilation surfaces, where we broaden allowable transition maps to include dilations of the plane. Such surfaces do not have natural metrics; however, one can ask: “Are natural analogs of geodesic representatives in this context?” Relatedly, translation surfaces which are not closed (e.g., infinite genus surfaces) may or may not have geodesic representatives in every homotopy class. We will describe conditions on surfaces that guarantee that canonical representatives of homotopy classes of curves exist. In doing so, we realize that even less structure is needed: we describe a class of geometric structures on surfaces that are not modeled on the plane at all, but still have canonical curve representatives. This is joint work with Ferrán Valdez and Barak Weiss.

February 24: Ara Basmajian (Hunter College and the Graduate Center)
Homogeneous Riemann surfaces

We are interested in spaces that look the same from any point of the space (that is, ``homogeneous spaces"). Of course the notion of looking the same is dependent on the context. For example, the simply connected Riemann surfaces, namely the Riemann sphere, complex plane, and unit disc are conformally homogeneous Riemann surfaces. In fact, along with the punctured plane and the torus these are the only ones. On the other hand, given any surface it is not difficult to cook up a diffeomorphism between any two points of the surface. Hence one needs a notion that is not as strong as conformality and not as weak as differentiability. The key observation is that while smooth maps can distort infinitesimal circles to ellipses with unbounded eccentricity (the ratio of the major to the minor axis can be arbitrarily large), conformal maps do not distort infinitesimal circles at all. This leads to the notion of a homeomorphism being $K$-quasiconformal (has eccentricity bounded by $K$). Conformal homeomorphisms are 1-quasiconformal.

A Riemann Surface X is said to be K-quasiconformally homogeneous if for any two points x and y on it, there exists a K-quasiconformal self-mapping taking x to y. If such a K exists we say that X is a QCH Riemann surface. After introducing the basics, the focus of this talk will be on Riemann surface structures that are QCH, and their connections to the topology and hyperbolic geometry of the surface.

March 3: Richard Schwartz (Brown University)
Continued Fractions and the 4-Color Theorem

In this talk I will describe some experiments I did which look at explicit solutions to the famous four-color theorem on triangulations having non-negative combinatorial curvature. I will also spend some time explaining Thurston's paper, "Shapes of Polyhedra" concerning these kinds of triangulations. One thing I found, in a special case, is a neat connection to continued fractions. This connection lets you define a kind of isoperimetric ratio for quadratic irrationals.

March 10: Enrique Pujals (The Graduate Center of CUNY)
The $C^r$-stability conjecture for surface diffeomorphisms and renormalization

We will discuss the $C^r$-stability conjecture in the context of dissipative smooth diffeomorphisms of the disk and how renormalization may play a role to tackle that conjecture.

March 17: Saeed Zakeri (Queens College and the Graduate Center)
Hausdorff limits of external rays: the topological picture

We characterize Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials with connected Julia sets. This is a basic question in the context of geometric limits of conformal dynamical systems, but it also arises naturally in the study of limbs of the connectedness loci of polynomial maps (the higher degree analogs of the Mandelbrot set). Further insight is provided by investigating the local behavior of external rays as they approach a near-parabolic point. Joint work with C. L. Petersen.

March 24: Willie Rush Lim (Stony Brook University)
From Herman Rings to Herman Curves

A maximal invariant domain of a rational map is called a Herman ring if it is an annulus on which the map is conjugate to irrational rotation. By adapting the near-degenerate machinery, we show that Herman rings of bounded type rotation number of the simplest configuration satisfy a priori bounds, that is, the boundaries are quasicircles with dilatation independent of the moduli. As a result, we can study the limits of degenerating Herman rings and construct general examples of rational maps admitting a "Herman curve" (a rotation curve that is not contained in the closure of any rotation domain) of arbitrary degree and combinatorics.

March 31: Andrey Gogolev (The Ohio State University)
Rigidity of 3-dimensional Anosov flows

I will sketch a proof of the following theorem. If two 3-dimensinal volume preserving Anosov flows are continuously conjugate, then either the conjugacy is smooth or the flows are constant roof suspensions. Joint work with F. Rodriguez Hertz.

April 21: Anthony Quas (University of Victoria)
Stability of Banach space Lyapunov exponents

Motivated by an application to oceanography, I will discuss the stability of Lyapunov exponents of cocycles of operators on Banach spaces. Joint work with Cecilia Gonzalez-Tokman.

May 12: Edson de Faria (Stony Brook University)
Rigidity and Orbit-flexibility of Circle Maps

Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Swiatek that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, I will show in this talk that the set of geometric equivalence classes is uncountable when f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in (0, 1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. This talk is based on joint work with Pablo Guarino.

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