Complex Analysis and Dynamics Seminar

Spring 2016 Schedule

Feb 5: Jun Hu (Brooklyn College and Graduate Center of CUNY) Rational Maps from Quotient Maps of Finite Kleinian Groups and Their Julia Sets

Quotient maps of finite Kleinian groups are rational maps with two or three critical values. They are also called rational maps with half symmetries. We explore some one-parameter families of such maps with a smilar classification of Julia sets in the one-parameter families of singularly perturbed monomials investigated by Devaney and his collaborators. This is joint work with Francisco G. Jimenez and Oleg Muzician.

Feb 19: Matthieu Arfeux (Stony Brook University) Trees of Spheres and Berkovich Spaces

In the world of complex dynamics on the Riemann sphere, we study the action of complex rational maps of a given degree modulo the natural action by conjugacy of the group of Mobius transformations. As the quotient space is not compact, one is interested in giving a good compactification of this space that underlines some interesting dynamical behavior. For this, two tools from algebraic geometry have been introduced: the Deligne-Mumford compactification of the moduli space of stable curves and Berkovich spaces. I will explain the vocabulary used in my thesis to deal with the first tool and will compare it to the second one. No prior knowledge of either of these topics will be assumed.

Feb 26: The seminar will feature two talks: 2:00-2:45 Scott Schmieding (University of Maryland) Dynamics of Isolated Invariant Sets

A compact invariant set for a homeomorphism is called isolated if it is the maximal invariant set of some neighborhood of itself. A system (i.e., a homeomorphism of a compact metric space) can be isolated in dimension $n$ if it is topologically conjugate to the restriction of a homeomorphism of an $n$-dimensional compact manifold to an isolated set. We show that the restrictions of homeomorphisms in dimension $n$ to compact invariant sets can be isolated in dimension $n + 1$. Consequently, every Cantor system can be isolated in dimension $3$ and higher. We demonstrate some classes of Cantor systems which can be isolated in dimension $2$, but show also that there are Cantor systems which cannot be isolated in dimension $2$. This is joint work with Mike Boyle.

3:00-3:45 Lucia Simonelli (University of Maryland) Absolutely Continuous Spectrum for Parabolic Flows/Maps

This talk will discuss the recent developments in the study of the spectral properties of parabolic flows and maps. More specifically, it will focus on the techniques used to determine the spectrum of the time-changes of the horocycle flow and the effort to describe conditions under which a general parabolic flow would be expected to have absolutely continuous spectrum.

Mar 18: Hideki Miyachi (Osaka University) Levi Form of Extremal Length Functions on Teichmuller Space

Extremal length is a basic conformal invariant in Complex Analysis. In this talk, I will give formulas for the Levi forms of the extremal length functions, and show that the extremal length functions are plurisubharmonic under the standard complex structure on the Teichmuller space. Our formula is described by Thurston’s symplectic form with a suitable train track. If time permits, I will also discuss an alternative approach to Gardiner’s differential formula and applications of our formula.

Apr 1: Viveka Erlandsson (University of Fribourg / Aalto University) Counting Curves on Hyperbolic Surfaces

In this talk I will discuss the growth of the number of closed geodesics of bounded length, as the length grows. More precisely, let $c$ be a closed curve on a hyperbolic surface $S=S(g,n)$ and let $N_c(L)$ denote the number of curves in the mapping class orbit of $c$ with length bounded by $L$. By the work of Mirzikhani it is known that in the case $c$ is simple this number is asymptotic to $L^{6g-6+2n}$. Here we consider the case when $c$ is an arbitrary closed curve, i.e. not necessarily simple. This is joint work with Juan Souto.

Apr 15: Carsten Petersen (Roskilde University / Stony Brook) On the (Filled-)Julia Sets of Orthogonal Polynomials

The fields of research holomorphic dynamics in one variable on the one hand and orthogonal polynomials on the other hand share the field of complex numbers with its immense toolbox of complex analysis, harmonic analysis and potential theory. However, there has not been a lot of exchange between the two fields. In a joint project with Christian Henriksen (Technical University Of Denmark), Henrik L. Pedersen (Copenhagen University) and Jacob S. Christiansen (Lund University), we work on expanding the connections between the two fields. As a first result, we have related the dynamical properties of the sequence of orthogonal polynomials for a probability measure with compact non-polar support in $\mathbb C$ to the potential-theoretic properties of the measure. In this talk I will describe and discuss these results.

May 6: Mikhail Hlushchanka (Jacobs University) Expanding Thurston Maps and Their Iterated Monodromy Groups

I will talk about connections between dynamical systems, geometry, and algebra which are caused by an algebraic object called the iterated monodromy group (IMG). IMG is a self-similar group associated to every branched covering $f$ of the 2-sphere (in particular to every rational map). First introduced by Nekrashevych, it encodes combinatorial information about the map $f$ and its dynamics in a computationally efficient way.
I will first introduce a special class of branched covering maps on the 2-sphere, called expanding Thurston maps (they include all post-critically finite rational maps with Julia set equal to the whole Riemann sphere), and discuss its dynamical properties. Then I will present a method for proving exponential growth of IMGs and illustrate it for a particular rational expanding Thurston map $f$. The proof is based on the "geometry" of the tilings associated to the map $f$, which are studied by Bonk and Meyer. Moreover, the method can be adjusted for the IMGs of some infinite families of rational maps with Julia set equal to the Riemann sphere or a Sierpinski carpet. This is a joint project with Daniel Meyer (University of Jyväskylä).