# Complex Analysis and Dynamics Seminar

## Spring 2013 Schedule

#### Feb 8: Joseph Maher (College of Staten Island and Graduate Center of CUNY) Statistics for Teichmuller Geodesics

We describe two ways of picking a geodesic "at random" in a space, one coming from the standard Lebesgue measure on the visual sphere, and the other coming from random walks. The spaces we are interested in are hyperbolic space and Teichmuller space, together with some discrete group action on the space. We investigate the growth rate of word length as one moves along the geodesic, and we show these growth rates are different depending on how one chooses the geodesic. This is joint work with Vaibhav Gadre and Giulio Tiozzo.

#### Feb 15: Araceli Bonifant (University of Rhode Island) Antipode Preserving Cubic Rational Maps and Herman Rings

In this talk, I will discuss a family of cubic rational maps which carry antipodal points of the Riemann sphere to antipodal points, with emphasis on the abundance of Herman rings.

#### Feb 22: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY) Compactifying Teichmuller Space; Horocycles and Wedges

There is a horoball compactification of Teichmuller space based on extremal length that corresponds to Mumford's compactification of moduli space. Elaborating this idea and Grotzsch's argument leads to the following topics:
1. The Reich-Strebel inequalities
2. The minimum Dirichlet principle for measured foliations
3. Horoballs and wedges
5. Julia's lemma
6. Cylindrical (Jenkins-Strebel) differentials
7. Dehn multitwists
8. Masur's theorem that Teichmuller's space is not CAT(0)
9. Analogies to the Denjoy-Wolf theorem on iterating holomorphic self maps of the disc
10. Measured foliations and Thurston's compactification

#### Mar 1: Alex Kheyfits (Bronx Community College of CUNY) Valiron-Titchmarsh Theorem for Subharmonic Functions in ${\mathbb R}^n$ with Masses on a Half-Line

Let $f(z)$ be an entire function of order $\rho,\; 0<\rho <1$, with negative zeros and let $n(r)$ be the counting function of its zeros. If the finite limit $(*) \lim_{r \to \infty} r^{-\rho}n(r)=\Delta$ exists, then it is straightforward to prove the existence of the limit $$\lim_{r \to \infty} r^{-\rho} \log f(r e^{i \theta}) = \frac{\pi \mathbf{\Delta}}{\sin \pi \rho}e^{i \rho \theta},\;\; \theta \in (-\pi, \pi).$$ The Valiron-Titchmarsh theorem gives the tauberian converse of this statement, namely, if the limit over the positive half-line $$\lim_{r \to \infty} r^{-\rho} \log f(r) = \frac{\pi \mathbf{\Delta}}{\sin \pi \rho}$$ exists, then the limit $(*)$ exists. In this talk, the theorem is extended to subharmonic functions in $\mathbb{R}^n,\; n\geq 3$, with the Riesz associated masses on a ray.

#### Mar 8: Clifford Earle (Cornell University) Teichmueller Space of a Quadratic Differential

The Teichmuller space of a Fuchsian group $G$ is defined by Bers as the set of quadratic differentials in universal Teichmuller space that are $G$-invariant. We reverse the process: Starting with a quadratic differential $\phi$ in universal Teichmuller space, we study the group of all $g$ in $\text{PSL}(2,{\mathbb R})$ such that $\phi$ is $g$-invariant. There is an unexpected (by me) connection with an old paper of Charles Kalme.

#### Mar 15: Jun Hu (Brooklyn College and Graduate Center of CUNY) Thurston's Earthquake Measure Characterization of the Asymptotic Teichmuller Space on the Hyperbolic Plane

Let $MLb(D)$ be the collection of Thurston bounded measured geodesic laminations on the hyperbolic plane $D$. We introduce an equivalence relation on $MLb(D)$ such that the earthquake measure map induces a bijection between the asymptotic Teichmuller space $AT(D)$ and the quotient space $AMLb(D)$ of $MLb(D)$ under the equivalent relation. Furthermore, we introduce a topology on $AMLb(D)$ under which the bijection is a homeomorphism between $AT(D)$ and $AMLb(D)$ with respect to the Teichmuller metric on $AT(D)$. Corresponding results are also developed for a bijection and then a homeomorphism between the tangent space $AZ(S^1)$ of $AT(D)$ at a base point and $AMLb(D)$ with respect to the asymptotic cross-ratio norm on $AZ(S^1)$ and the topology on $AMLb(D)$. This is joint work with Jinhua Fan.

#### Mar 22: Joel Zablow (Long Island University) Families of Relations in Mapping Class Groups and Beyond, Parametrized by Knots

For an oriented surface $F$ with mapping class group $MCG(F)$, it is known that if a mapping class $f$ fixes a simple closed curve $c$, then $f$ commutes with Dehn twists about $c$. For a given non-separating curve $c$ in $F$, we give a construction for creating families of such commutation relations, where the mapping class $f$ can involve arbitrarily long products of elements in $MCG(F)$. These families are parametrized by knots (here, mainly pretzel knots) admitting particular "colorings" by element of Dehn quandles. In these quandles, there is a notion of two elements having "intersection number" 0 or 1. We show the above construction generalizes to produce such relations in other groups associated to quandles with such intersection number 1 elements, e.g. $SL(2,{\mathbb Z})$, $SL(2,{\mathbb R})$, braid groups, symmetric groups, symplectic groups, and certain Artin groups.

#### Apr 5: Saeed Zakeri (Queens College and Graduate Center of CUNY) On Margulis Cusps of Hyperbolic $4$-Manifolds

We study the geometry of the Margulis region associated with an irrational screw-translation $g$ acting on ${\mathbb H}^4$. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which generalizes the notion of an invariant horoball for a translation in dimensions $\leq 3$. The boundary of the Margulis region is described by a function $B_\alpha : [0,\infty) \to {\mathbb R}$ which solely depends on the irrational rotation angle $\alpha \in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an asymptotically universal upper bound for $B_\alpha(r)$ as $r \to \infty$ for arbitrary $\alpha$, as well as lower bounds when $\alpha$ is Diophatine and optimal bounds when $\alpha$ is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic $4$-manifolds. Among other things, we prove that such cusps are bi-Lipschitz rigid. This is joint work with Viveka Erlandsson.

#### Apr 12: Rubi Rodriguez (Catholic University of Chile) Decomposition of Jacobian and Abelian Varieties with Group Actions

We will describe the decomposition of Abelian varieties with automorphism group $G$ as a product of $G$-simple subabelian varieties. We will also give geometric interpretations for the factors in the case of Jacobian varieties of Riemann surfaces with non-trivial automorphism group.

#### Apr 19: Eric Bedford (Indiana University) Dynamics of Automorphisms and Pseudo-Automorphisms of Complex Manifolds

We will survey the compact complex surfaces with automorphisms of positive entropy. Then we look at the higher dimensional situation and discuss the existence of pseudo-automorphisms with positive entropy.

#### Apr 26: Sergiy Merenkov (University of Illinois at Urbana-Champaign) Uniqueness Properties of Schottky Maps

If $D$ is a domain in the complex plane, a relative Schottky set in $D$ is the residual set obtained by removing from $D$ open geometric discs whose closures are pairwise disjoint. A Schottky map is a local homeomorphism between relative Schottky sets that is conformal (in an appropriate sense) and whose derivative is continuous. In this talk, I will discuss uniqueness properties of such maps. The main result states that under mild geometric assumptions a Schottky map from an open subset of a relative Schottky set $S$ into $S$ is the identity provided that it fixes a point and its derivative is $1$ at this point.

#### May 3: Kentaro Ito (Nagoya University) Convergence and Divergence of Kleinian Punctured Torus Groups

In this talk, we give a necessary and sufficient condition for a sequence of Kleinian punctured torus groups to converge. This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method due to Anderson and Canary (Invent. Math., 1996). Thus we obtain a complete description of the set of points at which the space of Kleinian punctured torus groups self-bumps.

#### May 10: Robert Suzzi Valli (Graduate Center of CUNY) Non-simple Closed Geodesics on Orbifold Surfaces

Let $\mathbb{H}$ be the hyperbolic plane and suppose $\Gamma$ is a Fuchsian group. In general the quotient $X=\mathbb{H}/\Gamma$ is an orbifold surface. In the case that $\Gamma$ contains elliptic elements, their fixed points in $\mathbb{H}$ project to cone points on $X$. The presence of cone points requires a finer notion of paths and homotopy on $X$ with the goal of defining the orbifold fundamental group, $\pi_1(X,b)$, and obtaining an isomorphism between $\pi_1(X,b)$ and $\Gamma$ (analagous to the surface case). With this at our disposal we will study once self-intersecting closed geodesics on $X$ which are disjoint from the cone points, called figure eight geodesics. We identify the shortest figure eight geodesic on a triangle group orbifold ($\mathbb{H}/\Gamma$, where $\Gamma$ is a triangle group), and then use this to find the shortest figure eight geodesic among all orbifold surfaces without cone points of order two.

#### May 17: Sandra Hayes (City College of CUNY) The Real Dynamics of Bieberbach's Example

Bieberbach constructed in 1933 domains in ${\mathbb C}^2$ which were biholomorphic to ${\mathbb C}^2$ but omitted an open set. The existence of such domains was unexpected because the analogous statement for the one-dimensional complex plane is false. The special domains Bieberbach considered are given as basins of attraction of a cubic Henon map. This classical method of construction is one of the first applications of dynamical systems to complex analysis. In this talk, the boundaries of the real sections of Bieberbach's domains will be shown to be smooth. The real Julia sets of Bieberbach's map will be calculated explicitly and illustrated with computer generated graphics.
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