# Complex Analysis and Dynamics Seminar

## Spring 2010 Schedule

#### Jan. 29: Ara Basmajian (Hunter College and Graduate Center of CUNY) Universal Length Bounds for Non-Simple Closed Geodesics

We consider the relationship between the length of a closed geodesic on a hyperbolic surface and its self-intersection number. In the 1993 paper The stable neighborhood theorem and lengths of closed geodesics it was shown that a closed geodesic with intersection number k has length bounded from below by a constant Mk that goes to infinity with k. The Mk are universal constants in that they only depend on k and not on any particular hyperbolic structure. In this talk we show that the Mk grow like Log k. Our techniques are elementary and involve using a quantitative version of the Margulis lemma.

#### Feb. 5: Sarah Koch (Harvard University) Böttcher Coordinates inCm

A well-known theorem of Böttcher asserts that an analytic germ f:(C,0) → (C,0) which has a superattracting fixed point at 0, more precisely of the form f(z) = a zk + o(zk) for some non-zero a, is analytically conjugate to the power map za zk by an analytic germ (C,0) → (C,0) which is tangent to the identity at 0. In this talk, we generalize this result and give a Böttcher criterion for analytic maps in several complex variables.

#### Feb. 19: Trevor Clark (Toronto/Stony Brook University) Regular or Stochastic Dynamics in Higher Degree Families of Unimodal Maps

About twenty years ago, Palis conjectured that typical dynamical systems should possess good statistical properties. Through the work of Avila, Lyubich, de Melo and Moreira, this has been proven for unimodal maps with a non-degenerate critical point. I will show how to remove the condition on the critical point in analytic families of unimodal maps; along the way proving that the hybrid classes in the space of unimodal maps yield a lamination near all but countably many maps in the family. The essential difference in the higher degree case is the presence of non-renormalizable maps without "decay of geometry." The key to their study is the use of a generalized renormalization operator, which has much in common with the usual renormalization operator.

#### Mar. 5: Reza Chamanara (Brooklyn College of CUNY) Rigidity of Inversive Distance Disk Patterns

The study of configurations of disks with prescribed combinatorial and geometric patterns is an important part of combinatorial geometry. The most well-known example is the theory of circle packing with its numerous applications in discrete analytic function theory. The basis of the theory of circle packing is the so called Koebe's theorem which states that given any triangulation K of the sphere S2, there is a circle packing PK in S2 with the tangency pattern prescribed by K. Moreover, the disks in PK have mutually disjoint interiors and PK is unique up to application of a Mobius map. Theorems by Andreev and Rivin on the existence and uniqueness of convex compact or convex ideal hyperbolic polyhedra can be interpreted as theorems about configurations of disks with prescribed patterns of intersection. I will discuss the possible extensions of such results to patterns of pairwise disjoint disks on the sphere. In particular, I will show how such patterns define a weighted planar graph. Moreover, I will explain why the weighted graph determines the disk pattern up to application of a Mobius map.

#### Mar. 12: Robert Devaney (Boston University) Dynamics of Family of Maps zn + C/zn: Why the Case n = 2 is Crazy

In this talk we describe some of the interesting dynamics associated to the family of complex maps zn + C/zn, where C is a complex parameter and n > 1. It turns out that the case n = 2 is very different from the case n > 2. For example, when n > 2 there is a "McMullen domain" in the parameter space which is surrounded by infinitely many "Mandelpinski" necklaces, but there is no such structure when n = 2. Also, when n = 2, as C tends to 0, the Julia sets of these maps converge to the closed unit disk, but this never happens when n > 2. So the dynamical behavior of these maps near the parameter C = 0 is much more complicated when n = 2.

#### Mar. 19: Gabino Gonzalez-Diez (UAM, Spain) On Beauville Surfaces and the Genus of the Curves Arising in Their Construction

A Beauville surface is a 2-dimensional compact complex manifold of the form S = (C1 x C2) / G, where C1 and C2 are compact Riemann surfaces of hyperbolic type and G is a finite group acting freely on the product C1 x C2 in such a way that each of the factors is preserved by the action and, moreover, the quotient Ci / G is an orbifold of genus zero with three cone points. Beauville surfaces were introduced by Catanese following an initial example of Beauville in which C1 = C2 is the Fermat curve of genus 6 with the equation X05+X15+X25 = 0 and G is isomorphic to Z5 x Z5.
In this talk, I shall discuss questions such as which groups G and which genera g1, g2 of C1, C2 can arise in the construction of Beauville surfaces. In particular, I will show that g1 and g2 have to be at least 6 and that if g1 = g2 = 6 then S agrees with (one of the two) Beauville examples. The proof of this fact will rely on methods belonging to the theory of Fuchsian groups and Riemann surfaces.

#### Mar. 26: Xiao Jun Huang (Rutgers University) A Bishop Surface with a Vanishing Bishop Invariant

Bishop surfaces are generically embedded surfaces in the complex Euclidean space of dimension two. The surfaces have been playing important roles in many recent studies of complex analysis of several variables and symplectic geometry. In this talk, I will focus on the equivalence problem for such surfaces, as well as its connection with classical dynamics and hyperbolic geometry. I will also discuss a recent joint work with W. Yin on the solution of a problem of Moser and Moser-Webster.

#### Apr. 9: Jian Song (Rutgers University) The Kahler-Ricci Flow Through Singularities

We will show the Kahler-Ricci flow on projective varieties can be uniquely continued through divisorial contractions and flips if they exist. In particular, we prove the analytic minimal model program with Ricci flow for complex projective surfaces.

#### Apr. 16: Moira Chas (Stony Brook University) Structures Related to Intersection of Curves on Surfaces

Given a surface S, one can consider the set P of free homotopy classes of oriented closed curves. This is the set of equivalence classes of maps from the circle into the surface, where two such maps are equivalent if the corresponding directed curves can be deformed to one another. Given a free homotopy class C one can ask what the minimum number of times, counted with multiplicity, that a curve in that class intersects itself is. We study how this minimal self-intersection number may vary with the word length, which is the minimal number of letters required for a description of C in terms of the standard generators of the fundamental group and their inverses. Analogously, given two classes, one can study the minimum number of times representatives of these classes intersect and how this quantity is distributed in terms of word length.
In this talk, several problems (and some solutions) related to minimal intersection and self-intersection will be discussed. We will address such questions as: the possible maximal self-intersection for a given length, the number of conjugacy classes with given self-intersection and given length, distribution (in terms of the self-intersection) of the number of classes of a given length.
One part of this work is joint with Anthony Phillips and another part is joint with Steve Lalley.

#### Apr. 23: Melkana Brakalova (Fordham University) Conformality and Homogeneity at a Point

We will discuss results on conformality and homogeneity and some of their applications.

#### Apr. 30: Shou-Wu Zhang (Columbia University) Calabi Theorem and Algebraic Dynamics

In this talk, I will first state a theorem of Calabi for both complex and p-adic manifolds and show its applications to dynamical systems on complex varieties in terms of preperiodic points. Then I will discuss dynamical Manin-Mumford conjectures with both supporting examples and counterexamples. This is a report on joint works with Xinyi Yuan, and with Ghioca and Tucker respectively.

#### May 7: Clifford Earle (Cornell University) Holomorphic Families of Riemann Surfaces

Certain holomorphic families of hyperbolic Riemann surfaces are familiar from the theory of Teichmuller spaces. Their base spaces are the Teichmuller spaces, and their total spaces, known as the Teichmuller curves, are fibered over the Teichmuller spaces so that the fiber over each point t is the Riemann surface that t represents. These families are uniquely determined by their special role among all holomorphic families. We shall start with the basic definitions, explain the special role of the Teichmuller families, and use it to prove some rigidity and uniqueness properties of general families. One uniqueness property is that a holomorphic family over a simply connected parameter space is determined up to equivalence by its fibers. We shall also describe families over quotients of Teichmuller spaces and explain their role in constructing "degenerating families" over some quotients of augmented Teichmuller spaces. Finally, we shall describe two degenerating holomorphic families of 4-punctured spheres over the punctured unit disk that have isomorphic fibers over each point but are not equivalent.
This is joint work with Al Marden and is part of our continuing study of analytic properties of augmented Teichmuller spaces and their quotients.

#### May 14: Sudeb Mitra (Queens College of CUNY) Quasiconformal Motions and Families of Mobius Groups

The concept of quasiconformal motions was first introduced in a paper by Sullivan and Thurston, where they proved an extension theorem for quasiconformal motions over an interval. In this talk, we will discuss some new properties of quasiconformal motions of a closed set E in the Riemann sphere, over connected Hausdorff spaces. As a spin-off, we strengthen the result of Sullivan and Thurston, and show that if a quasiconformal motion of E over an interval has a certain group-equivariance property, then the extended quasiconformal motion can be chosen to have the same group-equivariance property. The main goal of this talk is to discuss isomorphisms of continuous families of Mobius groups arising from a group-equivariant quasiconformal motion of E over a path-connected Hausdorff space. This is an analogue for quasiconformal motions, of Bers's theorem on holomorphic families of isomorphisms of Mobius groups. The techniques used, crucially exploit a certain "universal" property of the Teichmuller space of the closed set E.
This is joint work with Yunping Jiang and Hiroshige Shiga.

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