# Complex Analysis and Dynamics Seminar

## Fall 2012 Schedule

#### Sep 7: Jayadev Athreya (University of Illinois at Urban-Champaign / Yale) Gap Distributions for Saddle Connections

We study the distribution of gaps between directions for holonomy vectors of saddle connections on translation surfaces. We give a theorem for generic translation surfaces as well as explicit computations of limiting distributions in specific cases. Joint work with J. Chaika, and also J.Chaika and S. Lelievre.

#### Sep 21: Viveka Erlandson (Graduate Center of CUNY) The Margulis Region and Screw Parabolic Elements of Bounded Type

Given a discrete subgroup of the isometries of $n$-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. While in dimensions $2$ and $3$ this region is a horoball, it has in general a more complicated shape due to the existence of screw parabolic elements in higher dimensions. In fact, in a discrete group acting on hyperbolic $4$-space containing a screw parabolic element with irrational rotation, the corresponding Margulis region does not contain a horoball. In this talk we describe the asymptotic behavior of the boundary of the Margulis region when the irrational screw parabolic is of bounded type. As a corollary, we show that the region is quasi-isometric to a horoball.

#### Sep 28: Hiroshige Shiga (Tokyo Institute of Technology / Graduate Center of CUNY) Making Examples of Riemann surfaces in the Length-Spectrum Teichmuller Theory

In the length-spectrum Teichmuller theory, I'm interested in constructing Riemann surfaces which have some characteristic properties. In this talk, I shall give examples of Riemann surfaces which reflect special phenomena of the length-spectrum Teichmuller space.

#### Oct 5: Francisco Jimenez (Graduate Center of CUNY) Modified Length Spectrum Metric on the Teichmuller Space of a Riemann Surface with Boundary

The length spectrum function defines a metric on the reduced Teichmuller space $T^R(S_0)$ of a bordered Riemann surface $S_0$. However, it does not separate points in the Teichmuller space $T(S_0)$ of $S_0$ and hence fails to define a metric there. In this talk, we introduce a modified length spectrum function that does define a metric on $T(S_0)$. We then show that if two points in $T(S_0)$ are close in the Teichmuller metric, they are close in the modified length spectrum metric. We also show that the converse statement is not true, which implies that these two metrics on $T(S_0)$ are not topologically equivalent. This is joint work with Jun Hu.

#### Oct 12: Andrew Sanders (University of Maryland) Minimal Surfaces in Hyperbolic 3-Space and Hausdorff Dimension of Limit Sets

The quasi-Fuchsian space is the deformation space of complete, convex co-compact hyperbolic metrics on a closed surface times the real line. The Hausdorff dimension of the limit set of the corresponding group is a real analytic function on quasi-Fuchsian space whose fine scale structure is difficult to understand. We will introduce a deformation space constructed by Taubes whose general element consists of a minimal immersion of a disk into hyperbolic 3-space. We define a Hausdorff dimension function on this space and show how it can be used to construct some of the first explicit one-parameter families of quasi-Fuchsian groups where the Hausdorff dimension of the limit set is constant.

#### Oct 19: Christian Wolf (City College of CUNY) Geometry and Entropy of Rotation Sets

For a continuous map $f$ on a compact metric space, we study the geometry and entropy of the generalized rotation set Rot$(\Phi)$. Here $\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and Rot$(\Phi)$ is the set of all $\mu$-integrals of $\Phi$ and $\mu$ runs over all $f$-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ${\mathbb R}^m$. We study the question of whether every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set $K$ in ${\mathbb R}^m$ a potential $\Phi=\Phi(K)$ with Rot$(\Phi)=K$.
Next, we study the relation between Rot$(\Phi)$ and the set of all statistical limits Rot$_{Pt}(\Phi)$. We show that in general these sets differ but also provide criteria that guarantee Rot$(\Phi)=$ Rot$_{Pt}(\Phi)$. Finally, we study the entropy function $w\mapsto H(w)$ for $w \in$ Rot$(\Phi)$. We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems, $H(w)$ is determined by the growth rate of those hyperbolic periodic orbits whose $\Phi$-integrals are close to $w$. We also show that for systems with strong thermodynamic properties (subshifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function $w\mapsto H(w)$ is real-analytic in the interior of the rotation set. This is joint work with Tamara Kucherenko.

#### Nov 30: Jay Mireles James (Rutgers University) Parameterization of Mixed Stability" Invariant Manifolds Associated with Fixed Points of Symplectic Dynamical Systems

I will discuss the existence of certain invariant manifolds associated with fixed points of analytic symplectic diffeomorphisms. These manifolds are not defined in terms of forward or backward asymptotic convergence to the fixed point. Instead, they are defined by the property of being tangent to certain mixed-stable" linear invariant subspaces of the differential (i.e., a subspace spanned by some combination of stable and unstable eigenvectors). The method of proof is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the symplectic geometry and requires some Diophantine properties of the linearization. The theorem has an a-posteriori format (i.e., given an approximate solution with good condition number, there is an exact solution close by).

#### Dec 7: David Constantine (Wesleyan University) Group Actions and Compact Clifford-Klein Forms of Homogeneous Spaces

A compact Clifford-Klein form of the homogeneous space $J \backslash H$ is a compact manifold $J \backslash H / \Gamma$ constructed using a discrete subgroup $\Gamma$ of $H$. I will briefly survey the existence problem for compact forms, and will show how the existence of an action by a large (higher-rank) semisimple group proves that many spaces do not have compact forms. I will also make some remarks on a conjecture of Kobayashi on the scarcity of compact forms.
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