Complex Analysis and Dynamics Seminar

Fall 2012 Schedule

Aug 31: No meeting

Sep 7: Jayadev Athreya (University of Illinois at UrbanChampaign / 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
quasiisometric to a horoball.

Sep 28: Hiroshige Shiga (Tokyo Institute of Technology / Graduate Center of CUNY)
Making Examples of Riemann surfaces in the LengthSpectrum Teichmuller Theory
In the lengthspectrum 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 lengthspectrum 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 3Space and Hausdorff Dimension
of Limit Sets
The quasiFuchsian space is the deformation space of
complete, convex cocompact 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 quasiFuchsian
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
3space. We define a Hausdorff dimension function on this space and
show how it can be used to construct some of the first explicit
oneparameter families of quasiFuchsian 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 nonuniformly 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 realanalytic in the
interior of the rotation set. This is joint work with Tamara
Kucherenko.

Oct 26: Moon Duchin (Tufts University)
TBA

Nov 2: Seminar canceled

Nov 9: Dragomir Saric (Queens College and Graduate Center of CUNY)
TBA

Nov 16: Linda Keen (Lehman College and Graduate Center of CUNY)
TBA

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 ``mixedstable" 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 aposteriori 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 CliffordKlein Forms of Homogeneous Spaces
A compact CliffordKlein 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 (higherrank) 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.
