Complex Analysis and Dynamics Seminar

Spring 2011 Schedule

Feb 4: Hrant Hakobyan (Kansas State University)
Conformal Dimension and Moduli

Feb 11: No meeting

Feb 18: Zeno Huang (CUNY Staten Island)
Almost Fuchsian 3Manifolds and Teichmuller Geometry
Almost Fuchsian 3folds are a class of quasiFuchsian hyperbolic manifolds introduced by
Uhlenbeck and it has been discovered that this class has deep connections to several topics
involving the geometry and topology of hyperbolic 3manifolds, Teichmuller theory as well as
geometric PDEs. We discuss several results concerning incompressible minimal surfaces,
geometrical quantities of the 3manifolds and Teichmuller geometry.

Feb 25: John Loftin (Rutgers University at Newark)
Minimal Lagrangian Surfaces in CH^{2}
For Σ a closed Riemann surface equipped with a hyperbolic metric and a small cubic differential U,
we produce a minimal Lagrangian immersion of the universal cover of Σ into CH^{2} by solving the integrability condition
Δ u  4 U^{2} e^{2u}  4 e^{u} + 2 = 0
on Σ. The development of a natural frame provides a representation of π_{1}Σ into SU(2,1).
If U is small enough, then the minimal Lagrangian surface is embedded, and the induced representation is complexhyperbolic quasiFuchsian.
This theory is a close analog of the theory of minimal surfaces in quasiFuchsian hyperbolic 3manifolds, as originally studied by
Uhlenbeck and further explored by Huang, Lucia and others. In particular, Uhlenbeck solves the integrability condition
Δ u  4 V^{2} e^{u}  4 e^{u} + 2 = 0
for V a small quadratic differential on a closed Riemann surface with hyperbolic background metric.
This produces a minimal surface in hyperbolic 3space which is invariant under an induced representation
of π_{1}Σ into PSL(2,C). We compare our results to known results in the
quasiFuchsian case and make conjectures on what should be true for complexhyperbolic quasiFuchsian representations into SU(2,1).
This is joint work with Ian McIntosh.

Mar 4: Albert Fathi (École Normale Supérieure de Lyon)
Lyapunov Functions: Towards an AubryMather Theory of Homeomorphisms?
This is a joint work with Ettore Minguzzi and Pierre Pageault.
We will call Lyapunov function a function that is nonincreasing
along orbits. By looking at simple dynamical systems on the circle,
we will see that there are systems which are topologically conjugate
and have Lyapunov functions with various regularity. This will lead us
to define barriers analogous to the well known Peierls barrier or
to the Mañé potential in Lagrangian systems. That will produce by
analogy to Mather's theory of Lagrangian Systems an Aubry set
which is the generalized recurrence set introduced in the 60's by
Joe Auslander (via transfinite induction) and a Mañé set which is
essentially Conley's chain recurrent set.

Mar 11: Ethan Akin (The City College of CUNY)
Good Measures on Cantor Space
Certain measures on the Cantor Set have nice homogeneity properties, which
lead to nice dynamic constructions. I don't want to give too much away,
but all this is not as boring as it may sound at first.

Mar 18: Dragomir Saric (Queens College and Graduate Center of CUNY)
Uniform Weak* Topology and Eathquakes
A quasisymmetric map of the unit circle is induced by a unique earthquake of the hyperbolic plane
with bounded earthquake measure. The earthquake measure map from the universal Teichmuller space into
the space of measured laminations which assigns to each normalized quasisymmetric map its earthquake
measure is a bijection. We introduce uniform weak* topology on the space of measured laminations of
the hyperbolic plane and show that the earthquake measure map is a homeomorphism. As an application,
we show that the set of quasisymmetric maps induced by earthquakes with discrete measured laminations
is dense in the universal Teichmuller space. This is joint work with H. Miyachi.

Mar 25: No meeting

Apr 1: Martin Bridgeman (Boston College)
Volume Identities for Hyperbolic Manifolds with Boundary
We show that in any dimension n greater than 1, there is a real valued function
F_{n} such that the volume of any finite volume hyperbolic nmanifold M
with totally geodesic boundary is the sum of the values of F_{n} on the orthogeodesic
length spectrum of M. For n=2 the function F_{n} is the Rogers
Lfunction and the summation identities give dilogarithm
identities on the moduli space of surfaces. We will also discuss
volume identities for geometrically finite Kleinian groups.

Apr 8: Ali Gokturk (Brown University)
Comparison of Teichmuller and WeilPetersson Geodesics
Let S be a surface with genus g and n punctures and let x(S) = 3g
+ n denote the complexity of S. In this talk we prove that in
the Teichmuller space T(S), Teichmüller geodesics and WeilPetersson
geodesics with the same pair of endpoints are fellowtravelers with respect
to the WeilPetersson metric if and only if x(S)<6. More precisely, we show
that if x(S)<6 then there is a constant N such that for any X,Y in T(S), the
Teichmuller geodesic connecting X to Y lies in an Nneighborhood (in the
WeilPetersson metric) of the WeilPetersson geodesic connecting X to Y. On
the other hand, we show the opposite of the above statement for surfaces with
x(S) >5.

Apr 15: Roger Vogeler (Central Connecticut State University)
MultiplyTransitive Actions on Cusps
Suppose M is a cusped finitevolume hyperbolic threemanifold. Then Isom(M) induces a permutation
action on the cusp set. In extreme cases this action will be ktransitive. I will show some examples
for k=1 and k=3. As the main result, I will describe a construction which shows that for k=2
there is no upper bound on the number of cusps. The main tools in the construction are triangle groups,
finite fields, and pseudoAnosov surface automorphisms.

Apr 29: Richard Schwartz (Brown University)
Outer Billiards, Continued Fractions, and Polyhedron Exchange Maps
An outer billiard is a dynamical system defined on the plane,
relative to a convex shape. When the shape is a convex polygon whose
vertices have rational coordinates, there exists a dynamically invariant
tiling of the plane associated to the system. When the coordinates of
the vertices are irrational, it sometimes happens that there is a more
complicated kind of partition of the plane which, in some sense, is a
limit of tilings. I will explain how I use this point of view to solve
one of the main questions about outer billiards, the existence of
unbounded orbits. For the case of bilaterally symmetric quadrilaterals
(i.e. kites) the dimension of the union of unbounded orbits is closely
related to the continued fraction expansion of the parameter that
controls the kite.

May 6: Dan Thompson (Penn State University)
Uniqueness of Equilibrium States: BetaShifts, the Bowen Property and Beyond
This joint work with Vaughn Climenhaga (Maryland) establishes uniqueness of equilibrium states for
1) a large class of shift spaces which includes every betashift;
2) a large class of potential functions which strictly includes those with the Bowen property.
As an application, our method yields new results in the theory of thermodynamic formalism for
piecewise monotonic interval maps. Our method allows us to handle a variety of systems without a
Markov structure, and it covers a class of potentials that are well behaved away from a 'small' set;
for example, an indifferent fixed point or a point of discontinuity. This work extends the techniques
which we developed in a recent preprint, available at http://arxiv.org/abs/1011.2780, which gave a positive
answer to the question "Is every subshift factor of a betashift intrinsically ergodic?"
This question was included in Mike Boyle's article Open problems in symbolic dynamics, and was the
original motivation for the development of these techniques.

May 13: Oleg Muzician (Graduate Center of CUNY)
Rational Maps with Half Symmetries, Julia Sets and Mandelbrot Sets
We first work out explicit formulas for nontrivial rational maps of minimal degree that are invariant under precompositions by the elements of finite Kleinian groups, and we also exhibit such rational maps with real coefficients. Then through computergenerated pictures we explore the Julia sets of such maps in some oneparameter families and the Mandelbrot sets in the parameter spaces. We observe that the classifications of the Julia sets of the maps in these families have a great deal of similarity with the ones of singularly perturbed rational maps studied by C. McMullen, and more recently and extensively by R. Devaney and his team. This is joint work with Jun Hu and Francisco Jimenez.

May 20: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY)
Extremal Annuli and Extremal Punctured Discs
We explore the relationship between extremal annuli, extremal punctured discs,
extremal univalent functions, natural metrics on Teichmuller space and extension of holomorphic motions.
