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 3-folds are a class of quasi-Fuchsian 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 3-manifolds, Teichmuller theory as well as
geometric PDEs. We discuss several results concerning incompressible minimal surfaces,
geometrical quantities of the 3-manifolds and Teichmuller geometry.
Almost Fuchsian 3-Manifolds and Teichmuller Geometry
Feb 25: John Loftin (Rutgers University at Newark)
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 CH2 by solving the integrability condition
Minimal Lagrangian Surfaces in CH2
Δ u - 4 |U|2 e-2u - 4 eu + 2 = 0
on Σ. The development of a natural frame provides a representation of π1Σ into SU(2,1).
If U is small enough, then the minimal Lagrangian surface is embedded, and the induced representation is complex-hyperbolic quasi-Fuchsian.
This theory is a close analog of the theory of minimal surfaces in quasi-Fuchsian hyperbolic 3-manifolds, 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 eu + 2 = 0
for V a small quadratic differential on a closed Riemann surface with hyperbolic background metric.
This produces a minimal surface in hyperbolic 3-space which is invariant under an induced representation
of π1Σ into PSL(2,C). We compare our results to known results in the
quasi-Fuchsian case and make conjectures on what should be true for complex-hyperbolic quasi-Fuchsian representations into SU(2,1).
This is joint work with Ian McIntosh.
Mar 4: Albert Fathi (╔cole Normale SupÚrieure de Lyon)
This is a joint work with Ettore Minguzzi and Pierre Pageault.
We will call Lyapunov function a function that is non-increasing
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 Ma˝Ú 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 Ma˝Ú set which is
essentially Conley's chain recurrent set.
Lyapunov Functions: Towards an Aubry-Mather Theory of Homeomorphisms?
Mar 11: Ethan Akin (The City College of CUNY)
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.
Good Measures on Cantor Space
Mar 18: Dragomir Saric (Queens College and Graduate Center of CUNY)
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.
Uniform Weak* Topology and Eathquakes
Mar 25: No meeting
Apr 1: Martin Bridgeman (Boston College)
We show that in any dimension n greater than 1, there is a real valued function
Fn such that the volume of any finite volume hyperbolic n-manifold M
with totally geodesic boundary is the sum of the values of Fn on the orthogeodesic
length spectrum of M. For n=2 the function Fn is the Rogers
L-function and the summation identities give dilogarithm
identities on the moduli space of surfaces. We will also discuss
volume identities for geometrically finite Kleinian groups.
Volume Identities for Hyperbolic Manifolds with Boundary
Apr 8: Ali Gokturk (Brown University)
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), TeichmŘller geodesics and Weil-Petersson
geodesics with the same pair of endpoints are fellow-travelers with respect
to the Weil-Petersson 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 N-neighborhood (in the
Weil-Petersson metric) of the Weil-Petersson geodesic connecting X to Y. On
the other hand, we show the opposite of the above statement for surfaces with
Comparison of Teichmuller and Weil-Petersson Geodesics
Apr 15: Roger Vogeler (Central Connecticut State University)
Suppose M is a cusped finite-volume hyperbolic three-manifold. Then Isom(M) induces a permutation
action on the cusp set. In extreme cases this action will be k-transitive. 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 pseudo-Anosov surface automorphisms.
Multiply-Transitive Actions on Cusps
Apr 29: Richard Schwartz (Brown University)
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.
Outer Billiards, Continued Fractions, and Polyhedron Exchange Maps
May 6: Dan Thompson (Penn State University)
This joint work with Vaughn Climenhaga (Maryland) establishes uniqueness of equilibrium states for
Uniqueness of Equilibrium States: Beta-Shifts, the Bowen Property and Beyond
1) a large class of shift spaces which includes every beta-shift;
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 beta-shift 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)
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 computer-generated pictures we explore the Julia sets of such maps in some one-parameter 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.
Rational Maps with Half Symmetries, Julia Sets and Mandelbrot Sets
May 20: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY)
We explore the relationship between extremal annuli, extremal punctured discs,
extremal univalent functions, natural metrics on Teichmuller space and extension of holomorphic motions.
Extremal Annuli and Extremal Punctured Discs