Dynamics and Complex Analysis Research Seminar
Organized by Yunping Jiang (email@example.com) with Linda Keen and Fred Gardiner
Department of Mathematics, CUNY Graduate Center
Fall of 2014 Schedule
Wednesday, 1:30pm-3:00pm (Math Thesis Room, Rm. 4214-03):
February 4, Organization and Sudeb Mitra, Holomorphic and quasiconformal motions
February 11, Sudeb Mitra, Holomorphic and quasiconformal motions continued
February 18, Tanya Firsova (Kansas State University),
Title: Geometric proof of the \lambda-lemma.
Abstract: We relate the technique of filling totally real manifolds
with boundaries in pseudo convex domains to
the holomorphic motion.That replaces the major
technical part in the proof of the \lambda-lemma by
a transparent geometric argument. This is a joint
work with Eric Bedford.
She will give another talk on Friday, Feb 20.
Title: Critical locus for complex Henon maps.
Abstract: Henon maps are maps of the form (x,y)\to (p(x)+ay, x).
Dynamically non-trivial automorphisms of C^2 are conjugate
to compositions of Henon maps. For one-dimensional holomorphic
maps the dynamics of the map is too a large extent determined by
the orbits of the critical points. Since Henon maps are biholomorphisms
of C^2, they do not have critical points in the classical sense. However,
there is a way to define an appropriate analog, called critical locus.
Critical locus is a Riemann surface. I'll give a topological description of
the critical locus for Henon maps that are small perturbations of
quadratic polynomials with disconnected Julia set. This proves the
conjecture of J. Hubbard. I'll also show that the critical loci are
quasiconformally equivalent. The last part of the talk is a joint work
with Misha Lyubich.
February 25, Yoshihiko Shinomiya (Waseda Univ.)
Title: Veech groups of Veech surfaces and periodic points
Abstract: Flat surfaces are surfaces with singular Euclidean structures.
The Veech group of a flat surface is the Fuchsian group
consisting of all matrices inducing affine mappings of the flat surface.
Our main interested is the case where Veech groups are co-finite.
Then, flat surfaces are called Veech surfaces.
In general, it is difficult to determine Veech groups of Veech surfaces.
We will give relations between the signatures of Veech groups as
Fuchsian groups and geometrical values of Veech surfaces.
The geometrical values are given by widths, heights and moduli of cylinder
decompositions of Veech surfaces. As an application of these relations,
we estimate the numbers of periodic points of non-arithmetic Veech surfaces.
A Veech surface is arithmetic if the Veech group is commensurable to PSL(2, Z)
and is non-arithmetic if the Veech group is not commensurable to PSL(2, Z).
It is known that periodic points are dense in arithmetic Veech surfaces.
On the other hand, a non-arithmetic Veech surface has only finitely many periodic points.
Our estimation depends only on the topological types of Veech surfaces
and signatures of the Veech groups.
March 4, Yunping Jiang, Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems
March 11 Linda Keen, Dynamics on tangent family I
March 18, Marten Fels, On abstract Hubbard trees and an abstract Mandelbrot set I
March 25, Marten Fels, On abstract Hubbard trees and an abstract Mandelbrot set II
April 8, Spring Recess
April 15, Nishan Chatterjee, Asymptotical Beltrami coefficients for circle homeomorphisms
April 22, Linda Keen, Dynamics on tangent family II
April 29, Zhiqiang Li, Weak expansion properties of expanding Thurston maps
Anstract: Thurston maps are a class of branched covering maps on the 2-sphere that arose
in W.~Thurston's characterization of postcritically finite rational maps.
By imposing a natural expansion condition, M.~Bonk and D.~Meyer investigated
a subclass of Thurston maps known as expanding Thurston maps, which turned out
to enjoy nice topological, metric, and dynamical properties. In this talk we will investigate this notion
of expansion further by discussing some weak expansion properties of such maps.
More precisely, we will sketch a proof to show that an expanding Thurston map
is asymptotically $h$-expansive as defined by M.~Misiurewicz if and only if it has
no periodic critical points, and moreover, no expanding Thurston map is $h$-expansive
as defined by R.~Bowen. If time permits, we will discuss some applications to the distribution
of periodic points with respect to some natural invariant measures, and to a question of K.~Pilgrim.
May 6, Santanu Nandi, Dynamics on tangent family
May 13, John Adamski, On C^1 expanding circle endomorphisms