**Dynamics
and Complex Analysis Research Seminar**

Organized by
Yunping Jiang (yunping.jiang@qc.cuny.edu)
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 system, I. Quasicircles and Dominos

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 1, No Event or TBA

April 8, Spring Recess

April
15, Fred Gardiner, More
on Teichmueller's and Caratheodory's metrics.

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, Yunping Jiang, Simultaneous uniformization for uniformly quasisymmetric
circle dynamical system II, Transformation on measures