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



Spring of 2014 Schedule

Fall of 2013 Schedule

Spring of 2013 Schedule