Complex Analysis and Dynamics Seminar at CUNY's GC

## Complex Analysis and Dynamics Seminar

### Fall 2021 (in person+virtual):

#### Nov 19: Enrique Pujals (Graduate Center of CUNY)Dynamics of Hénon Maps with Zero Entropy

I will focus on surface diffeomorphisms with zero entropy: Can the dynamics of these 'simple' systems be described? How does it bifurcate to positive entropy systems? These questions will be discussed for a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. It includes the dynamics of any Hénon diffeomorphism with Jacobian smaller than 1/4.

For that type of systems, first we will try to prove that any Hénon map with zero entropy and Jacobian smaller than 1/4 "is renormalized" (this is part of joint work with S. Crovisier and C. Tresser).

Later, we will discuss work in progress with Sylvain Crovisier, Misha Lyubich and Jonguk Jang about the bounded geometry of those renormalizable systems.

#### Dec 17: David Aulicino (Brooklyn College and Graduate Center of CUNY)Siegel-Veech Constants of Cyclic Covers of Marked Tori

We consider tori with $n>1$ marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order $d$. Given a translation surface, the number of cylinders with waist curve of length at most $L$ grows like $L^2$. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by $L^2$, the limit as $L$ goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points $n$. All necessary background will be given and a connection to combinatorics will be presented. In fact, the results extend when the tori are replaced by generic translation surfaces that are generic in their strata. This is joint work with Martin Schmoll.