# Complex Analysis and Dynamics Seminar

## Fall 2014 Schedule

#### Sep 5: Max Fortier Bourque (Graduate Center of CUNY)Moving one Riemann Surface Inside Another

In this talk, I will sketch a proof of the following parametric homotopy principle: If two holomorphic embeddings between Riemann surfaces are homotopic, then they are isotopic through holomorphic embeddings. The proof uses a generalization of Teichmuller's theorem for quasiconformal embeddings, the geometry of quadratic differentials, and Gardiner's variational formula for extremal length.

#### Sep 12: Feng Luo (Rutgers University)A Dilogarithm Identity on the Moduli Space of Curves

We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface. This is joint work with Ser Peow Tan.

#### Sep 19: Yunhui Wu (Rice University) Translation Lengths of Parabolic Isometries of CAT(0) Spaces and its Application to the Geometry and Topology of Hadamard Manifolds

A CAT(0) space is a complete path-metric space with a certain inequality property. In this talk, we will discuss the translation length of parabolic isometries of CAT(0) spaces. As an application, we will connect several open problems and conjectures on Hadamard manifolds and moduli spaces of closed surfaces.

#### Oct 10: Athanase Papadopoulos (Université de Strasbourg)A Survey on Thurtson's Metric

I will survey some classical and some new results on Thurston's metric on Teichmuller space, and present some generalizations and open questions.

#### Oct 17: Nikita Selinger (Stony Brook University)Classification of Thurston Maps with Parabolic Orbifolds

In a joint work with M. Yampolsky, we give a classification of Thurston maps with parabolic orbifolds based on our previous results on characterization of canonical Thurston obstructions. The obtained results yield a partial solution to the problem of algorithmically checking combinatorial equivalence of two Thurston maps.

#### Oct 31: Yaar Solomon (Stony Brook University)Separated nets in ${\mathbb R}^d$ with a bounded displacement to ${\mathbb Z}^d$

Given a separated net $Y$ in ${\mathbb R}^d$, we study the question of whether $Y$ can be mapped injectively on ${\mathbb Z}^d$ with every point moving at most $M$ (for some constant $M$). This notion gives rise to an equivalence relation on the set of separated nets, called bounded displacement (BD) equivalence, which is more delicate than the biLipschitz equivalence relation. There is a correspondence between separated nets and tilings of ${\mathbb R}^d$, and nets that correspond to periodic tilings are clearly BD to ${\mathbb Z}^d$. We answer the above question for substitution tilings, which form a nice class of tilings that are often non-periodic. All the relevant background and definitions will be given in the talk.

#### Nov 7: Ara Basmajian (Hunter College and Graduate Center of CUNY)Geometric Structures on Infinite Type Surfaces

While the geometric theory of finite type surfaces is well developed, the study of hyperbolic geometric structures on infinite type surfaces (that is, infinitely generated fundamental group) is in its infancy. In this talk we first describe some of the known results about the geometry and topology of such surfaces and then consider surfaces $X$ constructed by gluing pairs of pants along their cuffs. When finitely many pants are used the geometric completion of $X$ is well understood (namely, attach a hyperbolic funnel to each boundary geodesic of $X$). In joint work with Dragomir Saric, we consider the case with infinitely many pairs of pants and give criteria for the resulting hyperbolic structure to be geometrically complete. These constructions lead to new phenomena for the deformation theory of such surfaces.

#### Nov 14: Rodrigo Trevino (Courant Institute, New York University) Flat Surfaces, Bratteli Diagrams and Adic Transformations

I will survey some recent developments in the theory of flat surfaces of finite area and translation flows, including both compact and (infinite genus) non-compact surfaces. In particular, I will concentrate on a new point of view based on a joint paper with K. Lindsey, where we develop a close connection of Bratteli diagrams and flat surfaces. I will also state a criterion for unique ergodicity in the spirit of Masur's criterion which holds in this very general setting and which implies Masur's criterion in moduli spaces of (compact) flat surfaces. No knowledge of anything will be assumed, and the talk will non-technical and full of examples.

#### Nov 21: Steven Frankel (Yale University)Quasigeodesic Flows and Dynamics at Infinity

A flow is called quasigeodesic if each flowline is uniformly efficient at measuring distances on the large scale. In a hyperbolic 3-manifold, quasigeodesic flows are exactly the ones that one can study "from infinity." We will illustrate how the 3-dimensional dynamics of a quasigeodesic flow is reflected in a simpler 1-dimensional discrete dynamical system at infinity: the universal circle. This is a topological circle, equipped with an action of the fundamental group, that lies at the edge of the orbit space of the flow. We will see that one can find closed orbits in a flow by looking at the action on the universal circle. We will also show that the universal circle provides a generalization of the well-known Cannon-Thurston theorem.
The universal circle for a quasigeodesic flow (due to Calegari) has a relative for a pseudo-Anosov flow (Calegari-Dunfield). The content of this talk completes a large part of Calegari's program to show that every quasigeodesic flow on a closed hyperbolic manifold can be deformed, keeping it quasigeodesic, to a pseudo-Anosov flow.

#### Dec 5: Nessim Sibony (University of Paris-Sud, Orsay)Nevanlinna's Theory and Holomorphic Dynamics

I will discuss some analogies between the second main theorem in Nevanlinna's theory and results in holomorphic dynamics. The two main examples will be equidistribution results for endomorphisms of ${\mathbb P}^k$ and equidistribution results for singular foliations by Riemann-surfaces in ${\mathbb P}^2$. This is joint work with T.C. Dinh.

#### Dec 12: The seminar will feature two talks: 1:50-2:50 Marian Gidea (Yeshiva University)Recent Progress in the Arnold Diffusion Problem

In 1964, V.I. Arnold conjectured that integrable Hamiltonian systems subjected to typical small perturbations always have some trajectories that travel a significant distance in the phase space. The problem has been later on classified into an a priori unstable case, when the unperturbed Hamiltonian depends on action-angle and hyperbolic variables, and an a priori stable case, when the unperturbed Hamiltonian depends on action-angle variables only. We will present a geometric/topological approach to this problem that relies on two ingredients: the existence of hyperbolic invariant manifolds, and Poincare recurrence. We will discuss applications to both the a priori unstable and the a priori stable case.

#### 2:55-3:55 Edson Vargas (University of Sao Paulo)Invariant Measures for Critical Coverings of the Circle

We study ergodic properties of a critical double covering of the circle, say $f$. This is a smooth double covering of the circle which has only one critical point, which we assume to be of finite order $> 1$. Examples of these maps are the Arnold maps $f_b$, induced by $$x \mapsto b + 2x + \frac{1}{\pi} \sin(2 \pi x).$$ We assume that $f$ is topologically conjugate to the double covering $L_2$, induced by $x \mapsto 2x$. Although Lebesgue measure on the circle is invariant by $L_2$, we prove that it may happen that f has no absolutely continuous invariant measure (acim). One cause of this kind of behavior is a strong recurrence of the critical point. We can study this from a combinatorial point of view and, as a consequence, we get that there is an uncountable set of parameters $b$ such that the critical covering $f_b$ has no acim. These type of results were obtained before in the context of unimodal maps by H. Bruin, J. Guckenheimer, F. Hofbauer, S. Johnson, G.Keller, T. Nowicki, S. van Strien and others. In the critical covering case there is no dynamical symmetry around the critical point and this cause some new combinatorial difficulties which need to be understood.
Back to the seminar page