January 14: Alden Walker (Chicago)
Title: Surface subgroups of HNN extensions
Abstract: This is joint work with Danny Calegari. I'll describe a combinatorial condition on a surface map into an HNN extension of a free group which certifies that the surface map is pi_1-injective. Using this condition together with probabilistic and computational methods, we show that an HNN extension of a free group by a random endomorphism contains a surface group with probability one. As a special case, we answer a question of Sapir by finding a particular surface subgroup. One of the inspirations for this work is the question of Gromov asking if every one-ended hyperbolic group contains a surface subgroup.
January 23: Christian Bonatti (Bourgogne)
Title: Building transitive Anosov flows on 3-manifolds
Abstact: In joint work with F. Beguin and B. Yu, we
develop a method
for building Anosov flows on 3-manifolds by gluing filtrating
neighborhoods of hyperbolic sets. We present several applications,
including proofs of the following:
* There exist Anosov flows admitting infinitely many non-isotopic,
transverse tori.
* For any N>0, there exists a 3-manifold M admitting at least N
different (up to toplogical equivalence) Anosov flows.
We also discuss the structure of Anosov flows with transverse tori and
describe a canonical way to decompose such an Anosov flow by cutting
along transverse tori.
January 28: Steve Hurder (UIC)
Title: The dynamics of Kuperberg flows
Abstract: In 1993 the Seifert Conjecture was resolved by Krystyna Kuperberg, who constructed, for any 3-manifold, a smooth flow with no periodic orbits. Kuperberg introduced a completely new type of ``aperiodic plug'' for flows, that was used to trap their periodic orbits. The dynamical properties of the flows constructed via her method have remained only partly understood. The work described in this talk, which is joint with Ana Rechtman, explores their properties in greater depth. We introduce the notion of a "zippered lamination", and show that there exists an invariant zippered lamination for a generic Kuperberg flow. The study of the dynamics of zippered laminations leads to a precise description of the topology and dynamical properties of the minimal set for a generic Kuperberg flow, including the presence of positive entropy and chaotic behavior.
February 4: Carlos Matheus (Paris)
Title: A Galois-theory simplicity criterion for the Kontsevich-Zorich exponents of square-tiled surfaces
Abstract: The dynamics of the Kontsevich-Zorich cocycle over the
Teichmuller geodesic flow on the moduli spaces of Abelian/quadratic
differentials is an important example of renormalization dynamics for
interval exchange transformations, translation flows and billiards in
rational polygons. In particular:
a) after H. Masur and W. Veech, we know that recurrence properties of
Teichmuller flow imply nice ergodic properties for interval exchange
transformations and translation flows,
b) after M. Kontsevich, A. Zorich and G. Forni, we know that the
Kontsevich-Zorich exponents (i.e., Lyapunov exponents of the
Kontsevich-Zorich cocycle) drive the deviations of ergodic averages of
interval exchange transformations and translation flows,
c) more recently, V. Delecroix, P. Hubert and S. Lelievre used the
precise knowledge of the Kontsevich-Zorich exponents in certain
situations (coming from a profound result of A. Eskin, M. Kontsevich and
A. Zorich) to confirm a conjecture of the physicists J. Hardy and J.
Weber on the abnormal rate of diffusion of typical trajectories in
typical realization of the so-called Ehrenfest wind-tree model of Lorenz
gases.
In this talk, we will discuss the simplicity property of the
Kontsevich-Zorich exponents (a qualitative property saying that their
multiplicities are 1) for certain invariant measures naturally
associated to square-tiled surfaces (i.e., rational points in the moduli
space of Abelian differentials). In particular, we will show a
simplicity criterion based on some techniques of A. Avila and M. Viana
(who showed the simplicity of the Kontsevich-Zorich exponents with
respect to the so-called Masur-Veech measures) and the features of
Galois groups of the characteristic polynomials of the matrices
representing the Kontsevich-Zorich cocycle over the support of the
measure. As an application of this criterion (and Faltings' theorem), we
will show the simplicity of the Kontsevich-Zorich exponents of all but
(possibly) finitely many square-tiled surfaces of genus 3 in the minimal
stratum H(4) if a certain conjecture of V. Delecroix and S. Lelievre
holds. These results are a joint work with M. Moeller and J.-C. Yoccoz.
February 11: Chen Meiri (Chicago)
Title: Thin hyperbolic monodromy groups for the hypergeometric differential equation.
Abstract: We will gives examples of families of thin hyperbolic monodromy groups for the hypergeometric differential equation. More precisely, each group is an infinite index subgroup of the integral orthogonal group of a rational quadratic form of signature (n,1). The difficulty in constructing such examples comes from the fact that there in no general way to decide if a finite set of a f.g group generates a finite index subgroup. This is joint work with Elena Fuchs and Peter Sarnak.
February 18: Francois Ledrappier (Notre Dame)
Title: Entropy and rigidity for compact manifolds
Abstract: We introduce and discuss asymptotic growth rates defined on the universal cover of compact manifolds. We show some general inequalities. We discuss some equality cases when the manifold has no focal points.
February 25: Vincent Delecroix (Paris)
Title: Weak mixing for billiards in regular polygons
Abstract: A dynamical system is called weak mixing if it admits no rotation as factor. As the billiard in a rectangle is isomorphic to a translation flow on a flat torus, it admits two independent (continuous) rotation factors and is not weak mixing. In this talk, we investigate that property for billiard in regular polygons and prove that weak mixing is prevalent. Our main theorem applies more generally for so called "non-arithmetic Veech surfaces".
Wednesday, February 27 4-5pm: Dzmitry Dudko (Goettingen) Room 308
Title: Expanding Thurston maps and matings
Abstract: We show that if a Thurston map f has no Levy obstruction and f is not covered by a torus endomorphism, then f is isotopic to an expanding map. As an application, we show that the geometric mating of two hyperbolic polynomials is a (topological) dynamical system on a 2-sphere if and only if there is no periodic ray connection.
March 4: Yuri Lima (Weizmann)
Title: Ergodic properties of skew products in infinite measure
Abstract: Attached.
March 11: Laura DeMarco (UIC)
Title: Degenerations of complex dynamical systems
Abstract: Given a family of rational maps or polynomials, a basic problem is to understand what happens (dynamically) as the family degenerates. In this talk, I will describe a limiting object as a "non-archimedean dynamical system" in terms of a rational function acting on a Berkovich space. In geometric terms, this is a dynamical system on an R-tree.
March 18: Steven Frankel (Cambridge)
Title: Quasigeodesic flows and pseudo-Anosov dynamics
Abstract: A flow is called quasigeodesic if each flowline is uniformly efficient at measuring distances on the large scale. We'll discuss quasigeodesic flows on closed hyperbolic 3-manifolds "from infinity." In particular, we'll see that quasigeodesic flows have pseudo-Anosov dynamics at infinity. Along the way we'll show that quasigeodesic flows give rise to group-invariant sphere-filling curves, generalizing the Cannon-Thurston construction for suspension flows.