January 13: Tushar Das (University of Wisconsin - La Crosse)
Title: Avatars of Poincaré-Bowen rigidity in conformal dynamics
Abstract: In 2009 Misha Kapovich, following up a on long line of research (Bowen, Sullivan, Tukia, Bishop-Jones, et al) proved that for a geometrically finite Kleinian group in Isom(Hn), if the topological dimension of the limit set equals its Hausdorff dimension, then the limit set is a geometric sphere. Earlier in 2003 Mayer and Urbanski proved a similar dichotomy for finite conformal iterated function systems (IFSs) using the concept of rectifiability of measures. We present our investigations with David Simmons and Mariusz Urbanski extending Kapovich's theorem to certain geometrically infinite groups and also our related rigidity results on the rational maps side of Sullivan's dictionary and for conformal IFSs. Time permitting we may talk about our generalization to analogues of Kleinian group actions on infinite-dimensional hyperbolic space, where the plot thickens...
Tuesday, January 21: Curt McMullen (Harvard)
Title: Entropy: graphs, surfaces, knots and number fields
Wednesday, January 22: Curt McMullen (Harvard)
Title: Entropy: graphs, surfaces, knots and number fields
Thursday, January 23: Curt McMullen (Harvard)
Title: Entropy: graphs, surfaces, knots and number fields
January 27: Sara Lapan (Northwestern)
Title: Attracting domains for holomorphic maps tangent to the identity
Abstract: One of the guiding questions behind the study of holomorphic dynamics is: given a germ of a holomorphic self-map of C^m that fixes a point (say the origin), can it be expressed in a simpler form? If so, then the dynamical behavior of the map can be more easily understood. In general, we want to know how points near the origin behave under iteration by the map. More specifically, we want to know when there exists a domain whose points are attracted to the origin under iteration by the map and, if such a domain exists, when its points converge tangentially to a given direction v. In dimension one, the Leau-Fatou Flower Theorem tells us of the existence of such domains. In higher dimensions, Hakim showed that given some assumptions on the map and the direction v, a domain of attraction whose points converge to the origin along v does exist. In this talk, we will begin by discussing the existence of attracting domains in dimension one. Then we will discuss what happens in higher dimensions and introduce Hakim's theorem. We will focus the discussion on a collection of maps in C^2 that does not satisfy some of the assumptions in Hakim's theorem; in particular, maps with a unique (and non-degenerate) characteristic direction. It turns out that such maps will also have a domain of attraction. We will discuss this result as well as some of the techniques used in its proof.
February 3: Ralf Spatzier (Michigan)
Title: Smooth Rigidity of Quotients of Projective Actions
Abstract: In joint work with Alex Gorodnik, we consider smooth actions of lattices in higher rank semi simple groups which are topological quotients of projective actions. Under a mild condition, we show that the conjugacy is automatically smooth. This is differentiable version of similar results in the measurable and continuous case respectively by Margulis and Dani.
February 10: Zhenghe Zhang (Northwestern University)
Title: Entropy variation for smooth systems
Abstract: In 1983 Herman proved that the Lyapunov exponents of Schrödinger cocycles are positive for all energies, provided the potentials are trigonometric polynomials. In 1991 Sorets and Spencer generalized this result for one-frequency non-constant real analytic potentials and for large coupling constants. The result in multi-frequency real analytic case was established by Bourgain, Goldstein and Schlag in early 2000s. All these results mentioned above follow essentially from the subharmonicity of Lyapunov exponents as functions of phase. In this talk I will present a joint work with Yiqian Wang in which, based on a different mechanism, we show the corresponding result for a certain class of C^2 potentials and for one dimensional Diophantine frequency.
February 17: Pat Hooper (CUNY)
Title: Detecting vanishing periodic billiard paths
Abstract: We consider billiards in polygons with labeled edges. Periodic billiard paths in distinct polygons are combinatorially equivalent if they hit the same sequence of sides. Given a path in the space of polygons we ask: Which combinatorial types of periodic billiard paths exist up to but not including a particular point in the path. Answering such questions offers the opportunity to generalize a theorem of Rich Schwartz: There is a sequence of triangles converging to the 30-60-90 triangle so that the length of the shortest periodic billiard path tends to infinity. Our approach involves taking limits of affinely stretched infinite translation surfaces, and these arguments hinge on some recent work in which places a topology on the space of all pointed infinite translation surfaces. The limiting surfaces often have unexpected symmetries which we exploit in our arguments.
February 24: Giulio Tiozzo (ICERM)
Title: Galois conjugates of entropies of quadratic maps
Abstract: The problem of characterizing the algebraic numbers arising from dynamical systems has recently drawn considerable attention. One of the first contexts in which this question makes sense is in the family of real quadratic polynomials; in this case, W. Thurston considered the set of all Galois conjugates of entropies of real quadratic polynomials with finite critical orbit, and found out that this set displays an extremely rich fractal structure. We shall prove that such a set (or rather, its closure) is path-connected and locally connected. Pictures will be provided.
March 3: Diana Davis (Northwestern)
Title: Cutting sequences on Bouw-Möller surfaces
Abstract: We will investigate a dynamical system that comes from geodesic trajectories on flat surfaces. We will start with known results for the square torus and the double pentagon surface, and then discuss new results for Bouw-Möller surfaces, made from many polygons.