September 15: Elise Goujard (Rennes)
Title: Counting closed geodesics on flat surfaces
Abstract: Counting periodic trajectories in polygonal billiards is related in some cases to counting closed geodesics on corresponding flat surfaces. The asymptotic of the number of closed geodesics on a flat surface is given by a constant called Siegel-Veech constant. For a flat surface defined by a quadratic differential we explain how this constant is related to the volumes of moduli spaces of quadratic differentials, extending the work of Masur-Zorich and Athreya-Eskin-Zorich. We illustrate this correspondence with an example of small complexity, for which we compute the volume explicitly.
September 29: Kathryn Lindsey (U. Chicago)
Title: Families of dynamical systems: flat and complex
Abstract: In both the fields of flat surfaces and complex polynomial dynamics, the collection of all "finite type" dynamical systems admits a natural stratification - for flat surfaces the stratification is based on the number and type of cone points, while polynomials are stratified according to their degree. In both fields, one may ask "how does restricting to a single stratum limit the range of dynamical properties exhibited by the systems?" I will present several theorems that address aspects of this question in the contexts of flat surfaces and complex polynomial dynamics, and I will attempt to draw parallels between the two fields. Specific topics I will discuss include shapes of Julia sets, infinite interval exchange maps, and characterizations of lattice surfaces.
October 6: Chong-Qing Cheng (Nanjing/U Toronto)
Title: Arnold diffusion in nearly integrable Hamiltonian systems of three degrees of freedom
Abstract: We prove that Arnold diffusion is a generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$H(x, y) = h(y) + \varepsilon P(x, y), \ x\in \mathbb T^3, y\in \mathbb R^3.$$ Under typical perturbation \varepsilon P, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided E is bigger than the minimum of the average action. The preprint is available at [arxiv.org]
October 13: Seonhee Lim (Seoul Univ. and UC Berkeley)
Title: Martin boundary and Brownian motion on hyperbolic manifolds
Abstract: We show that the heat kernel $p(t,x,y)$ on the universal cover of a closed manifold of negative curvature is asymptotically of the form $$e^{-\lambda t} t^{-3/2} C(x,y)$$, where $\lambda$ is the bottom of the spectrum of the Laplacian and $C(x,y)$ is a positive constant depending only on $x$ and $y$. We also show that the $\lambda$-Martin boundary coincides with its Gromov boundary. We use the uniform Harnack inequality and the uniform two-mixing of the geodesic flow for suitable Gibbs-Margulis measures. This is a joint work with F. Ledrappier.
Octobet 20: Boris Khesin (Toronto)
Title: Integrability and non-integrability of pentagram maps
Abstract: We define pentagram maps on polygons in any dimension, which extends R. Schwartz's definition of the 2D pentagram map. Many of those maps turn out to be discrete integrable dynamical systems, while the corresponding continuous limits of such maps coincide with equations of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We discuss their geometry, Lax forms, and interrelations between recent pentagram generalizations. This is a joint work with Fedor Soloviev (Univ. of Toronto).
October 27: Kiran Parkhe (Technion)
Title: Groups of polynomial growth and 1D dynamics
Abstract: Let $M$ be a connected one-manifold, and $G$ a group of homeomorphisms of $M$ which is finitely-generated and virtually nilpotent, i.e., which has polynomial growth. We prove a structure theorem which says, roughly, that the manifold decomposes into wandering regions (in which no $G$-orbit is dense), and minimal regions (in which every $G$-orbit is dense); and on the latter, the action is actually abelian. As a corollary, if $G$ is a group of polynomial growth of degree $d$, then for any $\alpha < 1/d$, any continuous $G$-action on $M$ is conjugate to an action by $C^{1 + \alpha}$ diffeomorphisms. This strengthens a result of Farb and Franks.
November 3: Jacopo De Simoi (University of Toronto)
Title: Decay of correlations in fast-slow partially hyperbolic systems
Abstract: We show existence of an open class of smooth partially hyperbolic local diffeomorphisms of the two-torus which admit an unique SRB measure satisfying exponential decay of correlations. This is joint work with Carlangelo Liverani.
November 10: Howard Masur (U. Chicago)
Title: Ergodic Theory of Interval Exchange Transformations
Abstract: An interval exchange transformation (IET) is a map of an interval to itself defined by cutting the interval into d pieces and rearranging them by translations. IET arise in the study of rational billiards and translation surfaces. An IET with 2 intervals is equivalent to a rotation of a circle. In that case it is classical that there is a dichotomy: if the rotation angle is rational then every orbit is periodic or it is irrational and every orbit is dense and what is stronger; every orbit is equidistributed on the circle; a property called unique ergodicity. Once d is at least 4 there exist examples of IET with dense orbits which are not uniquely ergodic. I will discuss the issue of whether a non ergodic IET may still possess some orbit that is equidistributed. Along the way I will introduce the process of Rauzy induction, a generalization of the continued fraction algorithm, which is one of the main tools in studying IET. This is joint work with Jon Chaika.
November 17: Miguel Walsh (University of Oxford)
Title: Norm convergence of nonconventional ergodic averages
Abstract: Consider a group of measure preserving transformations acting on a probability space. The limiting behavior of the nonconventional ergodic averages associated with this action has been the subject of much attention since the work of Furstenberg on Szemerédi's theorem. We will discuss this problem, and how to establish a general convergence result for these averages.
November 24: Terry Soo (University of Kansas)
Title: A monotone Sinai theorem
Abstract: Let X be the space of all bi-infinite sequences of nonnegative integers less than some finite N, and endow X with the shift map T, so that Tx(i) = x(i+1). A self-map f on X is equivariant if f(Tx) = Tf(x), and monotone if f(x)(i) is no greater than x(i). Let mu and nu be product measures on X. Sinai proved that if the entropy of nu is less than mu, then there exists an equivariant map so that push-forward of mu is nu; in joint work with Anthony Quas, we show that if we also assume that the entropy inequality is strict and mu stochastically dominates nu, then Sinai's theorem can be realized via a monotone map.
Thursday, December 11: Serge Cantat (Université de Rennes I)
Title: Smooth invariant measures for automorphisms of surfaces
Abstract: Let X be a smooth complex projective surface, and let f be an automorphism of X. Assume that the topological entropy of f is positive. Then f has infinitely many saddle periodic points, and these points are equidistributed with respect to an f-invariant probability measure. One can characterize the pairs (X,f) for which this measure is smooth. (Based on a joint work with C. Dupont).