January 12: Pengfei Zhang (University of Houston)
Title: Homoclinic points for generic convex billiards
Abstract: Kupka-Smale Theorem implies that, $C^r$-generically, each homoclinic orbit, if exist, is the transversal intersection of invariant manifolds. It is an open problem that whether the homoclinic orbits exist $C^r$-generically. In this talk I will give a brief introduction of some recent progresses for surface diffeomorohisms, and describe the dynamical billiards on convex domains. We prove that $C^r$ generically, there exists some homoclinic orbit for every hyperbolic periodic point of the billiard system. This is a joint work with Zhihong Jeff Xia.
February 9: Thomas Barthelme (Penn State)
Title: Counting orbits of Anosov flows in free homotopy classes
Abstract: (Joint work with Sergio Fenley) Since Margulis and Bowen gave an estimate of the growth rate of periodic orbits of Anosov flow, there has been a lot of research furthering counting questions. If one consider only Anosov flows, these developments have been either into giving more precise estimates or into counting periodic orbits given a homological constraint, i.e., counting periodic orbits that are in the same fixed homology class. I will talk here about a third direction: Despite what one might think when considering the most classical examples of Anosov flows, a lot of Anosov flows (maybe most) in 3-manifolds are such that some periodic orbits are freely homotopic to infinitely many other. It is therefore legitimate to ask whether one can give an estimate of the growth rate of periodic orbits inside an infinite free homotopy class. I will explain how one can use the geometry and topology of Anosov flows in 3-manifolds to obtain such estimates. As a corollary, we get an answer to the following question, asked by Plante and Thurston in 1972: If M is a manifold supporting an Anosov flow, does the number of conjugacy classes in the fundamental group grows exponentially fast with the length of the shortest orbit representative?
February 12: Anatole Katok (Penn State)
Title: FLEXIBILITY OF ENTROPIES AND LYAPUNOV EXPONENTS
Abstract: Motivation for the program that I will outline in this talk comes form various recent results concerning smooth actions of higher-rank abelian groups as well as the older Zimmer program concerning actions of ``large'' non-abelian groups. Among rigidity phenomena established for those actions are strong restrictions of arithmetic nature on the values of Lyapunov exponents for positive entropy invariant measures as well as on values of topological and measure-theoretic entropies. On the other hand, for classical rank one systems, i.e diffeomorphisms and flows, one does not expect any restrictions of similar nature, other than those that come from rigidity of topological orbit structure form the discrete time case. Surprisingly though this is easier said than done. There is a variety of local results coming from various constructions of Anosov and non-uniformly hyperbolic systems but already such a simply sounding question as characterizing all possible triples of positive numbers that appear as Lyapunov exponents of volume-preserving Anosov diffeomorphisms (wrt. to the volume measure) on the three-dimensional torus seems to require some serious new ideas beyond combination of known methods. I will describe the only non-trivial global result so far, namely, characterization of pairs of numbers that appear as values of topological and measure-theoretic (Liouville) entropies for geodesic flow on a surface of genus greater than one with a Riemannian metric of a fixed total area. Even there the answer is not known if one restricts to the metrics in a fixed conformal class. This is a joint work with Alena Erchenko.
February 16: Dan Cuzzocreo (Smith College)
Title: Parameter space structures for rational maps
Abstract: $F_{n, d, \lambda} = z^n + \lambda/z^d$ give 1-parameter, $n+d$ degree families of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial $z^n$. Despite the high degree, symmetries cause these maps to have just a single free critical orbit and thus to form a natural 1-dimensional slice. Due to some similarities with polynomial maps, these families give some of the best-understood examples of non-polynomial rational dynamics in arbitrarily high degree. In this talk we give a survey of some recent results about these maps, with a focus on characterizing some of the fractal structure in the parameter space.
March 2: Kelly Yancey (University of Maryland)
Title: Minimal Self Joinings of Substitutions Arising from IETs
Abstract: In this talk we will discuss substitution systems that have the property of minimal self joining. Then we will focus our attention on self-similar interval exchange transformations and their associated substitutions. We will show that 3-IETs have MSJ. This is joint with Giovanni Forni.
March 9: Giulio Tiozzo (Yale University)
Title: Growth of infinite graphs and core entropy of quadratic polynomials
Abstract: The core entropy of quadratic polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. A classical tool to compute the entropy of a dynamical system is the clique polynomial (recently used by McMullen to study the entropy of pseudo-Anosov maps). We will develop an infinite version of the clique polynomial for infinite graphs, and use it to study the symbolic dynamics of Hubbard trees. Using these methods we will prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.