April 7: Ayse Sahin (DePaul)
Title: Special representations of group actions and low complexity tilings of groups
Abstract: Generalizing a result of Alpern, Rudolph showed that the orbits of any measurable, measure preserving R^d action can be measurably tiled by 2^d rectangles and asked if this number of tiles is optimal. In joint work with B. Kra and A. Quas, we show that d+1 rectangular tiles suffice. Furthermore, for d=2, for flows with completely positive entropy, this bound is optimal.
April 21: Amie Wilkinson (U. Chicago)
Title: Robust mechanisms for chaos in smooth dynamics
May 5: Andrey Gogolev (Binghamton)
Title: Anosov bundles
Abstract: Consider a non-trivial fiber bundle M -> E -> B whose total space E is compact and whose base B is simply connected. Can one equip E with a diffeomorphism or a flow which preserves each fiber and whose restriction to each fiber is Anosov? In a joint work with Tom Farrell we answer this question negatively and give an application to geometry of negatively curved bundles. However, in a joint work with Pedro Ontaneda and Federico Rodriguez Hertz, we construct non-trivial bundles that admit fiberwise Anosov diffeomorphisms that permute the fibers.
May 12: Oleg Ivrii (Harvard)
Title: The Weil-Petersson metric in complex dynamics
Abstract: In this talk, I describe an analogue of the Weil-Petersson metric on "main cardioids" of parameter spaces of complex dynamical systems proposed by McMullen. I will show that the metric is incomplete, and examine its metric completion. I will also describe various "asymptotic symmetries" arising from degenerations of dynamical systems to vector fields and decorated rescaling limits, which could be seen as a partial analogue of the mapping class group.
May 19: Bryna Kra (Northwestern)
Title: Periodicity, complexity and shifts
Abstract: A beautiful example of a global property being determined by a local one is the Morse-Hedlund Theorem; it gives the relation between the global property of periodicity for an infinite word in a finite alphabet and local information on the complexity of the word. I will discuss higher dimensional versions of this problem, and the use of local conditions on complexity to determine global properties of the configuration, and applications of this to the automorphism group of shifts. This is joint work with Van Cyr.
June 2: Sarah Koch (Michigan)
Title: Twisted matings of polynomials
Abstract: Given two suitable polynomials of degree d, we can form the mating of the polynomials by gluing their Julia sets together in a dynamically meaningful way; this is a topological construction that yields a branched cover from an oriented 2-sphere to itself. If the mating is equivalent to a rational map we say that the geometric mating of the polynomials exists. We define {\em{twisted matings}} of polynomials, and we prove that for the basilica polynomial $P(z)=z^2-1$, for any $n>0$, all of the twisted matings of $P^{\circ n}$ with itself are (geometrically) classified by the periodic cycles of $z\mapsto z^2$, of length $n$.
June 9: Mikhail Lyubich (Stony Brook)
Title: Geometry of Feigenbaum Julia sets
Abstract: Feigenbaum quadratic maps appear naturally through cascades of bifurcations of attracting cycles. Geometry of their Julia sets J is quite fascinating and intimately related to the Universality and Rigidity phenomena in geometry anddynamics. We consider the following Basic Trichotomy for these sets:
1) Lean case: HD(J)< 2;
2) Balanced case: HD(J)=2 but area(J)=0;
3) Black Hole case: area(J)>0.
We show that all three options are realizable in the Feigenbaum class. In particuar, this provides us with ``tame" and ``observable" Julia sets of positive area (with explicit topological models and computable images). Existence of such Julia sets goes against intuition coming from hyperbolic geometry and theory of Kleinian groups. It is a joint work with Artur Avila.