I present joint work with B. Weiss that describes a concrete operation on words that allows one to generate symbolic representations of Anosov-Katok diffeomorphisms. We show that each A-K diffeomorphism can be represented this way and that each symbolic system generated by this operation can be realized as an A-K diffeomorphism.

A real number x is called diophantine if its distance to rationals p/q is large relative to q -- more precisely, if for every d > 0 there is a positive C such that for every reduced rational p/q, we have |x - p/q| > Cq^{-2-d}, or equivalently |qx-p| > Cq^{-1-d}. Almost all reals have this property. Furthermore, almost every pair (x_1, x_2) has the property that for every d > 0 there is a C such that |q_1x_1+q_2x_2 -p| > C||q||^{-2(1+d)} for all p, q_1, q_2. In this talk, we discuss a noncommutative analog of this property for elements of SO(3). Namely, a pair (A,B) is called diophantine if there exists a constant D such that for every positive integer n and every reduced word W of length n in A, B, A^{-1}, B^{-1}, we have ||W - E|| > D^{-n}, where E is the identity matrix. It is conjectured that almost every such pair (in the sense of Haar measure) is diophantine. We will present a paper of Kaloshin and Rodnianski, in which the weaker bound D^{-n^2} is obtained.

I present joint work with B. Weiss that describes a concrete operation on words that allows one to generate symbolic representations of Anosov-Katok diffeomorphisms. We show that each A-K diffeomorphism can be represented this way and that each symbolic system generated by this operation can be realized as an A-K diffeomorphism.

We will discuss some problems (related to piecewise isometris, sums and products of Cantor sets, dynamics of the Fibonacci trace map etc.) that are in the scope of current interests of the dynamical systems seminar. Many of the problems can be considered as potential research projects by the interested graduate students. 

We consider product of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by the product of two Cantor sets. We also discuss the connection between our problem and the ”intersection of two Cantor sets” problem, which is a problem considered in several papers before.