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.

We sketch a proof of a theorem due to Kleinbock, and generalizing previous work of Dani and of Margulis and Kleinbock, regarding the size of the set of bounded orbits of a mixing flow on a homogeneous space. We then discuss connections to number theory, specifically the fact, proved in the same paper of Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.

Given a Lie group G and a lattice \Gamma in G, we consider a flow on G/\Gamma induced by the action of a one-parameter subgroup of G. If this flow is mixing then a generic orbit is dense, but nevertheless one can discuss the dimension of the set of exceptions. We discuss work of S. G. Dani, in which such estimates are made in certain cases and a connection to diophantine approximation is established, and also generalizations due to D. Kleinbock and G. Margulis. In particular, we outline a dynamical proof, due to Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.