# A symbolic representation of Anosov-Katok Diffeomorphisms III

## Speaker:

## Institution:

## Time:

## Location:

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.

# Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures

## Speaker:

## Institution:

## Time:

## Host:

## Location:

Inspired by a recent work of Marcin Sabok, we define a criterionfor a homogeneous metric structure X that implies that its automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space, the Lebesgue probability measure algebra, and the Hilbert space, regarded as metric structures, thus proving that their automorphism groups share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group with a left-invariant, complete metric, is trivial, and we verify it for the Urysohn space, and the Hilbert space.

# Diophantine properties of elements of SO(3)

## Speaker:

## Institution:

## Time:

## Location:

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.

# Estimating the Fractal Dimension of Sets Determined by Nonergodic Parameters.

## Speaker:

## Institution:

## Time:

## Location:

In 1969, William Veech introduced two subsets K_1(*θ*) and K_0(*θ*) of R/Z which are defined in terms of the continued fraction expansion of *θ*. These subsets are known to give information about the dynamics of certain skew products of the unit circle. We show that the Hausdorff dimension of K_i(*θ*) can achieve any value between zero and one.