Diophantine properties of elements of SO(3)

Speaker: 

Ryan Broderick

Institution: 

UC Irvine

Time: 

Tuesday, February 10, 2015 - 1:00pm to 2:00pm

Location: 

RH 440R

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: 

Joseph Squillace

Institution: 

UC Irvine

Time: 

Tuesday, February 3, 2015 - 1:00pm to 2:00pm

Location: 

RH 440R

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.

A symbolic representation of Anosov-Katok Diffeomorphisms

Speaker: 

Matt Foreman

Institution: 

UC Irvine

Time: 

Tuesday, January 13, 2015 - 1:00pm to 2:00pm

Location: 

RH 440R

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 symbolic representation of Anosov-Katok Diffeomorphisms II

Speaker: 

Matt Foreman

Institution: 

UC Irvine

Time: 

Tuesday, January 20, 2015 - 1:00pm to 2:00pm

Location: 

RH 440R

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.

Selected problems in dynamical systems

Speaker: 

Anton Gorodetski

Institution: 

UC Irvine

Time: 

Tuesday, December 9, 2014 - 1:00pm to 2:00pm

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. 

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