Dynamical foliations for system isotopic to Anosov diffeomophisms

Speaker: 

Todd Fisher

Institution: 

Brigham Young University

Time: 

Thursday, June 5, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH 440R

We discuss the existence of foliations that are invariant under the dynamics for systems that are isotopic to Anosov diffeomorphisms. Specifically, we examine partially hyperbolic diffeomorphisms with one dimensional center that are isotopic to a hyperbolic toral automorphism and contained in a connected component. We show in this case there is a center foliation. We will also discuss more general cases where there is a weak form of hyperbolicity called a dominated splitting. This is joint work with Jerome Buzzi, Rafael Potrie, and Martin Sambarino.

Global Rigidity of Anosov Z^r Actions on Tori and Nilmanifolds

Speaker: 

Zhiren Wang

Institution: 

Yale University

Time: 

Tuesday, April 29, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH 440R

As part of a more general conjecture by Katok and Spatzier, it was asked if all smooth Anosov Z^r-actions on tori, nilmanifolds and infranilmanifolds without rank-1 factor actions are, up to smooth conjugacy, actions by automorphisms. In this talk, we will discuss a recent joint work with Federico Rodriguez Hertz that affirmatively answers this question.

On the Statistical Properties of Direct Product Systems

Speaker: 

Marks Ruziboev

Institution: 

ICTP/SISSA

Time: 

Tuesday, May 27, 2014 - 1:00pm

Host: 

Location: 

RH 440R

We consider direct product of finitely many Young towers with the tails decaying at certain rate and show that the product map admits a Young tower whose tail can be estimated in terms of the rates of component towers. It has been shown that many systems admit such a towers and our results therefore imply statistical properties such as decay of correlations, central limit theorem, large deviations, local limit theorem for large class of product systems.

Dimension estimates for sets of uniformly badly approximable systems of linear forms

Speaker: 

Ryan Broderick

Institution: 

Northwestern University

Time: 

Tuesday, May 13, 2014 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

A number is called badly approximable if there is a positive constant c such that |x-p/q| > c/q^2 holds for all rationals p/q, so that close approximation by rationals requires relatively large denominators. The set of such numbers is Lebesgue-null but has full Hausdorff dimension. This set can be viewed as the union over c of the set BA(c) of numbers which satisfy the above inequality for the fixed constant c. J. Kurzweil obtained dimension bounds on BA(c), which were later improved by D. Hensley. We will discuss recent work, joint with D. Kleinbock, in which we use homogeneous dynamics to produce dimension bounds for a higher-dimensional analog.

Products of Cantor sets and Spectral Properties of Labyrinth Model

Speaker: 

Yuki Takahashi

Institution: 

UC Irvine

Time: 

Tuesday, April 1, 2014 - 1:00pm to 2:00pm

Location: 

RH 440R

We prove that the product of two Cantor sets of large thickness is an interval in the case when one of them contains the origin. We apply this result to the Labyrinth model of a two-dimensional quasicrystal, where the spectrum is known to be the product of two Cantor sets, and show that the spectrum becomes an interval for small values of the coupling constant. We also consider the density of states measure of the Labyrinth model, and show that it is absolutely continuous with respect the Lebesgue measure for most values of coupling constants.

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