Time

Feb 21 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Donald Robertson (Ohio State University)

Abstract

A coarse description of sub-sigma-algebras that characterize the behavior of averages arising from commuting actions of an amenable group will be given. The averages generalize those used in the ergodic proof of Szemeredi’s theorem. The description will be used to prove an ergodic version of Roth’s theorem for amenable groups.

Time

Nov 29 2012 – 3:00pm – 3:50 pm

Location

MW154

Speaker

Paul Reschke (UIC)

Abstract

By a theorem due to Gromov and Yomdin, the entropy of an endomorphism of a compact Kahler manifold is determined by its cohomological actions. For the case of automorphisms of compact Kahler surfaces, I will explain how the cohomological interpretation of entropy leads to characterizations of Salem numbers and I will present a variety of examples of such characterizations. I will then discuss other applications of the cohomological interpretation of entropy to dynamical questions on Kahler manifolds.

Time

Oct 18 2012 – 3:00pm – 3:50 pm

Location

MW154

Speaker

Joe Rosenblatt (University of Illinois, Department of Mathematics)

Abstract

Directional ergodicity and directional weak mixing of the action of two commuting transformations S and T can be analyzed by looking at extensions in which S and T are embedded in a two real variable flow. For a suitable class of extensions, the directional behavior observed is determined not by the extension itself, but by intrinsic spectral properties of the original action by S and T.

Time

Oct 25 2012 – 3:00pm – 3:50 pm

Location

MW 154

Speaker

David Koslicki (Mathematical Biosciences Institute, OSU)

Abstract

Inspired by concepts from ergodic theory, we give new insight into coding sequence (CDS) density estimation for the human genome. Our approach is based on the introduction and study of topological pressure: a numerical quantity assigned to any finite sequence based on an appropriate notion of ‘weighted information content’. For human DNA sequences, each codon is assigned a suitable weight, and using a window size of approximately 60,000bp, we obtain a very strong positive correlation between CDS density and topological pressure. The weights are selected by an optimization procedure, and can be interpreted as quantitative data on the relative importance of different codons for the density estimation of coding sequences. This gives new insight into codon usage bias which is an important subject where long standing questions remain open. Inspired again by ergodic theory, we use the weightings on the codons to define a probability measure on finite sequences. We demonstrate that this measure is effective in distinguishing between coding and non-coding human DNA sequences of lengths approximately 5,000bp. We emphasize that topological pressure is a flexible tool and we expect it to be useful for the investigation of many other features of DNA sequences such as interspecies comparison of codon usage bias. We give a first result in this direction, investigating CDS density in the mouse genome and comparing our results with those for the human genome. This is joint work with Dan Thompson (OSU).

Time

Nov 8 2012 – 3:00pm – 3:50 pm

Location

MW154

Speaker

Yunping Jiang (CUNY)

Abstract

In the study of modern dynamical systems, an invariant measure is an important topic and in the study of modern complex analysis, the quasisymmetric condition on a map is an important topic. In this talk, I will combine these two topics together introducing a new interesting topic, symmetric invariant measure. I will discuss the existence and the uniqueness of a symmetric invariant measure.  I will also discuss some ergodicity problem for symmetric measures and a related topic, smooth rigidity and symmetric rigidity.

Time

Nov 15 2012 – 3:00pm – 3:50 pm

Location

MW154

Speaker

Ryan Peckner (Princeton)

Abstract

The squarefree flow is a natural dynamical system whose topological and ergodic properties are closely linked to the behavior of squarefree numbers. We prove that the squarefree flow carries a unique measure of maximal entropy and describe the structure of the associated measure-preserving dynamical system. Our method involves first studying approximations arising from finite collections of prime numbers, then taking a limit under Ornstein’s dˉ-metric in order to consider all primes simultaneously. This is accomplished by proving uniform Gibbs bounds for a sequence of sofic systems and constructing explicit joinings between them in order to directly estimate their dˉ-distances.

Time

Feb 14 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Kenichiro Yamamoto (Tokyo Denki University)

Time

Mar 7 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

William Mance (University of North Texas)

Abstract

We show that the set of numbers that are Q-distribution normal but not simply Q-ratio normal has full Hausdorff dimension.  We further show under some conditions that countable intersections of sets of this form still have full Hausdorff dimension even though they are not winning sets (in the sense of W. Schmidt).  As a consequence of this, we construct many explicit examples of numbers that are simultaneously distribution normal but not simply ratio normal with respect to certain countable families of basic sequences.  Additionally, we prove that some related sets are either winning sets or sets of the first category.

Time

Mar 21 2013 – 3:00pm – 4:00 pm

Location

MW154

Speaker

Joseph Rosenblatt (University of Illinois at Urbana-Champaign)

Abstract

Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation
among all possible averages. We consider 1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function, 2) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average, and 3) oscillatory behavior of these norm approximations.

Time

Apr 9 2013 – 1:50pm – 2:50 pm

Location

CH240

Speaker

Jon Chaika (University of Chicago)

Abstract

Recently Eskin-Mirzakhani-Mohammadi have proven a number of
powerful results about the SL_2(R) orbits of abelian differentials and
the SL_2(R) ergodic measures on the stratum. We discuss some results
motivated and enabled by this work. One result is that for
every abelian differential there is a measure on the stratum, such
that after rotating in almost every direction, the geodesic flow
equidistributes for this measure on the stratum. Another result is
that for any surface the conclusion of Oseledets multiplicative
ergodic theorem applies for the Kontsevich-Zorich cocycle. This has an
application, being explored by others, to the windtree model. This is
joint work with Alex Eskin.

Time

Apr 18 2013 – 3:00pm – 3:50 pm

Location

MW154

Speaker

Younghwan Son (Ohio State)

Abstract

Mixing properties are important invariants in ergodic theory. In recent decades the theory of quasicrystals has facilitated the study of mixing properties in tiling dynamical systems. In this talk we will survey some results and problems regarding weakly, mildly, and strongly mixing tiling dynamical systems.