Dynamics Seminar: Spring 2006

The Dynamics seminar will meet on Friday afternoons at 2:30 in Clearihue D126. All are welcome. The list of seminars is as follows. For further information, please contact me at  

Date Speaker Title
13/1 Wael Bahsoun (UVic) Chaotic dynamical systems: a probabilistic approach I
20/1 Wael Bahsoun (UVic) Chaotic dynamical systems: a probabilistic approach II
27/1 Chris Bose (UVic) Approximation of Invariant Measures in Ergodic Theory I
3/2 Chris Bose (UVic) Approximation of Invariant Measures in Ergodic Theory II
10/2 Matthijs Vos (UVic Biology) Systems of many differential equations are usually unstable Systems of many ecological species are usually stable How to resolve the paradox of diversity-stability relations in ecology
17/2 Robert Moody (UVic) Dynamics in the Theory of Diffraction in Systems with Long-range Aperiodic Order
24/2 No seminar No seminar
Tuesday 28/2 Izabella Laba (UBC)  [Note: Different Time/Place: DSB C116 on Tuesday] Distance sets: Combinatorics and Fourier Analysis
10/3 Ciprian Demeter (UCLA)   Breaking the duality in the Return Times Theorem
17/3 Ian Putnam (UVic) Unique ergodicity for interval exchange transformations

Date: 13/1 and 20/1
Speaker: Wael Bahsoun (UVic)
Title: Chaotic dynamical systems: a probabilistic approach
Abstract: We study chaotic dynamical systems probabilistically by examining the following questions.

Let (X,B,m) be a measure space and T:X -> X be a measurable nonsingular transformation.

1. Does T admit an invariant probability measure which is absolutely continuous (acim for short) with respect to the ambient measure?

2. If 1. is satisfied, does the acim persist under small perturbation of T?

3. If 1. is satisfied, can we find the density of the acim?

4. If we can not find the density of the acim, can we approximate it by a computable density and find a bound on the approximation error?

To answer these questions we study the Perron-Frobenius operator associated with T.

In the first lecture, I will introduce the Perron-Frobenius operator and state its properties. Moreover, I will give specific examples where 1. and 3. are satisfied.

In the second lecture, I will state (and prove if time permits) stability of the isolated eigenvalues of transfer operators (including Perron-Frobenius) satisfying a certain inequality. This will give answers to questions 2 and 4.

Date: 27/1 and 3/2
Speaker: Chris Bose (UVic)
Title: Approximation of Invariant Measures in Ergodic Theory
Abstract: The set of invariant measures for a transformation T on a measurable space X contains all the information about asymptotic behavior of the T-orbits on X. Unfortunately, except for a few special examples T, we don't have closed form expressions for these measures and therefore we are unable to compute these asymptotics.

One way to try to get around the problem is to derive a closed form expression for an APPROXIMATION to the invariant measure in question. Such approximations would then be used to compute ESTIMATES on the asymptotics.

Lecture 1:

Many methods for measure approximation have been suggested over the years -- I will review some of these ideas and attempt to organize them into a cohesive framework involving either

1. long orbit calculations

2. Finite-rank projection methods

3. Convex optimization methods.

Lecture 2:

I will present in some detail a case study from the convex optimization approach: maximal entropy approximations, and show how rigourous convergence results can be obtained in quite general settings.

Date: 10/2
Speaker: Matthijs Vos (UVic Biology)
Title: Systems of many differential equations are usually unstable Systems of many ecological species are usually stable How to resolve the paradox of diversity-stability relations in ecology
Abstract: Explaining the brimming complexity of highly diverse systems such as tropical rainforests and coral reefs is of fundamental interest to ecologists. They have tried to understand the existence and persistence of species-rich communities using systems of coupled ordinary differential equations. These models generally predict simple communities to be stable, while more complex communities show population cycles and chaotic behaviour that make species prone to extinction (because of very low minimum population densities).

Simple theory thus predicts that complex communities cannot exist or persist for long. However, they clearly do so in nature.

In this talk I discuss the characteristics that differentiate ecological communities from randomly assembled mathematical communities. I show how introducing ecological realism into the equations can reverse the sign of diversity-stability relations from negative to positive.

Date: 17/2
Speaker: Robert Moody (UVic)
Title: Dynamics in the Theory of Diffraction in Systems with Long-range Aperiodic Order
Abstract: The distinguishing feature of a system with long-range aperiodic order (e.g. Penrose tilings or quasicyrstals) is the distinctive nature of its diffraction, namely the prominent appearance of many bright spots, or Bragg peaks as they are called. An important technique in the study of long-range aperiodic order is the use of dynamical systems. The associated spectral theory is closely associated with the diffraction and has been an important tool in understanding it. However the connection between diffraction and dynamics is not particularly straightforward and it is not yet fully understood. In the talk we will introduce all the concepts, defining diffraction, showing how we obtain dynamical systems from aperiodic systems, and giving some indication of the way in which dynamical systems play an useful part in the theory. We then look more closely at the relation between diffraction and dynamics and survey some recent work with Xinghua Deng, still in progress, that we hope shed some useful light on this.
Date: 28/2
Speaker: Izabella Laba (UBC)
Title: Distance sets: combinatorics and Fourier analysis
Abstract: Given a set E in Rn, what can we say about the lower bounds on the size of the set Delta(E) of all possible distances between pairs of points in E, depending on the size of E itself? For discrete sets, this is a notorious unsolved combinatorial problem due to Erdös; there is also a continuous version, due to Falconer, where the best results so far have been obtained by Fourier-analytic methods. Although the main conjectures remain open, recent years have seen significant progress on these and other closely related questions. In this talk, I will review the recent work on the distance set problem, its variants (non-Euclidean distance functions, averaging estimates), and connections to other well known problems such as the restriction conjecture.
Date: 10/3
Speaker: Ciprian Demeter (UCLA)
Title: Breaking the duality in the Return Times Theorem
Abstract: We prove Bourgain's Return Times Theorem for a range of exponents p and q that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain's averages. As an immediate corollary we obtain a Wiener-Wintner type of result for the ergodic Hilbert series. The methods of proof are a combination of time-frequency analysis, classical Fourier analysis and ergodic theory.

This is joint work with Michael Lacey, Terence Tao and Christoph Thiele.

Date: 17/3
Speaker: Ian Putnam (UVic)
Title: Unique ergodicity for interval exchange transformations
Abstract: An interval exchange transformation is a bijection of the unit interval which is picecwise a translation. Clearly, such a map must leave Lebesgue measure invariant, but it is a rather subtle issue of when this is the only invariant probability measure. I will discuss some aspects of this question.