Dynamics Seminar Spring 2010

The Dynamics seminar will meet on Thursday afternoons at 2:30 in DSB C118. All are welcome. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title

4/2/2010 Anthony Quas Uniform distribution, Lattices and 3log(2)-pi^2/8

11/2/2010

18/2/2010 No seminar (reading break)

25/2/2010 John Griesmer (UBC) Sumsets with one dense and one sparse summand

4/3/2010 Mahsa Allahbakhshi (UVic) Class Degree and Relative Maximal Entropy

11/3/2010 Robert Moody (UVic) Going backwards from diffraction to dynamics

18/3/2010 Omer Angel (UBC) (Colloquium) Local and global properties of graphs

25/3/2010 Brian Marcus (UBC) Limiting entropy and Independence entropy for Z^d-shift spaces

1/4/2010 Ronnie Pavlov (UBC) Estimating the entropy of some Z^2 shifts of finite type with probabilistic methods

8/4/2010 Terry Soo (UBC) Deterministic Poisson Thinning

15/4/2010 Wael Bahsoun (Loughborough)
Note: Different Room: C116 Quasi-Invariant measures, escape rates and the effect of the hole

Date: 4/2/2010
Speaker: Anthony Quas
Title: Uniform distribution, Lattices and 3log 2-pi^2/8
Abstract: Starting out from some questions concerning piecewise isometries we are led to the study of the distribution of "visible points" in subsets of the square lattice

Date: 11/2/2010
Speaker:
Title:
Abstract:

Date: 25/2/2010
Speaker: John Griesmer (UBC)
Title: Sumsets with one dense and one sparse summand
Abstract: We use ergodic theory to analyze sets of the form A+B:={a+b|a in A, b in B}, where A and B are sets of integers and B has positive density. This continues a line of research initiated by R. Jin, who showed that A+B is piecewise syndetic whenever A and B have positive density. Subsequently Bergelson, Furstenberg, and Weiss (BFW) strengthed Jin's conclusion to "A+B is piecewise Bohr", and Beiglbock, Bergelson, and Fish generalized the latter result to the setting where the integers are replaced by an amenable group. In this talk, We will show how to connect properties of sumsets to properties of measure preserving systems. We then generalize the BFW result to sumsets A+B where A has zero density and B has positive density, exploiting a result of Boshernitzan, Kolesnik, Quas, and Wierdl on ergodic averages over sparse sets of times. Finally we pose some questions raised by our investigations.

Date: 4/3/2010
Speaker: Mahsa Allahbakhshi (UVic)
Title: Class Degree and Relative Maximal Entropy
Abstract: Given a factor code $\pi$ from a shift of finite type $X$ onto an irreducible sofic shift $Y$, and a fully supported ergodic measure $\nu$ on $Y$, we give an explicit upper bound on the number of ergodic measures on $X$ which project to $\nu$ and have maximal entropy among all measures in the fiber $\pi^{-1}(\nu)$. This bound is invariant under conjugacy. We relate this to an important construction for finite-to-one symbolic factor maps.

Date: 11/3/2010
Speaker: Robert Moody (UVic)
Title: Going backwards from diffraction to dynamics
Abstract: Many dynamical systems arising from aperiodic tilings (e.g. Penrose tilings) or aperiodic point sets give rise to ergodic dynamical systems with pure point spectra. They also have pure point diffraction, a fact that is quite different but closely related. The diffraction is actually a positive and centrally symmetric measure on the Fourier dual to the original space of the tiling or point set. Pure pointedness simply means that the measure is a point measure, i.e. is atomic. The question we address here is whether every pure point, positive, and centrally symmetric measure on the Fourier dual of a locally Abelian group G (e.g. a finite dimensional real space) is the diffraction of "something" in G. In other words, is anything that looks potentially like the diffraction of something actually the diffraction of something.

Date: 18/3/2010
Speaker: Omer Angel (UBC)
Title: Local and global properties of graphs
Abstract: I will discuss the relation between the local structure and local properties of a graph (such as vertex degrees and allowed neighborhoods and recurrence of random walk) and global properties such as size and planarity. For example, an elegant result of Benjamini and Schramm states (loosely) that (finitary) planar graphs are uniformly recurrent. The talk relates to works with Szegedy, Sheffield, Silberman, and Wilson.

Date: 25/3/2010
Speaker: Brian Marcus (UBC)
Title: Limiting entropy and Independence entropy for Z^d-shift spaces
Abstract: (Joint work with Erez Louidor and Ronnie Pavlov, UBC)
Topological entropy is a fundamental invariant for Z^d-shift spaces. When d = 1 and the shift space is a shift of finite type, it is easy to compute the topological entropy. In higher dimensions, it is much more difficult to compute, and exact values are known in only a handful of cases. However, a limiting entropy, as the dimension d approaches infinity, defined as follows, can be computed in many cases.
A 1-dimensional shift space X determines Z^d-shift spaces X^*d for each d; namely, X^*d is the Z^d-shift space where every row in every coordinate direction satisfies the constraints of X. Let h_d(X) denote the topological entropy of X^*d. Then h_d(X) is non-increasing in d and its limit is denoted hlim(X).
We introduce a notion of independence entropy for any Z^d-shift space X, denoted hind(X), which is explicitly computable if X is a 1-dimensional shift of finite type (more generally a one-dimensional sofic shift). We show that h_lim(X)>= h_ind(X) = h_ind(X^d) for all d. We can prove that equality holds in many cases, and it may well hold in general.

Date: 1/4/2010
Speaker: Ronnie Pavlov (UBC)
Title: Estimating the entropy of some Z^2 shifts of finite type with probabilistic methods
Abstract: In symbolic dynamics, a Z^d shift of finite type (or SFT) is the set of all ways to assign elements from a finite alphabet A to all sites of Z^d, subject to local rules about how elements of A are allowed to appear near each other. For instance, the Z^d golden mean SFT has alphabet {0,1} and consists of all infinite configurations in {0,1}^{Z^d} in which no two adjacent 1s appear. The (topological) entropy of any Z SFT is easily computable (it is the log of an algebraic number). However, for d>1, the situation becomes more complex. There are in fact only a few non examples of Z^2 SFTs whose entropies have explicit closed forms. For any Z^2 SFT from a certain class which includes the golden mean shift (for which no explicit closed form for the entropy is known), we use some probabilistic techniques, including Markov random fields and percolation theory, to give a sequence of approximations to the entropy which converge at an exponential rate. This implies that the entropy of any SFT in this class is computable in polynomial time.

Date: 8/4/2010
Speaker: Terry Soo (UBC)
Title: Deterministic Poisson Thinning
Abstract: Given a homogeneous Poisson point process it is well known that selecting each point independently with some fixed probability gives a homogeneous Poisson process of lower intensity. This is often referred to as thinning. In this talk we will discuss the following question. Can thinning be achieved without additional randomization; that is, as a deterministic function of the point process, can we choose a subset of the points so that the chosen points from a Poisson process of lower intensity? On the infinite line and plane (and in any other higher dimension) we can colour a Poisson point process red and blue, so that each colour class forms a Poisson point process; furthermore, the function can be chosen as an isometry-equivariant finitary factor (that is, if an isometry is applied to the points of the original process, the points are still coloured the same way). Thus using only local information, without a central authority or additional randomization, points of a Poisson process can split into two groups, each of which are still Poisson. On a set of finite volume (say the unit disc or circle), we find that even without considerations of equivariance, thinning can not always be achieved as a deterministic function of the Poisson process and the existence of such a function depends on the intensities of the original and resulting Poisson processes. We prove a necessary and sufficient condition on the two intensities for the existence of such a function. The condition exhibits a surprising lack of monotonicity. (Joint work with Omer Angel, Alexander Holroyd, and Russell Lyons)

Date: 15/4/2010
Speaker: Wael Bahsoun (Loughborough)
Title: Quasi-Invariant measures, escape rates and the effect of the hole
Abstract: A result by Keller and Liverani on the stability of non-essential spectrum of transfer operators can be used to study existence of absolutely continuous conditionally invariant measures (accim) for interval maps with holes. In particular its says that if a mixing Lasota-Yorke map is perturbed by introducing a `sufficiently small' hole in the phase space, then the resulting open dynamical system admits an accim. In this talk we show how the condition `sufficiently small' can be verified rigorously on a computer. In particular, for a given Lasota-Yorke map, we use Ulam's method on the closed dynamical system $T$ to give a computable size of the hole $H$ for which the open dynamical system $T_H$ must admit an accim. This first part of the talk is only concerned with size of the hole. If time permits, we will also talk about the effect of the position of the hole on the escape rate. Recently, it has been observed that the escape rate is affected not only by the size of the hole but also by its position in the phase space. We will discuss how our methods can be used to complement such interesting results.
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