# Dynamics seminar

Title: Z^2 skew products

Speaker: Chris Hoffman, Washington

Date and time:
02 Apr 2014,
3:30pm -
4:30pm

Location: Clearihue A303

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Many constructions in ergodic theory involve skew products. The use of skew products are much more common for constructing actions of Z than for actions of other groups. In this talk I will describe how to make the Z^2 analogs of some common skew products. Along the way we will find connections to several models statistical mechanics as well as questions in computability.

Title: Four Cycle Free Graphs and Entropy Minimality.

Speaker: Nishant Chandgotia, UBC

Date and time:
26 Mar 2014,
3:30pm -
4:30pm

Location: Clearihue A303

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Entropy minimality was introduced by Coven and Smítal (1992) as a property of dynamical systems which is stronger than topological transitivity and weaker than minimality. A topological dynamical system (X,T) is said to be entropy minimal if all closed T-invariant subsets of X have entropy strictly less than (X,T). In the sphere of symbolic dynamics this translates to the following: a shift-space is entropy minimal if by exclusion of any globally allowed pattern we incur a drop in the topological entropy. In one dimension it is well known that a nearest neighbour shift of finite type is entropy minimal if it is irreducible; the same is known in higher dimensions under strong irreducibility. In this talk we will discuss a class of nearest neighbour shift of finite type which appear as the space of graph homomorphisms from Z^d to graphs without four cycles; for instance, we will see why the space of 3-colourings of Z^d is entropy minimal even though it does not have any of the nice mixing properties.

Title: Numerical Experiments on the Modulus of Continuity of Matrix Functions

Speaker: Terry Loring, New Mexico

Date and time:
19 Mar 2014,
3:30pm -
4:30pm

Location: Clearihue A303

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If f is a real-valued function on the unit interval, we can extend its domain to matrices that are Hermitian with eigenvalues in the unit interval. Now assume f is continuous. It is uniformly continuous, on its original domain. Is f also uniformly continuous for matrices? That is, can we find eta tending to zero at zero so that, for 0<=X<=I and 0<=Y<=I, ||f(X)-f(Y)| <= eta(||X-Y||) holds for the spectral norm? Ando showed this is false for f(t)=|t-1/2| but true when f is operator monotone. A related question is uniformly bounding ||f(X)A-Af(X)|| in terms of ||XA-AX|| whenever ||A||<=1. The conjecture ||sqrt(X)A-A\sqrt(X)||<= sqrt(||XA-AX||) has been around for decades. We are able to prove this in an important special case. I will discuss the Monte Carlo methods used when computing billions of examples and present some conjectures based on the data we collected. This is joint work with my graduate student, Fredy Vides.

Title: One-dimensional solenoids

Speaker: Sarah Saeidi Gholikandi, Tehran, UVic

Date and time:
05 Mar 2014,
3:30pm -
4:30pm

Location: CLE A303

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I will talk about one-dimensional solenoids and their homology, as defined for Smale spaces by Putnam. First I will start with definition of Smale spaces and show how shifts of finite type spaces are examples, which play an important role in this homology. Then I will go into solenoids by giving some examples. We will show how there are natural shifts of finite type which factor onto them, which is used in computing their homology. Since the process of computing this homology is complicated, I will not enter into the details of the computations and just give the results. This is joint work with M.Amini and I.F Putnam

Title: Some algebraic dynamical systems and Markov partitions for them

Speaker: Ian Putnam, University of Victoria

Date and time:
26 Feb 2014,
3:30pm -
4:30pm

Location: CLE A303

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Abstract: I will describe some interesting dynamical systems of an algebraic nature. In many ways, they are similar to the well-known hyperbolic automorphisms of tori, such as Arnold's cat map. The new ingredient is the use of the field of p-adic numbers in place of the real numbers. My guess here is that most people may not be very familiar with p-adic numbers so I will give a short introduction to them, their wonderful properties and what about them might appeal to people in dynamical systems. After describing the systems, I will discuss the problem of finding convenient symbolic codings for them (i.e. Markov partitions) which was done for hyperbolic toral automorphisms by Adler and Weiss. This is joint work with Nigel Burke (UVic/ Cambridge). From these codings, we are able to compute the homology, as I defined in earlier work, but this will be a brief footnote to the rest of the talk.

Title: Heat Flow and Brownian Motion on NxN Matrix Lie Groups, and the Large-N Limit

Speaker: Todd Kemp, UCSD

Date and time:
03 Feb 2014,
3:30pm -
4:30pm

Location: Clearihue A316

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Abstract: Heat flow is one of the most pervasive concepts in mathematics. Understood from antiquity as a solution to a PDE, the mid-Twentieth Century development of stochastic processes gave a new understanding, rigorously completing Einstein's picture of heat flowing through random collisions of particles with aggregate behavior described by (a mean value of) Brownian motion. This picture makes sense on Riemannian manifolds, and led to a revolution in understanding geometry through heat flow.

In this talk, I will discuss some of the standard theory of heat flow on classical Lie groups, focusing on unitary groups *U*_{N} and general linear groups *GL*_{N}. I will then describe my recent work on the large-*N* limits of the Brownian motions on these groups, their fluctuations, and applications to random matrix theory and operator algebras.

Title: New results for the two-stage contact process

Speaker: Eric Foxall, University of Victoria

Date and time:
29 Jan 2014,
3:30pm -
NaN:pm

Location: CLE A303

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Abstract: The two-stage contact process, first introduced by Steve Krone, is a stochastic growth model in which each individual must reach a mature stage before reproducing, and is a natural generalization of the contact process. We first give some background on the contact process, outlining some important results of the 80's and 90's including complete convergence, We then introduce the two-stage process, for which the same results hold. We indicate why this should be so, by showing that the two-stage process possesses the same useful properties as the contact process. Finally, we describe an important difference in the two-stage process, namely the existence of a critical value of the maturation rate below which survival is not possible. This fact is in qualitative contrast to the corresponding two-stage branching process.

Title: A coarse-grained stochastic multi-type particle interacting model for tropical convection: nearest neighbour interactions

Speaker: Boaulem Khouider, University of Victoria

Date and time:
22 Jan 2014,
3:30pm -
4:30pm

Location: CLE A30

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Particle interacting systems on a lattice are widely used to model complex physical processes that occur on much smaller scales than the observed phenomenon one wishes to model. However, their full

applicability is hindered by the curse of dimensionality so that in most practical applications a mean field equation is derived and used. Unfortunately, the mean field limit does not retain the inherent variability of the microscopic model. Recently, a systematic methodology is developed and used to derive stochastic coarse-grained birth-death processes which are intermediate between the microscopic model and the mean field limit, for the case of the one-type particle-Ising system. Here we consider a stochastic multicloud model for organized tropical convection introduced recently to improve the variability in climate models. Each lattice site is either clear sky of occupied by one of three cloud types. In earlier work, local interactions between lattice sites were ignored in order to simplify the coarse graining procedure that leads to a multi-dimensional birth-death process; Changes in probability transitions depend only on changes in the large-scale atmospheric variables. Here the coarse-graining methodology is extended to the case of multi-type particle systems with nearest neighbour interactions and the multi-dimensional birth-death process is derived for this general case. The derivation is carried under the assumption of uniform redistribution of particles within each coarse grained cell given the coarse grained values. Numerical tests show that despite the coarse graining the birth-death process preserves the variability of the microscopic model. Moreover, while the local interactions do not increase considerably the overall variability of the system, they induce a significant shift in the climatology and at the same time boost its intermittency from the build up of coherent patches of cloud clusters that induce long time excursions from the equilibrium state.