# Dynamics seminar

Title: Dynamical systems modulo universal null sets

Speaker: Mike Boyle, Maryland, and Distinguished PIMS Visitor, UBC

Date and time:
02 Jul 2014,
3:00pm -
4:00pm

Location: DSB C108

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The systems we consider are homeomorphisms of compact metric spaces, shifts of finite type, countable state Markov shifts, surface diffeomorphisms which are C^{1+} (have Holder continuous derivative), and isomorphisms of standard Borel spaces.

The "almost-Borel" viewpoint is to consider a system with respect to all its invariant Borel probabilities, neglecting sets which have measure zero for all of them. We consider issues around the classification of systems in this category, and give complete (or close to complete) results for the Markov shifts and C^{1+} surface diffeomorphisms.

The basis for this comes from Mike Hochman, who showed that various familiar systems have a universality property: all Borel systems of smaller entropy can be embedded into them (modulo those universal null sets). Such a system has a universal part (for this category), supporting its nonatomic ergodic measures of less than maximum entropy, and a top part supporting the measures of maximum entropy. The strictly universal part is uniquely determined up to Borel isomorphism (modulo the universal null sets). Thus the entropy and the almost-Borel structure of the top part determine the system up to almost-Borel isomorphism. The universal part is fantastically complicated, but it is homogeneous. The idiosyncratic part at the top is sometimes easily classified.

We extend Hochman's results to more general systems. This requires considering universality and entropy relative to periods. From Hochman's previous work and proper definitions, one gets a complete almost-Borel classification of finite entropy countable state Markov shifts (NOT necessarily irreducible).

The main payoff: up to a set (often empty) supporting some zero entropy measures, every C^{1+} surface diffeomorphism is almost Borel isomorphic to a countable state Markov shift. This result rests on the remarkable symbolic dynamics of Omri Sarig, and an analysis of their factors under this symbolic dynamics with respect to measures with entropy "maximal at a period".

One of the basic open questions about generalizing these results arises from the work of Quas and Soo.

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.