Dynamics Seminar Spring 2013

The Dynamics seminar will meet on Friday afternoons at 2:30 in DSB C116. All are welcome. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title
11/1/2013 Eric Foxall (UVic) Counting Paths on a Directed Graph
18/1/2013 Anthony Quas (UVic) Multiplicative ergodic theorems and Ulam's method
25/1/2013 Chris Bose (UVic) Maximum entropy approach to open dynamical systems
1/2/2013 Ian Putnam (UVic) A homology theory for Smale spaces
8/2/2013 Terry Soo(UVic) A Krieger generator theorem for toral automorphisms
15/2/2013   No seminar: Reading Break - epsilon
22/2/2013   No seminar: Reading Break
1/3/2013   Seminar? AQ away 
7/3/2013 Reinhard Illner (UVic) Fun with hard sphere collisions
15/3/2013 Felipe Garcia Ramos (UBC) Measure theoretical continuity and equicontinuity, and weak convergence of measures
19/3/2013 Alexander Holroyd (Microsoft) Colloquium: Random Matching
22/3/2013 YitWah Cheung (San Francisco State) Ergodic theory of billiards in rational polygons
5/4/2013 John Antonioli (UVic) Factor maps and compensation functions

Date: 11/1/2013
Speaker: Eric Foxall (UVic)
Title: Counting Paths on a Directed Graph
Abstract: We consider an arbitrary directed graph on finitely many vertices, on which each edge is assigned a positive length and a positive weight.  The asymptotic growth is given for the weighted count of paths of length at most t, as t approaches infinity.
Date: 18/1/2013
Speaker: Anthony Quas (UVic)
Title: Multiplicative ergodic theorems and Ulam's method
Abstract: Ulam's method is a very successful and practical method for computing densities of absolutely continuous invariant measures of transformations. In this talk, we recall our recent work on Multiplicative ergodic theorems and indicate how this work can be used to give a version of Ulam's method for non-autonomous dynamical systems. This is joint work with Gary Froyland and Cecilia Gonzalez-Tokman.
Date: 25/1/2013
Speaker: Chris Bose (UVic)
Title: Maximum entropy approach to open dynamical systems. 
Abstract: An open dynamical system is a measure preserving transformation on a state space with a 'hole'. As orbits enter the hole, they are lost from the system. Example: a billiard table.  Typically, Lebesgue almost every orbit enters the hole eventually, so there can be no absolutely continuous invariant measure. Instead we look for a quasi-invariant measure, meaning, a measure stationary after conditioning on the state of not being in the hole.  In this setting one has a Perron-Frobenius operator and P-F equation whose solution gives the density of a quasi-invariant measure. 

Borrowing shamelessly from Anthony's careful setup last week in the closed dynamics setting, I hope to describe the basic functional analysis setup and a fundamentally different method for calculating (= approximating) the quasi-invariant density using optimization instead of discretization and projection.  

Keywords:  Open dynamics, Perron-Frobenius operator, density, maximum entropy principle, moment constraints.

Date: 1/2/2013
Speaker: Ian Putnam
Title: A homology theory for Smale Spaces
Abstract: In the 1960's, Stephen Smale initiated an ambitious program to understand the dynamics of a certain class maps on smooth manifolds, which he called Axiom A. David Ruelle gave a definition of a Smale space to describe, in purely topological terms, the dynamics of an Axiom A system on its non-wandering set. I will discuss these definitions and give several concrete examples. Later, Anthony Manning proved that the zeta function for such systems was rational and Rufus Bowen conjectured that this was due to the existence of an underlying homology theory. A partial solution to this was given by Krieger and also Bowen and Franks when they constructed a very beautiful invariant for shifts of finite type. I will discuss this invariant and then show how it can be extended to all Smale spaces as a homology theory, as predicted by Bowen.
Date: 8/2/2013
Speaker: Terry Soo (UVic)
Title: A Krieger generator theorem for toral automorphisms
Abstract: The Krieger generator theorem says that every invertible measure- preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. In joint work with Anthony Quas, we extend Krieger’s theorem to include toral automorphisms (which are not necessarily hyperbolic) to give a positive answer to a question of Lind and Thouvenot.
Date: 1/3/2013
Speaker:
Title:
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Date: 7/3/2013 (note unusual Date and Place: Cornett A-128)
Speaker: Reinhard Illner (UVic)
Title: Fun with hard sphere collisions
Abstract: Recently, Peter Dukes alerted me to a way of computing the digits of Pi via a "billiard" game. I will use this remarkable process to introduce the collision transformation, discuss its properties and provide a geometric proof that the number of collisions in a system of N hard spheres (any N, arbitrary diameters and masses) in all space is always finite. This answers a question originally asked by Sinai.
Date: 15/3/2013
Speaker: Felipe Garcia Ramos (UBC)
Title: Measure theoretical continuity and equicontinuity, and weak convergence of measures
Abstract: The main objects of interest of this talk are topological dynamical systems on measure metric spaces that are equicontinuous with respect to a measure. We will talk about different notions of continuity and equicontinuity with respect to a measure and characterize them. We will see how to check for weak convergence of sequences of measures of a certain family.
Date: 22/3/2013
Speaker: YitWah Cheung (San Francisco State)
Title: Ergodic theory of billiards in rational polygons
Abstract: In this talk, I will discuss the problem of studying  billiard trajectories in a polygon whose interior angles are  rational multiples of pi.  This leads naturally to the study of  translation surfaces, which is another term for the notion  of a holomorphic 1-form on a (closed) Riemann surface.   The moduli space of such objects admits an action of  SL(2,R) with a finite invariant measure that can be used  to give information about the dynamics of billiard trajectories  in the original polygon; for example, a theorem of  Kerckhoff-Masur-Smillie tells us that for almost every  direction theta, every trajectory with initial direction theta  will be uniformly distributed.  In this talk, I will describe some  refinements of the general technique of exploiting dynamics  on the moduli space to obtain information about dynamics of  billiard trajectories.  This talk is intended for a general audience.
For previous semesters, see
Spring 2006
Fall 2006
Spring 2007
Spring 2008
Spring 2009
Spring 2010
Spring 2011
Spring 2012