Dynamics Seminar Spring 2012

The Dynamics seminar will meet on Friday afternoons at 3:30 in DSB C126. All are welcome. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title
13/1/2012 Cecilia González Tokman(UVic) A semi-invertible operator Oseledets theorem
20/1/2012    
27/1/2012 Anthony Quas Alpern theorems for higher-dimensional flows
3/2/2012 Cecilia González Tokman Ulam's method for systems with holes
10/2/2012   No seminar
17/2/2012   No seminar: Reading Break
20/2/2012 Ronnie Pavlov (Denver) (Note different date) 
2/3/2012 Siri-Malén Høynes Toeplitz dynamical systems and their K-theory
9/3/2012 Antoine Julien (UVic) Metric properties of tiling space
16/3/2012 Susie Wieler (UVic) Hyperbolic dynamical systems via inverse limits
23/3/2012 Tom Meyerovitch (UBC) Stationary Markov random fields and Gibbs measures
3/4/2012 Wael Bahsoun (Loughborough) Metastability of Interval Maps Note in DSB C112

Date: 13/1/2012
Speaker: Cecilia González Tokman (UVic)
Title: A semi-invertible operator Oseledets theorem
Abstract: Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting-- a decomposition into generalized Jordan blocks-- for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding maps randomly chosen from a set of cardinality of the continuum.

This is joint work with Anthony Quas.

Date: 27/1/2012
Speaker: Anthony Quas
Title: Alpern theorems for higher-dimensional flows
Abstract: We discuss Alpern theorems (a generalization of Rokhlin's lemma) for R^d actions in which the `towers' are rectangular boxes of prescribed sizes, proving both sufficient and necessary conditions on the number of boxes. (joint work with Bryna Kra and Ayşe Şahin).
Date: 3/2/2012
Speaker: Cecilia González Tokman
Title: Ulam's method for systems with holes
Abstract: Ulam's method provides a way of approximating absolutely continous invariant measures for dynamical systems. These approximations come from fixing a finite partition of the phase space and encoding the dynamics in an inter-set transition matrix. We review some of the existing results regarding convergence of these approximations as the diameter of the partition goes to zero, and present a (non-perturbative) extension of these results to the context of open systems, or systems with holes. (Joint work with C. Bose, G. Froyland and R. Murray.)
Date: 20/2/2012
Speaker: Ronnie Pavlov (Denver)
Title:
Abstract:
Date: 2/3/2012
Speaker: Siri-Malén Høynes
Title: Toeplitz dynamical systems and their K-theory
Abstract: We will show that the family of Toeplitz systems can be associated to simple dimension groups with non-trivial rational subdimension groups. Furthermore, we will present a class of examples which has a particularly nice Bratteli diagram presentation.
Date: 9/3/2012
Speaker: Antoine Julien (UVic)
Title: Metric properties of tiling space
Abstract: I will describe how the usual distance on tiling space provides a nice approach for some properties of the tiling, namely its complexity. In some example, the complexity and the Hausdorff dimension of the space are closely related.

I will also describe some results on bi-Lipschitz embedding which were obtained with Jean Savinien and Jean Bellissard.

Date: 16/3/2012
Speaker: Susie Wieler (UVic)
Title: Hyperbolic dynamical systems via inverse limits
Abstract: A Smale space is a dynamical system with canonical coordinates of contracting and expanding directions.  The basic sets for Smale’s Axiom A systems are a key class of examples.  We will give a brief overview of R.F. Williams' work on expanding attractors. In this case, the local stable sets are totally disconnected and the local unstable sets  are Euclidean. Williams gave a description of such a system as a stationary inverse limit, where the space in the limit is a branched manifold. We consider Smale spaces with totally disconnected local stable sets and no restrictions on the unstable sets. We give criteria on a topological stationary inverse limit which ensures the result is a Smale space and give examples. We also prove that any such Smale space is obtained through this construction.
Date: 23/3/2012
Speaker: Tom Meyerovitch (UBC)
Title: Stationary Markov random fields and Gibbs measures
Abstract: Markov random fields are higher-dimensional processes with a conditional independence, which can be viewed as analogs of Markov-chains in ine dimension. The Hammersley-Clifford theorem states that Markov Random fields are Gibbs measures with a nearest neighbor interaction, under an assumption on the support: the existence of a ``safe symbol''. In this talk I will present joint work (in progress) with Nishant Chandgotia, Guangyue Han, Brian Marcus and Ronnie Pavlov, investigating Markov random fields without assuming a ``safe symbol'' in the support.
Date: 3/4/2012
Speaker: Wael Bahsoun (Loughborough)
Title: Metastability of Interval Maps
Abstract: Gonzalez-Tokman, Hunt and Wright studied a metastable expanding system which is described by a piecewise smooth and expanding interval map. It is assumed that the metastable map has two invariant sub-intervals and exactly two ergodic invariant densities. Due to small perturbations, the system starts to allow for infrequent leakage through subsets (called holes) of the initially invariant sub-intervals, forcing the two invariant sub-systems to merge into one perturbed system which has exactly one invariant density. It is proved that the unique invariant density of the perturbed interval map can be approximated by a particular convex combination of the two invariant densities of the original interval map, with the weights in the combination depending on the sizes of the holes. In this talk I will present analogous results in two cases: 1. intermittent interval maps; 2. Randomly perturbed expanding maps. In the random case, if time permits, I will also present random versions of Keller-Liverani escape rate formulae. This is a joint work with Sandro Vaienti.   
For previous semesters, see
Spring 2006
Fall 2006
Spring 2007
Spring 2008
Spring 2009
Spring 2010
Spring 2011