Research Interests

I work in an area of mathematical research known as Ergodic Theory,  a subfield of a broad, modern research area, Dynamical Systems.  Dynamical systems provides many theoretical tools that are foundational for other areas of Science and Engineering, including Ordinary Differential Equations (in Biology, Medicine, Chemistry, Astronomy and Electrical Engineering), Partial Differential Equations (in all the above areas, plus Mathematical Physics and Earth Sciences), Symbolic Dynamics and Coding Theory (with deep connections to Computer Science) and so-on.  

Ergodic theory is distinguished within dynamical systems theory by its focus on invariant measures -- these describe the stable statistical properties of the dynamical orbits and the asymptotic or long-range behaviour of the system.  For example, an invariant measure can be used to answer the question:  `What fraction of time, in the long run, does a particular orbit in the system spend in a given region of the state space?'  Ergodic theory is now a mature and vibrant research area

Broadly, my work in this field can be divided into four main categories.

1. Nonsingular transformations and absolutely continuous invariant measures.

In many examples, the state space comes equipped with a natural, but non-invariant measure (such as volume in the case of dynamics on a Riemannian manifold) and one is interested in finding invariant measures that are obtained by integrating a density function against this natural measure.   These invariant densities in ergodic theory play an analogous role to the stationary vectors for Markov chains.  There is an analogue of the Markov Chain transition matrix called the Perron-Frobenius (P-F) operator that lifts the dynamics from the state space to the space of densities, an infinite-dimensional Banach space. Spectral decomposition of the P-F operator separates observables into dynamically relevant features such as invariant densities versus transients.  This spectral approach for P-F operators was developed first for one-dimensional expanding maps; recently the focus of research has moved to higher-dimensional hyperbolic maps.  Since hyperbolic maps are ubiquitous in applications, this is an active and important research area.

2. Open dynamical systems

Open systems extend the notion of traditional dynamical systems by allowing orbits to leave the phase space. Billiard dynamics on a table with a hole is a frequently cited, simple example. The associated P-F operator in this context is sub-Markovian and instead of an invariant density, one looks for a conditionally invariant density that describes the statistics of orbits leaving the system.

3. Intermittent Dynamical Systems

Intermittency refers to decay of transients at a rate slower than would be expected by the local mixing behaviour in the flow (for example, slower than the rate of local divergence of orbits).   Such systems can display sub-exponential decay of correlation by having orbits which occasionally get `stuck' in a relatively small part of the phase space. 

4. Non-autonomous systems and random maps

Up to this point, we have described systems where repeated application of some fixed transformation on the state space generates the deterministic dynamics.  A vastly more realistic situation allows the transformation to vary in some time-dependent way, leading to the notion of a non-autonomous dynamical system.

The mathematical construction is called a skew product and consists of a family of transformations indexed by a second dynamical system (the base or `timing' system) that controls the time-dependence.

When the base is an i.i.d. process (a special kind of measure-preserving transformation)  we call the system a random map.   In this case, one imagines choosing, at each time step, one transformation from the family according to the outcome of a coin flip, for example. Random maps lead to interesting applications of ergodic theory; they allow one to incorporate noise into idealized dynamical systems and/or account for measurement errors introduced when modelling physical systems.  A number of my recent publications study random maps. 

A recent paper with J. Horan and Q. Quas returns to the more general context of a skew
product system, showing that a central result in the theory concerning matrix cocycles
(the multiplicative ergodic theorem) cannot be refined to produce an equivariant block triangularization.

And then there is . . .  industrial and applied mathematics

From time to time I am fortunate to spend some time with genuine applied mathematicians working on real problems that might be of interest even to non-mathematicians!  A couple of recent sample projects: developing efficient computational approaches to find stable (large) molecular structures (with colleagues at the University of Guanajuato and CIMAT, Mexico) and mathematical modelling of forest fire spread (a MITACS funded project in collaboration with mathematicians in Edmonton, London (Ontario) and Montreal).