## 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).