My main interest is the study of ergodic theory and its applications.
Below an overview of the basic ideas in ergodic theory is given,
followed by some specific research interests.
See also my papers/preprints here
for formal details of results.

** Basic ideas. **
Given a discrete time map f:X-> X, of a space X into itself,
and preserving a measure m (so that m(f^{-1}}A)=m(A) for all subsets
A &sub X) ergodic theory provides a mechanism of understanding the limiting
behaviour of the time averages:

for some given x in X and as time n goes to infinity . Here Q:X->R denotes a real-valued observation on the system (for example an indicator function). We refer to a (f,X,m) ergodic system as one which is dynamically indecomposable: all f-invariant sets in X have zero m-measure. This implies that m-almost all orbits in X are dense. For purposes of applications, an important result is the Birkhoff Ergodic Theorem which states that space and time averages are equal. That is:

** Simple examples of ergodic systems. ** The coin tossing map
(or Bernoulli map) f:[0,1]->[0,1] given by f(x)=2x (mod 1) is ergodic
and the invariant measure is Lebesgue measure. Another example is an
irrational rotation of the circle, i.e the map
r:[0,1]->[0,1] given by r(x)=x+&alpha (mod 1), with &alpha an irrational number.

** Mixing ergodic systems. **
A mixing dynamical system is one for which future observations performed on
the system are (asymptotically) independent of those performed in the past.
Formally we say (f,X,m) is a mixing system if

dx/dt = &sigma (y-x)

dy/dt = &rho x - y - xz

dz/dt = xy - &beta z.

The Lorenz strange attractor

Related problems for which I am active in working on include understanding the rate of mixing for the Lorenz flow itself, and whether the Lorenz flow satisfies any statistical limit laws (such as the Central Limit Theorem). One can ask the same questions for other parameters, such as those close to &rho=32. In this parameter regime the attractor is not robust: slight changes in parameter destroy the strangeness. However it is hoped that "strangeness" together with the associated ergodic properties are preserved for a positive measure subset of these parameters.

The Hénon strange attractor

Related open problems would be to understand the ergodic properties of non-smooth Hénon-like maps. Such maps arise naturally as Poincaré maps in nonlinear ODE's (for example the Lorenz equations).