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:
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
The Hénon strange attractor