  The global macroscopic behaviour of a dynamical system is encoded in the
eigenfunctions of a certain transfer operator associated to it. For systems
with low dimensional long term dynamics, efficient techniques exist for a
numerical approximation of the most important eigenfunctions, cf. DeJu99a. They
are based on a projection of the operator onto a space of piecewise constant
functions supported on a neighborhood of the attractor - Ulam's method.
  In this paper we develop a numerical technique which makes Ulam's approach
applicable to systems with higher dimensional long term dynamics. It is based
on ideas for the treatment of higher dimensional partial differential equations
using sparse grids. We develop the technique, establish statements about its
complexity and convergence and present two numerical examples.
