Evolution of the Liouville density of a chaotic system

An area-preserving map of the unit sphere, consisting of alternating twists and turns, is mostly chaotic. A Liouville density on that sphere is specified by means of its expansion into spherical harmonics. That expansion initially necessitates only a finite number of basis functions. As the dynamical mapping proceeds, it is found that the number of non-negligible coefficients increases exponentially with the number of steps. This is in contrast to the behavior of a Schr\"odinger wave function, which requires, for the analogous quantum system, a basis of fixed size. {} 1996 The American Physical Society.