We develop a paradigm for studying high dimensional far from integrable
smooth
Hamiltonian systems by utilizing the singular billiard limit. The novel
singular perturbation tools thus developed may be used to study the
properties
of the emerging orbits in the smooth flow, even when these limit to
singularities of the corresponding billiard. We utilize these methods to
predict the properties of scattering by smooth steep potentials.
Moreover, we
use these tools to identify a mechanism for the destruction of ergodicity in
some multi-dimensional smooth steep systems that limit to uniformly
hyperbolic
Sinai billiards in arbitrarily large dimension. So far, the proposed
mechanism
is highly symmetric and is not directly linked to the smooth many particle
problem. Nonetheless, this non-intuitive result, which includes an explicit
construction of stable motion in smooth n degrees of freedom systems
limiting
to strictly dispersing n dimensional billiards, is now established.

Joint work with A. Rapoport and D. Turaev.