Vered Rom-Kedar and Dmitry Turaev
Big islands  in dispersing billiard-like potentials.
(685K, Postscript)

ABSTRACT.  We derive a rigorous estimate of the
 size of islands (in both phase space and parameter space) appearing in
smooth Hamiltonian approximations of scattering billiards.
The derivation includes the construction of a local return map near
 singular periodic orbits for an arbitrary scattering billiard and for
 the  general smooth billiard potentials.
Thus, {\it universality} classes for the local behavior
are found. Moreover, for
all scattering geometries and for many
 types of natural potentials which limit to the
 billiard flow as a parameter $\eps \goto 0$,  islands of
 {\it polynomial} size
in $\eps$ appear. This suggests that the loss of ergodicity via the
introduction of the physically relevant effect of smoothening of the potential
in modeling, for example, scattering molecules, may be of physically
noticeable effect.