Consider a family of smooth potentials V ε , which, in the limit ε → 0, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as ε → 0 to the original billiard, and provide asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials which limit to the multi-dimensional close to ellipsoidal billiards, we predict when the separatrix splitting persists for various types of potentials.

ABSTRACT. The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep is established for any finite n. Furthermore, the stability regions in the parameter space scale as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity properties of this limit system are destroyed in the more realistic smooth setting. The considered example is highly symmetric and is not directly linked to the smooth many particle problem. Nonetheless, the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established.

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It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.

Consider a classical two-dimensional scattering problem: a ray is scattered by a potential composed of several tall, repelling, steep mountains of arbitrary shape. We study when the traditional approximation of this nonlinear far-from-integrable problem by the corresponding simpler billiard problem, of scattering by hard-wall obstacles of similar shape, is justified. For one class of chaotic scatterers, named here regular Sinai scatterers, the scattering properties of the smooth system indeed limit to those of the billiards. For another class, the singular Sinai scatterers, these two scattering problems have essential differences: though the invariant set of such singular scatterers is hyperbolic (possibly with singularities), that of the smooth flow may have stable periodic orbits, even when the potential is arbitrarily steep. It follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards' deviation from a singular scatterer. Thus, even in this singular case, the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow. Finally, we see that corners have nontrivial and significant impact on the scattering functions.