Approximating Multi-Dimensional Hamiltonian Flows by Billiards

Abstract

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.

On topological entropy of billiard tables with small inner scatterers

“…For the now so-called Sinai billiard system, he proved that it has positive measure-theoretic (Kolmogorov-Sinai) entropy and is hyperbolic almost everywhere. See also [11,18,32,34] and also [14,26,[29][30][31] for relevant and recent results and references therein. In contrast, rays converge after reflecting from convex boundaries.…”

Stability in High Dimensional Steep Repelling Potentials

“…Fixing ∆t and letting ε → 0, the diagonal periodic orbit γ(t) on the interval [∆t, T − ∆t] approaches the boundary of the billiard domain only once, at t = T /2, hitting the radius-R sphere Γ n+1 in the normal direction. This is a regular reflection, therefore, according to [21] 8 , the flow map from any time moment before the reflection to any moment after the reflection is close to the corresponding map for the billiard flow. The closeness is along with k derivatives of the map (recall that V is C k+1 , k ≥ 1), i.e.…”

“…and (3.7) follows from [21] as explained above (the same results can be achieved by asymptotic integration of equation (3.2), namely following a simplified version of the below construction of C).…”

“…We further assume that V (z) decays sufficiently rapidly for large z (with accordance to the assumptions in [23,21,35]), so there exists some α > 0 such that…”

“…periodic orbits which are bounded away from the singularity set) is typically inherited by steep billiard-like potentials [34,21]. It follows that the Krylov-Sinai instability mechanism indeed controls the smooth dynamics but only for some limited time scale, after which the non-hyperbolic behavior which stems from the billiard singularities will prevail.…”