Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

We study billiard dynamics inside an ellipse for which the axes lengths arechanged periodically in time and an O(δ)-small quartic polynomialdeformation is added to the boundary. In this situation the energy of theparticle in the billiard is no longer conserved. We show a Fermi accelerationin such system: there exists a billiard trajectory on which the energy tends toinfinity. The construction is based on the analysis of dynamics in the phasespace near a homoclinic intersection of the stable and unstable manifolds ofthe normally hyperbolic invariant cylinder Λ, parameterised by theenergy and time, that corresponds to the motion along the major axis of theellipse. The proof depends on the reduction of the billiard map near thehomoclinic channel to an iterated function system comprised by the shifts alongtwo Hamiltonian flows defined on Λ. The two flows approximate theso-called inner and scattering maps, which are basic tools that arise in thestudies of the Arnold diffusion; the scattering maps defined by the projectionalong the strong stable and strong unstable foliations Wss,uu of thestable and unstable invariant manifolds Ws,u(Λ) at the homoclinicpoints. Melnikov type calculations imply that the behaviour of the scatteringmap in this problem is quite unusual: it is only defined on a small subset of Λ that shrinks, in the large energy limit, to a set of parallel lines t=const as δ→0.