  We study billiard dynamics inside an ellipse for which the axes lengths are
changed periodically in time and an $O(\delta)$-small quartic polynomial
deformation is added to the boundary. In this situation the energy of the
particle in the billiard is no longer conserved. We show a Fermi acceleration
in such system: there exists a billiard trajectory on which the energy tends to
infinity. The construction is based on the analysis of dynamics in the phase
space near a homoclinic intersection of the stable and unstable manifolds of
the normally hyperbolic invariant cylinder $\Lambda$, parameterised by the
energy and time, that corresponds to the motion along the major axis of the
ellipse. The proof depends on the reduction of the billiard map near the
homoclinic channel to an iterated function system comprised by the shifts along
two Hamiltonian flows defined on $\Lambda$. The two flows approximate the
so-called inner and scattering maps, which are basic tools that arise in the
studies of the Arnold diffusion; the scattering maps defined by the projection
along the strong stable and strong unstable foliations $W^{ss,uu}$ of the
stable and unstable invariant manifolds $W^{s,u}(\Lambda)$ at the homoclinic
points. Melnikov type calculations imply that the behaviour of the scattering
map in this problem is quite unusual: it is only defined on a small subset of
$\Lambda$ that shrinks, in the large energy limit, to a set of parallel lines
$t=const$ as $\delta\to 0$.
