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

## TL;DR

This paper demonstrates Fermi acceleration in a time-dependent elliptical billiard with boundary deformation, showing that particle energy can grow unboundedly through complex phase space dynamics involving homoclinic intersections and scattering maps.

## Contribution

It introduces a novel analysis of billiard dynamics near homoclinic intersections using scattering maps and phase space reduction, revealing mechanisms for energy growth in perturbed elliptical billiards.

## Key findings

- Existence of trajectories with unbounded energy growth.
- Analysis of scattering maps with unusual behavior at high energies.
- Application of Melnikov calculations to billiard dynamics.

## Abstract

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 $\delta\to 0$.