We study the phenomenon of unlimited energy growth for a classical particle moving in the annular billiard. The model is considered under two different geometrical situations: static and breathing boundaries. We show that when the dynamics is chaotic for the static case, the introduction of a time-dependent perturbation allows that the particle experiences the phenomenon of Fermi acceleration even when the oscillations are periodic.

A simplified version of a time-dependent annular billiard is studied. The dynamics is described using nonlinear maps and we consider two different configurations for the billiard, namely (i) concentric and (ii) eccentric cases. For the concentric case and for a null angular momentum, we confirm that the results for the Fermi-Ulam model are recovered and the particle does not experience the phenomenon of Fermi acceleration. However, on the eccentric case the particle demonstrates unlimited energy gain and Fermi acceleration is therefore observed.

We identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics (RTS), which is based on the concept of Poincar\'e recurrences. The projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near-integrable system, the annular billiard. These microstructures interfere in the effective transport of a particle in the phase space, which can be observed through RTS. This technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea.