In the previous chapter, we discussed the fundamental concepts for the study of two-dimensional discrete maps. This chapter is dedicated to the study of a system described by a two-dimensional discrete map known in the literature as the Fermi accelerator model, or simply the Fermi-Ulam model. We will present the historical context of the problem and then derive the equations that describe the dynamics of the model. We will also discuss some properties of the phase space, including a stability analysis of the fixed points. This chapter focuses solely on the description of the problem in the absence of dissipation, so that the phase space exhibits mixed characteristics with chaotic seas, invariant spanning curves, and stability islands. We will show that the chaotic sea exhibits the property of being scale invariant with respect to the system’s control parameter.