Chaotic dynamical systems associated with tilings of ${\mathbb{R}}^N$

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of ${\mathbb{R}}^N$, whose most familiar example is provided by the $N-$dimensional torus ${\text{T}}^N$. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.