  Once recognizing that point particles moving inside the extended version of
the rippled billiard perform L\'evy flights characterized by a L\'evy-type
distribution $P(\ell)\sim \ell^{-(1+\alpha)}$ with $\alpha=1$, we derive a
generalized two-dimensional non-linear map $M_\alpha$ able to produce L\'evy
flights described by $P(\ell)$ with $0<\alpha<2$. Due to this property, we name
$M_\alpha$ as the L\'evy Map. Then, by applying Chirikov's overlapping
resonance criteria we are able to identify the onset of global chaos as a
function of the parameters of the map. With this, we state the conditions under
which the L\'evy Map could be used as a L\'evy pseudo-random number generator
and, furthermore, confirm its applicability by computing scattering properties
of disordered wires.
