  We present two proofs of the fact, originally due to Reiner Martin, that any
fully irreducible hyperbolic element of $Out(F_N)$ acts on the projectivized
space of geodesic currents $\mathbb{P}Curr(F_N)$ with uniform north-south
dynamics. The first proof, using purely train-track methods, provides an
elaborated and corrected version of Reiner Martin's original approach. The
second proof uses the geometric intersection form of Kapovich and Lustig and
relies on unique ergodicity results from symbolic dynamics.
