This paper, which is the second of a series of three papers, studies dynamical properties of elements of
Out
(
F
n
)
\operatorname {Out}(F_{\mathtt n})
, the outer automorphism group of a nonabelian free group
F
n
F_{\mathtt n}
. We prove that, for every exponentially growing outer automorphism of
F
n
F_{\mathtt n}
, there exists a preferred compact topological space, the space of currents relative to a malnormal subgroup system, on which
ϕ
\phi
acts by homeomorphism with a North-South dynamics behavior.