North-South type dynamics of relative atoroidal automorphisms of free groups on a relative space of currents

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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.