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

Abstract

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

110 pages, 2 figures