Generalized north-south dynamics on the space of geodesic currents

Abstract

We prove uniform north-south dynamics type results for the action of $\varphi\in Out(F_{N})$ on the space of projectivized geodesic currents $\mathbb{P}Curr(S)=\mathbb{P}Curr(F_{N})$, where $\varphi$ is induced by a pseudo-Anosov homeomorphism on a compact surface S with boundary such that $\pi_{1}(S)=F_{N}$. As an application, we show that for a subgroup $H\le Out(F_N)$, containing an iwip, either $H$ contains a hyperbolic iwip or $H$ is contained in the image in $Out(F_N)$ of the mapping class group of a surface with a single boundary component.