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

TL;DR

This paper investigates the dynamics of exponentially growing outer automorphisms of free groups, demonstrating they act with North-South dynamics on a specific space of currents relative to a malnormal subgroup system.

Contribution

It establishes the existence of a preferred compact space of currents where these automorphisms exhibit North-South dynamics, extending understanding of their dynamical behavior.

Findings

Proves exponential automorphisms act with North-South dynamics

Identifies a space of currents where dynamics are well-behaved

Extends previous work on automorphism dynamics in free groups

Abstract

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