North-South dynamics of hyperbolic free group automorphisms on the space of currents

TL;DR

This paper demonstrates that hyperbolic automorphisms of free groups exhibit predictable North-South dynamics on the space of geodesic currents, extending understanding of their geometric and dynamical properties.

Contribution

It proves that such automorphisms act with generalized uniform North-South dynamics on the space of projectivized geodesic currents, under certain train track conditions.

Findings

Automorphisms exhibit North-South dynamics on geodesic currents

The dynamics are uniform and generalized for hyperbolic automorphisms

Results extend the understanding of automorphism actions on free group boundaries

Abstract

Let $\varphi$ be a hyperbolic outer automorphism of a non-abelian free group $F_N$ such that $\varphi$ and $\varphi^{-1}$ admit absolute train track representatives. We prove that $\varphi$ acts on the space of projectivized geodesic currents on $F_N$ with generalized uniform North-South dynamics.