A dense geodesic ray in the $Out(F_r)$-quotient of reduced Outer Space

TL;DR

This paper establishes the existence of a dense geodesic in the reduced Outer Space with the Lipschitz metric, paralleling Masur's theorem in Teichmüller space, and explores related convergence and eigenvector density results.

Contribution

It proves the existence of a dense geodesic in reduced Outer Space and shows the weak convergence of Brun's unordered algorithm, with implications for eigenvector density.

Findings

Existence of a dense geodesic in reduced Outer Space.

Weak convergence of Brun's unordered algorithm.

Density of Perron-Frobenius eigenvectors in positive cones.

Abstract

In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichm\"uller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron-Frobenius eigenvectors of positive integer $m \times m$ matrices is dense in the positive cone $\mathbb{R}^m_+$ (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the boundary of Outer Space is the limit of a flow line.