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

Abstract

In [16] Masur proved the existence of a dense geodesic in the moduli space for a surface. 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.