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

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×m matrices is dense in the positive cone Rm C (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.