A dense geodesic ray in the \(\mathrm{Out}(F_r)\)-quotient of reduced outer space (Q1650825)

Summary: In [Ann. Math. Stud. 97, 417--438 (1981; Zbl 0476.32027)] \textit{H. 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 \(\mathbf{R}^m_+\) (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.