Convergence of sublinearly contracting horospheres

Qing and Rafi (Morse boundaries I: CAT(0) spaces, 2019. arXiv:1909.02096) introduced the sublinearly contracting boundary for spaces. Every point of this is uniquely represented by a geodesic ray: ray b where every disjoint ball projects to subset whose diameter bounded sublinear function in terms ball’s distance origin. This paper analyzes bahaviour horofunctions associated such rays, example, we show that horospheres are convergent. As consequence analysis, any proper space X, visual \(\partial X\) defined visibility point.