Convergence of sublinearly contracting horospheres

TL;DR

This paper studies the behavior of horospheres linked to sublinearly contracting geodesic rays in CAT(0) spaces, showing their convergence and the visibility of boundary points defined by such rays.

Contribution

It provides a detailed analysis of horospheres associated with sublinearly contracting geodesics and establishes their convergence and visibility properties in CAT(0) spaces.

Findings

Horospheres associated with sublinearly contracting geodesics are convergent.

Points on the boundary defined by such geodesics are visibility points.

The results apply to all proper complete CAT(0) spaces.

Abstract

In \cite{QR19}, Qing, Rafi and Tiozzo introduced the sublinearly contracting boundary for CAT(0) spaces. Every point of this boundary is uniquely represented by a sublinearly contracting geodesic ray: a geodesic ray $b$ where every disjoint ball projects to a subset whose diameter is bounded by a sublinear function in terms of the ball's distance to the origin. This paper analyzes the bahaviour of horofunctions associated to such geodesic rays, for example, we show that horospheres associated to such horofunctions are convergent. As a consequence of this analysis, we show that for any proper complete CAT(0) space $X$, every point of the visual boundary $\partial X$ that is defined by a sublinearly contracting geodesic ray is a visibility point.