On nonuniqueness of geodesics in asymptotic Teichm\"uller space

In an infinite-dimensional Teichm\"uller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichm\"uller space $AT(\Delta)$. We introduce the notion of substantial point and non-substantial point in $AT(\Delta)$. The set of all non-substantial points is open and dense in $AT(\Delta)$. It is shown that there are infinitely many geodesics joining a non-substantial point to the basepoint. Although we have difficulty in dealing with the substantial points, we give an example to show that there are infinitely many geodesics connecting certain substantial point and the basepoint. It is also shown that there are always infinitely many straight lines containing two points in $AT(\Delta)$. Moreover, with the help of the Finsler structure on the asymptotic Teichm\"uller space, a variation formula for the asymptotic Teichm\"uller metric is obtained.