Ultralimit - Examples

Examples

1. Let (X,d) be a compact metric space and put (X_n,d_n)=(X,d) for every . Then the ultralimit is isometric to (X,d).
2. Let (X,d_X) and (Y,d_Y) be two distinct compact metric spaces and let (X_n,d_n) be a sequence of metric spaces such that for each n either (X_n,d_n)=(X,d_X) or (X_n,d_n)=(Y,d_Y). Let and . Thus A₁, A₂ are disjoint and Therefore one of A₁, A₂ has ω-measure 1 and the other has ω-measure 0. Hence is isometric to (X,d_X) if ω(A₁)=1 and is isometric to (Y,d_Y) if ω(A₂)=1. This shows that the ultralimit can depend on the choice of an ultrafilter ω.
3. Let (M,g) be a compact connected Riemannian manifold of dimension m, where g is a Riemannian metric on M. Let d be the metric on M corresponding to g, so that (M,d) is a geodesic metric space. Choose a basepoint p∈M. Then the ultralimit (and even the ordinary Gromov-Hausdorff limit) is isometric to the tangent space T_pM of M at p with the distance function on T_pM given by the inner product g(p). Therefore the ultralimit is isometric to the Euclidean space with the standard Euclidean metric.
4. Let be the standard m-dimensional Euclidean space with the standard Euclidean metric. Then the asymptotic cone is isometric to .
5. Let be the 2-dimensional integer lattice where the distance between two lattice points is given by the length of the shortest edge-path between them in the grid. Then the asymptotic cone is isometric to where is the Taxicab metric (or L1-metric) on .
6. Let (X,d) be a δ-hyperbolic geodesic metric space for some δ≥0. Then the asymptotic cone is a real tree.
7. Let (X,d) be a metric space of finite diameter. Then the asymptotic cone is a single point.
8. Let (X,d) be a CAT(0)-metric space. Then the asymptotic cone is also a CAT(0)-space.