User Tools

Site Tools

Trace: • tema12 • tema7 • tema9 • tema3
ebsd2021:tema9

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
ebsd2021:tema9 [2021/09/22 10:23] – escolaebsd2021:tema9 [2021/09/30 12:19] (current) – escola
Line 44: Line 44:
 $$ $$
 where $d_H$ denotes the Hausdorff distance. In other words, this defines a relation between geodesic rays.  where $d_H$ denotes the Hausdorff distance. In other words, this defines a relation between geodesic rays. 
-Note that being forward asymptotic is an equivalence relation. The set of equivalence classes of asymptotic geodesics is called //sphere at infinity// or also //boundary at infinity// and denoted by $X(\infty)$. Given a geodesic ray $\sigma\colon\mathbb R^+\to X$, denote by $\sigma(\infty)$ the equivalence class it is contained in.+Note that being forward asymptotic is an equivalence relation. The set of equivalence classes of asymptotic geodesics is called //sphere at infinity// or also //boundary at infinity// and denoted by $X(\infty)$. See the previous post (Tema 8). Given a geodesic ray $\sigma\colon\mathbb R^+\to X$, denote by $\sigma(\infty)$ the equivalence class it is contained in.
  
 **Lemma 1.** For every $\xi\in X(\infty)$ and $x\in X$ there is at most one geodesic ray such that $\sigma(0)=x$ and $\sigma(\infty)=\xi$. **Lemma 1.** For every $\xi\in X(\infty)$ and $x\in X$ there is at most one geodesic ray such that $\sigma(0)=x$ and $\sigma(\infty)=\xi$.
Line 68: Line 68:
  
 **Lemma 3.** For every $\xi\in X(\infty)$ and $x\in X$ there is a unique geodesic ray $\sigma_{x,\xi}\colon\mathbb R^+\to X$ such that $\sigma_{x,\xi}(0)=x$ and $\sigma_{x,\xi}(\infty)=\xi$. **Lemma 3.** For every $\xi\in X(\infty)$ and $x\in X$ there is a unique geodesic ray $\sigma_{x,\xi}\colon\mathbb R^+\to X$ such that $\sigma_{x,\xi}(0)=x$ and $\sigma_{x,\xi}(\infty)=\xi$.
 +
 +Note that Lemmas 1-3 correspond Lemma 1 in [[ebsd2021:tema8|]].
  
 **Lemma 4.** Let $\sigma\colon\mathbb R\to\infty$ be a geodesic, $y:=\sigma(0)$, and $\xi=\sigma(\infty)$. Then for every $x\in X$ the limit **Lemma 4.** Let $\sigma\colon\mathbb R\to\infty$ be a geodesic, $y:=\sigma(0)$, and $\xi=\sigma(\infty)$. Then for every $x\in X$ the limit
ebsd2021/tema9.1632316988.txt.gz · Last modified: 2021/09/22 10:23 by escola