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--- ebsd2021:gibbsstates [2021/10/17 10:36] – created escola
+++ ebsd2021:gibbsstates [2021/10/17 10:36] (current) – escola
@@ Line 1: / Line 1: @@
-We first recall the //Hopf parametrization// of  $T^1X$  in terms of  $X(\infty)$ . Given  $v\in T^1M$ , consider its unique geodesic  $\gamma=\gamma_v$  and let $v_-\eqdef \gamma(-\infty) $and$ v_+\eqdef\gamma(\infty)$.
+We first recall the //Hopf parametrization// of  $T^1X$  in terms of  $X(\infty)$ . Given  $v\in T^1M$ , consider its unique geodesic  $\gamma=\gamma_v$  and let $v_-:= \gamma(-\infty) $and$ v_+:=\gamma(\infty)$.
 Fix some  $x_0\in X$ . We can identify  $T^1X$  by
 \[
-	\big(X(\infty)\times X(\infty)\setminus\Delta\big)\times\bR,
+	\big(X(\infty)\times X(\infty)\setminus\Delta\big)\times\mathbb R,
 	\quad
 	(v_-,v_+,t),
 \]
-where $t\in\bR $is the algebraic distance between$ \gamma(0) $and the closed point of$ \gamma(\bR) $to$ x_0$.
+where $t\in\mathbb R $is the algebraic distance between$ \gamma(0) $and the closed point of$ \gamma(\mathbb R) $to$ x_0$.
 Define the // $F$ -gap seen from  $x$  between  $\xi$  and  $\eta$ // by
 \[
 	D^F_x(\xi,\eta)
-	\eqdef e^{ \lim_{t\to\infty} \frac12\big(\int_x^{\eta_t}\tilde F-\int_{\xi_t}^{\eta_t}\tilde F
+	:= e^{ \lim_{t\to\infty} \frac12\big(\int_x^{\eta_t}\tilde F-\int_{\xi_t}^{\eta_t}\tilde F
 			+\int_{\xi_t}^x\tilde F\big)},
 \]
 where  $t\mapsto\eta_t,t\mapsto\xi_t$  are geodesic rays with ends  $\eta,\xi$ , respectively. This defines the //gap map//
 \[
-	X\times X(\infty)\times X(\infty)\setminus\Delta\to\bR,\quad
+	X\times X(\infty)\times X(\infty)\setminus\Delta\to\mathbb R,\quad
 	(x,\xi,\eta)\mapsto D^F_x(\xi,\eta).
 \]
-**Remark.** If  $F\equiv -1$  then  $D^F_x=d_x$  is the visual distance ou Bourdon metric. For every $s\in\bR$ it holds
+**Remark.** If  $F\equiv -1$  then  $D^F_x=d_x$  is the visual distance ou Bourdon metric. For every $s\in\mathbb R$ it holds
 \[
 	D^{F-s}_x