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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