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--- ebsd2021:gibbscocycle [2021/10/17 10:28] – escola
+++ ebsd2021:gibbscocycle [2021/10/17 10:29] (current) – escola
@@ Line 1: / Line 1: @@
 Let  $M$  be a complete connected Riemannian manifold with pinched negative curvature. Let  $X=\tilde M$  and  $\pi\colon X\to M$  be a covering map with covering group  $\Gamma$ , viewed as a nonelementary discrete group of isometries of  $X$ .
-Let  $F\colon T^1M\to\mathbb R$  be a H\"older continuous potential and consider its  $\Gamma$ -invariant associated potential  $\tilde F\colon T^1X\to\mathbb R$ ,  $\tilde F:= F\circ \pi$ . Assume that  $F$  (and hence  $\tilde F$ ) is //symmetric//, that is, invariant with respect to the antipodal map  $\iota(v):= -v$ . Define
+Let  $F\colon T^1M\to\mathbb R$  be a Hölder continuous potential and consider its  $\Gamma$ -invariant associated potential  $\tilde F\colon T^1X\to\mathbb R$ ,  $\tilde F:= F\circ \pi$ . Assume that  $F$  (and hence  $\tilde F$ ) is //symmetric//, that is, invariant with respect to the antipodal map  $\iota(v):= -v$ . Define
 \[
 	d^F(x,y)
@@ Line 39: / Line 39: @@
 **Lemma. [Cocycle properties and  $\Gamma$ -invariance]** For any  $x,y\in X,\xi\in X(\infty),\gamma\in\Gamma$  it holds
-  * Unordered List Item $C_\xi^F(x,z)=C_\xi^F(x,y)+C_\xi^F(y,z)$ ,
+  *  $C_\xi^F(x,z)=C_\xi^F(x,y)+C_\xi^F(y,z)$ ,
-  * Unordered List Item $C_\xi^F(y,x)=-C_\xi^F(x,y)$ ,
+  *  $C_\xi^F(y,x)=-C_\xi^F(x,y)$ ,
-  * Unordered List Item $C_{\gamma\xi}^F(\gamma x,\gamma y)=C_\xi^F(x,y)$ .
+  *  $C_{\gamma\xi}^F(\gamma x,\gamma y)=C_\xi^F(x,y)$ .
 **Lemma.** The map  $C^F_{(\cdot)}(\cdot,\cdot)\colon X(\infty)\times X\times X\to \mathbb R$  is continuous.
-Moreover, if  $\tilde F$  is bounded (this is satisfied, for example, if  $X/\Gamma$  is compact), then  $C^F$  is locally H\"older continuous, for every  $x,y\in X$  the map
+Moreover, if  $\tilde F$  is bounded (this is satisfied, for example, if  $X/\Gamma$  is compact), then  $C^F$  is locally Hölder continuous, for every  $x,y\in X$  the map
 \[
 	\xi\mapsto C^F_\xi(x,y)
 \]
-is H\"older continuous (in the Bourdon metric), and for every  $\xi\in X(\infty)$  the map
+is Hölder continuous (in the Bourdon metric), and for every  $\xi\in X(\infty)$  the map
 \[
 	(x,y)\mapsto C^F_\xi(x,y)
 \]
-is H\"older continuous.
+is Hölder continuous.