✎ ebsd2021:gibbscocycle []

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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
\[
	d^F(x,y)
	:= \int_x^y\tilde F
	:= \int_0^{d(x,y)}\tilde F(\phi^t(v))\,dt,
\]
where $v$ is a unit tangent vector such that $\pi(v)=x$ and $\pi(\phi^{d(x,y)}(v))=y$, that is, the curve integral of $\tilde F$ along the geodesic arc from $x$ to $y$. Note that
\[
	d^F(x,y)=d^F(y,x).
\]
Define the //Gibbs cocycle// associated to $F$ by
\[
	(\xi,x,y)
	\mapsto C_\xi^F(x,y)
	:= \lim_{t\to\infty}\int_y^{\gamma(t)}\tilde F-\int_x^{\gamma(t)}\tilde F
	= \lim_{t\to\infty}d^F(y,\gamma(t))-d^F(x,\gamma(t)),
\]
where $\gamma$ is any geodesic ray with $\gamma(\infty)=\xi$.

**Lemma.** The Gibbs cocycle $C^F$ entirely determines $\tilde F$ as for every $v\in T^1X$
\[
	\tilde F(v)
	= \lim_{t\searrow0}C^F_{\gamma_v(\infty)}(\pi(v)),\pi(\phi^t(v)).
\]

**Remark.** Note that for $\tilde F=-1$, this Gibbs cocycle is simply the Busemann cocycle, that is, the horospherical distance $\beta_\xi(\cdot,\cdot)$ from $x$ to $y$ (relative to $\xi$). Note that
\[
	C_\xi^{F-s}(x,y)
	= C_\xi^F(x,y) + s\beta_\xi(x,y).
\]
Moreover, if $\gamma(0)=y$, $\gamma(\infty)=\xi$ and $x\in\gamma$, that is, $x$ is on the geodesic ray from $y$ to $\xi$, then
\[
	C_\xi^F(x,y)
	= \int_x^y\tilde F
\]

**Lemma. [Cocycle properties and $\Gamma$-invariance]** For any $x,y\in X,\xi\in X(\infty),\gamma\in\Gamma$ it holds
  * $C_\xi^F(x,z)=C_\xi^F(x,y)+C_\xi^F(y,z)$,
  * $C_\xi^F(y,x)=-C_\xi^F(x,y)$,
  * $C_{\gamma\xi}^F(\gamma x,\gamma y)=C_\xi^F(x,y)$.

**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 
\[
	\xi\mapsto C^F_\xi(x,y)
\]
is H\"older 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.

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