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 $$\big(X(\infty)\times X(\infty)\setminus\Delta\big)\times\mathbb R, \quad (v_-,v_+,t),$$ 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 $$D^F_x(\xi,\eta) := e^{ \lim_{t\to\infty} \frac12\big(\int_x^{\eta_t}\tilde F-\int_{\xi_t}^{\eta_t}\tilde F +\int_{\xi_t}^x\tilde F\big)},$$ where $t\mapsto\eta_t,t\mapsto\xi_t$ are geodesic rays with ends $\eta,\xi$, respectively. This defines the gap map $$X\times X(\infty)\times X(\infty)\setminus\Delta\to\mathbb R,\quad (x,\xi,\eta)\mapsto D^F_x(\xi,\eta).$$

Remark. If $F\equiv -1$ then $D^F_x=d_x$ is the visual distance ou Bourdon metric. For every $s\in\mathbb R$ it holds $$D^{F-s}_x = D^F_x\cdot d_x^s.$$

Let $\{\nu_x^F\}$ be the Patterson density of dimension $\delta$ for $(\Gamma,F)$. Fix $x_0\in X$ some base point, which provides the Hopf parametrization. The Gibbs measure on $T^1X$ associated with $\{\mu_x^F\}_{x\in X}$ is $$dm(v) =\frac{d\mu_{x_0}(v_-)\,d\mu_{x_0}(v_+)\,dt} {D^{F-\delta}_{x_0}(v_-,v_+)}.$$