Let us define, analogously to before, the critical exponent (relative to $\Gamma$ and $F$) (taking some $c>0$ large enough) by $$\delta=\delta(\Gamma,F) := \limsup_{n\to\infty}\frac1n\log\sum_{n-c\le d(\gamma x,x)\le n}e^{d^F(x,\gamma x)}.$$ One can show, as before, that the series $$P^F(s,x,y) := \sum_{\gamma\in\Gamma}e^{-sd^F(x,\gamma y)}$$ converges for $s>\delta(\Gamma,F)$ and diverges for $s<\delta(\Gamma,F)$, and that these properties and the value of $\delta$ do not depend on $x,y\in X$.

Generalizing the before defined Busemann densities, a family of positive finite measures $\{\mu_p^F\colon p\in X\}$ on $X\cup X(\infty)$ is a Patterson density of dimension $\delta$ (relative to $\Gamma$ and $F$) if it satisfies:

• $\displaystyle\frac{d\mu_x^F}{d\mu_y^F}(\xi)=e^{\displaystyle -C_\xi^{F-\delta}(x,y)}$ for almost every $\xi\in X(\infty)$,
• $\displaystyle\{\mu_p^F\colon p\in X\}$ is $\Gamma$-invariant, that is, for every $p\in X$ and $\gamma\in\Gamma$, it holds

$$\mu_{\gamma p}^F=\gamma_\ast\mu_p^F.$$

A Patterson density can be constructed as follows: Given $x\in X$, $s>\delta$, and $p\in X$, let $$\mu_{p,x,s}^F := \frac{\sum_{\gamma\in\Gamma}e^{-sd^F(p,\gamma x)}\delta_{\gamma x}} {P^F(s,x,x)}.$$ Choose $s_k\searrow \delta(\Gamma,F)$ and consider a weak$\ast$ limit $$\mu_p^F := \lim_{k\to\infty}\mu_{p,x,s_k}.$$

Lemma. If $\delta(\Gamma,F)<\infty$, then there exists at least one Patterson density of dimension $\delta(\Gamma,F)$ which has support $X(\infty)$.

The following Sullivan shadow lemma is a version of the shadow lemma.

Lemma. For every $s\ge \delta(\Gamma,F)$ and any compact set $K\subset X$ there are $\rho=\rho(K)>0$ and $b=b(K)>0$ such that for every $x,y\in\Gamma K$ $$\frac{1}{b}\,e^{d^F(x,y)-s d(x,y)} \le \mu_x^F\big({\rm pr}_y(B_\rho(x))\big) \le b\,e^{d^F(x,y)-s d(x,y)}$$

The shadow of a ball can be described by means of the Bourdon metric. Recall that $X$ is $\delta$-hyperbolic if for all $x,y,z,\omega\in X\cup X(\infty)$ it holds $$(x,z)_\omega \ge \min\{(x,y)_\omega,(y,z)_\omega\}-\delta.$$

Lemma. If $X$ is $\delta$-hyperbolic, then, given $\rho\ge4\delta$ and $C=e^{2\delta+\rho}$, for every $x,y\in X$ it holds $$B_{e^{-d(x,y)}}\big([x,y)\cap X(\infty)\big) \subset {\rm pr}_x(B_\rho(y)) \subset B_{Cd^{-d(x,y)}}([x,y)\cap X(\infty)\big).$$