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Let us define, analogously to before, the //critical exponent// (relative to $\Gamma$ and $F$) (taking some $c>0$ large enough) by
\[
	\delta=\delta(\Gamma,F)
	\eqdef \limsup_{n\to\infty}\frac1n\log\sum_{n-c\le d(\gamma x,x)\le n}e^{d^F(x,\gamma x)}.
\]
One can show, as before, that the series
\[
	P^F(s,x,y)
	\eqdef \sum_{\gamma\in\Gamma}e^{-sd^F(x,\gamma y)}
\]
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 $\xi\in X(\infty)$,\\[0.2cm]
  * $\displaystyle\{\mu_p^F\colon p\in X\}$ 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
\[
	\mu_{p,x,s}^F
	\eqdef \frac{\sum_{\gamma\in\Gamma}e^{-sd^F(p,\gamma x)}\delta_{\gamma x}}
		{P^F(s,x,x)}.
\]
Choose $s_k\searrow \delta(\Gamma,F)$ and consider a weak$\ast$ limit
\[
	\mu_p^F
	\eqdef \lim_{k\to\infty}\mu_{p,x,s_k}.
\]

**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  $x,y\in\Gamma K$
\[
	\frac{1}{b}\,e^{d^F(x,y)-s d(x,y)}
	\le \mu_x^F\big(\pr_y(B_\rho(x))\big)
	\le b\,e^{d^F(x,y)-s d(x,y)}
\]

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
\[
	B_{e^{-d(x,y)}}\big([x,y)\cap X(\infty)\big)
	\subset \pr_x(B_\rho(y))
	\subset B_{Cd^{-d(x,y)}}([x,y)\cap X(\infty)\big).
\]

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