We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere. We define the Poincaré exponent
δ
(
f
,
z
)
=
inf
{
α
≥
0
:
P
(
z
,
α
)
≤
0
}
\delta (f,z)=\inf \{ \alpha \ge 0: \mathcal {P}(z, \alpha ) \le 0\}
, where
P
(
z
,
α
)
:=
lim sup
n
→
∞
1
n
log
∑
f
n
(
x
)
=
z
|
(
f
n
)
′
(
x
)
|
−
α
.
\begin{equation*} \mathcal {P}(z, \alpha ):=\limsup _{n\to \infty }{1\over n}\log \sum _{f^n(x)=z} |(f^n)’(x)|^{- \alpha }. \end{equation*}
We prove that
δ
(
f
,
z
)
\delta (f,z)
and
P
(
z
,
α
)
\mathcal {P}(z, \alpha )
do not depend on
z
z
, provided
z
z
is non-exceptional.
P
\mathcal {P}
plays the role of pressure; we prove that it coincides with the Denker-Urbański pressure if
α
≤
δ
(
f
)
\alpha \le \delta (f)
. Various notions of “conical limit set" are considered. They all have Hausdorff dimension equal to
δ
(
f
)
\delta (f)
which is equal to the hyperbolic dimension of the Julia set and also equal to the exponent of some conformal Patterson-Sullivan measures. In an Appendix we also discuss notions of “conical limit set" introduced recently by Urbański and by Lyubich and Minsky.