Let $F$ be a rational map of degree at least 2 and $x \in \mathcal J(F)$ be a point in the Julia set of $F$. Define

$n(x, T) = \# \bigl \{ (n \ge 0, \, y \in X) \, : \, F^{\circ n}(y) = x, \, \log |(F^{\circ n})'(y)| < T\bigr \}$.

Assume $F$ admits a conformal measure $m$ of dimension $\alpha$ and an absolutely continuous invariant probability measure $\mu = \gamma \, dm$ with a positive Lyapunov exponent. We show that if we exclude a small list of exceptional rational maps, then

$n(x, T) \sim e^{\alpha T} \cdot \frac{\gamma(x)}{\alpha \int_{X} \log |F'(x)| d\mu}, \qquad \text{as }T \to \infty$.

for $\mu$ a.e.~$x \in X$. Using a rigidity theorem of Eremenko and van Strien, we establish a variant of the above result that also incorporates the argument of the derivative, as in the work by Oh and Winter.