**MSC:**- 30C85 Capacity and harmonic measure in the complex plane, See also {31A15}
- 31A15 Potentials and capacity, harmonic measure, extremal length, See also {30C85}

denotes the Hausdorff dimension of the associated Julia sets.

We derive a formula

which describes in a uniform way the scaling of this measure

at arbitrary elements of the Julia set.

Further\-more, we establish the Khintchine Limit Law for parabolic rational maps (the analogue

of the `logarithmic law for geodesics' in the theory of Kleinian groups),

and show that this law provides some efficient control for

the fluctuation of the $h$-conformal measure.

We then show that these results lead to some refinements of the description of this measure

in terms of Hausdorff and packing measures with respect to some gauge functions.

Also, we derive a simple proof of

the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally,

we obtain that the conformal measure is a regular doubling measure, we show that

its Renyi dimension and its information dimension is equal to $h$, and we compute

its logarithmic index.