The results of the previous chapter are not the last word about Sullivan conformal measures. Left alone, these measures would be a kind of curiosity. Their true power, meaning, and importance come from their geometric characterizations and their usefulness, one could even say indispensability, in understanding geometric measures on Julia sets, i.e., their Hausdorff and packing $h$-dimensional measures, where, we recall, $h=\HD(J(f))$. This is fully achieved in the present chapter. Having said that, this chapter can be viewed from two perspectives. The first is that we provide therein a geometrical characterization of the $h$-conformal measure $m_h$, which, with the absence of parabolic points, turns out to be a normalized packing measure, and the second is that we give a complete description of geometric, Hausdorff, and packing measures of the Julia sets $J(f)$. Owing to the fact that the Hausdorff dimension of the Julia set of an elliptic function is strictly larger than $1$, this picture is even simpler than for nonrecurrent rational functions.

In this chapter, we use the fruits of the, already proven, existence of Sullivan conformal measures with a minimal exponent and its various dynamical characterizations. Having compact nonrecurrence, we are able to prove in this chapter that this minimal exponent is equal to the Hausdorff dimension $\HD(J(f))$ of the Julia set $J(f)$, which we denote by $h$. We also obtain here some strong restrictions on the possible locations of atoms of such conformal measures. In the last section of this chapter, we culminate our work on Sullivan conformal measures for elliptic functions treated on their own. There and from then onward, we assume that our compactly nonrecurrent elliptic function is regular (we define this concept). For this class of elliptic functions, we prove the uniqueness and atomlessness of $h$-conformal measures along with their first fundamental stochastic properties such as ergodicity and conservativity.