A rational map-T of degree not less than two is known tp preserve a measure, called the con formal measure, equivalent to the Hausdprff measure of the sanio dimension as its Julia set J and supported there, with respect to which it is ergodic and even exact. As a consequence of Birkhoffs pointwise ergodic theorem almost every zin f with, respect to ihe coni'oimal measure has an orbit that is asymptotically distributed on J with respect to this measure. As a counterpoint to this, the following result is established in this paper. Let 2(z) = Qj-(z) denote the closure of the set [T"(z) : n = 1.2,...}. any expanding rational map T of degree at least two we set S(zo)={zJ:zoT(z)}. We show tluit for all z() ihe Hausdorff dimensions of S(i) and J are equal.
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