A zero–infinity law for well-approximable points in Julia sets

Abstract

Let T:J \to J be an expanding rational map of the Riemann sphere acting on its Julia set J and f:J\to \mathbb{R} denote a Hölder continuous function satisfying f(x) > \log|T^\prime(x)| for all x in J. Then for any point z_0 in J define the set D_{z_0}(f) of ‘well-approximable’ points to be the set of points in J which lie in the Euclidean ball

B\bigg(y,\exp\bigg(-\sum_{i=0}^{n-1} f(T^iy)\bigg)\bigg)

for infinitely many pairs (y,n) satisfying T^n(y)=z_0. In our 1997 paper, we calculated the Hausdorff dimension of D_{z_0} (f). In the present paper, we shall show that the Hausdorff measure \mathcal{H}^s of this set is either zero or infinite. This is in line with the general philosophy that all ‘naturally’ occurring sets of well-approximable points should have zero or infinite Hausdorff measure.