A zero–infinity law for well-approximable points in Julia sets
نویسنده
چکیده
Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J and f : J → R denote a Hölder continuous function satisfying f (x) > log |T ′(x)| for all x in J . Then for any point z0 in J define the set Dz0(f ) of ‘well-approximable’ points to be the set of points in J which lie in the Euclidean ball B ( y, exp ( − n−1 ∑ i=0 f (T y) )) for infinitely many pairs (y, n) satisfying T (y) = z0. In our 1997 paper, we calculated the Hausdorff dimension ofDz0(f ). In the present paper, we shall show that the Hausdorff measureHs 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.
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