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L etT : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J and f : J → R denote a Holder 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

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