Abstract: Denote by HD(J(f))the Hansdorff dimension of the Julia set J(f)of a rational function f.Our first result asserts that if f is an NCP map,and f_n→f horocyclically, preserving sub-critical relations,then f_n is an NCP map for all n》0 and J(f_n)→J(f)in the Hausdorff topology.We also prove that if f is a parabolic map and fn is an NCP map for all n》0 such that f_n→f horocyclically,then J(f_n)→J(f) in the Hansdorff topology, and HD(J(f_n))→HD(J(f)). 

Key words: Julia set, Hausdorff dimension, Markov partition, conformal measure