Abstract: Let f, g be two parabolic maps of degree ≥ 2. HD(J) denotes the Hausdorff dimension of the Julia set J and m f and m g denote the t-conformal measure supported on the Julia set J(f) and J(g) respectively. In this paper we show that if J(f) and J(g) are locally connected and f and g topologically conjugate, then HD(J(f)) = HD(J(g)), m_g = m_foh^-1 .

Key words: Julia set, net, conformal measure, Hausdorff dimension