lnvenL math. 85, 439-455 (1986) II/l ve~l tiovle$ mathematicae (C~ Springer-Verlag 1986 Riemann map and holomorphic dynamics Feliks Przytycki Institute of Mathematics, PolishAcademyof Sciencesul. Sniadeckich 8, PL-00-950 Warszawa, Poland wO. The main results Let A be an open, simply connected domain (a topological disc) in I~ (A =4= (r). Assume there exists a holomorphic mapping f defined on a neighbourhood U of FrA in the Riemann sphere $2= ~ w{oo), f: U~S 2, such that f(Uc~A)=A, f(FrA)=FrA and ~ f-"(Uc~clA)=FrA (i.e. FrA repels to the side of A). n=O (Assume that U is always sufficiently small so that f has no singularities in U c~ A.) Let R: DZ~A be a Riemann map (a conformal homeomorphism) of the unit disc onto A. Then there exists a holomorphic extension g of R-lofo R to a neighbourhood of S 1 and gls, happens to be an expanding map, i.e. there exists n>0 such that for every z~Sl, l(g")'(z)[>l. (Although the proofs are straightforward, we include them for sceptics in the last section, Sect. 7, together with a remark about examples of A and f satisfying the above assumptions.) Denote by _gthe non-tangential limit of R and by ~K= S 1 the domain where it exists. Denote by ~IJl (g) the space of all Borel, probability g-invariant, ergodic, positive entropy measures on S 1. It is known (see [P]) that for every/~ EgJI (g), K exists ~- almost everywhere, so the f-invariant measure R. (/~) on FrA can be considered. The functions log[g'[, log lf'[ are /z-, respectively R,(/0-integrable (see 1.5). Denote these integrals by )~u (g), g~.(~)(f)- (In this paper we consider derivatives with respect to the Riemannian metric on $2.) Define the Hausdorff dimension of any probability measure v: HD (v) = inf {HD (Y)" v (Y) --= 1 }, where HD (Y) denotes the Hausdorff dimension of the set Y. If v is preserved by a map q~, h v (q0 denotes entropy. ~rheorem 1. For every measure Iz ~1 (g),for p-almost every z ~ S 1, and for x ~D 2 ,~onverging non-tangentially to z, there exists the limit: log [R' (x) [ (R) (z) = lira nontan x~z -loglx-z[