Riemann map and holomorphic dynamics

Abstract

f ( F r A ) = F r A and ~ f " ( U c ~ c l A ) = F r A (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: D Z ~ A be a Riemann map (a conformal homeomorphism) of the unit disc onto A. Then there exists a holomorphic extension g of R l o f o 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 ~ S l , 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: