FUNDAMENTA MATHEMATICAE 186 (2005) Expanding repellers in limit sets for iterations of holomorphic functions by Feliks Przytycki (Warszawa) Abstract. We prove that for Ω being an immediate basin of attraction to an attract- ing fixed point for a rational mapping of the Riemann sphere, and for an ergodic invariant measure μ on the boundary Fr Ω, with positive Lyapunov exponent, there is an invariant subset of Fr Ω which is an expanding repeller of Hausdorff dimension arbitrarily close to the Hausdorff dimension of μ. We also prove generalizations and a geometric coding tree abstract version. The paper is a continuation of a paper in Fund. Math. 145 (1994) by the author and Anna Zdunik, where the density of periodic orbits in Fr Ω was proved. 1. Introduction. Let Ω be a simply connected domain in C and f be a holomorphic map defined on a neighbourhood W of Fr Ω to C. Assume f (W ∩ Ω) ⊂ Ω, f (Fr Ω) ⊂ Fr Ω and Fr Ω repells to the side of Ω, that is,  ∞ n=0 f −n (W ∩ Ω) = Fr Ω. An important special case is where Ω is an immediate basin of attraction of an attracting fixed point for a rational function. This covers also the case of a component of the immediate basin of attraction to a periodic attracting orbit, as one can consider an iterate of f mapping the component to itself. Distances and derivatives are considered in the Riemann spherical metric on C. Let R : D → Ω be a Riemann mapping from the unit disc onto Ω and let g be a holomorphic extension of R −1 ◦ f ◦ R to a neighbourhood of the unit circle ∂ D. It exists and it is expanding on ∂ D (see [P2, Section 7]). We prove the following Theorem A. Let ν be an ergodic g-invariant probability measure on ∂ D such that for ν -a.e. ζ ∈ ∂ D the radial limit  R(ζ ) := lim rր1 R(rζ ) exists. Assume that the measure µ :=  R ∗ (ν ) has positive Lyapunov exponent χ µ (f ). 2000 Mathematics Subject Classification : Primary 37F15; Secondary 37F35, 37D25. Key words and phrases : boundary of basin of attraction, iteration of rational map, Hausdorff dimension, hyperbolic dimension, coding tree, Pesin theory, Katok theory. Supported by Polish KBN grant 2P03A 03425. [85]