Conformal expanding repellers (abbreviation CERs) have already been defined in Chapter 6, and some basic properties of expanding sets and repellers in dimension one were discussed in Section 6.2. A more advanced geometric theory in the real one-dimensional case was covered in Chapter 7.

Now we have a new tool: the Frostman Lemma and related facts from Chapter 8. Equipped with the theory of Gibbs measures, and with the pressure function, we are able to develop a geometric theory of CERs, with Hausdorff measures and dimension playing the crucial role. We shall present this theory for C1+ε conformal expanding repellers in ℝd. The main case of our interest will be d = 2. Recall (Section 6.2) that the assumed conformality forces for d = 2 that f is holomorphic or anti-holomorphic, and for d ≥ 3 that f is locally a Möbius map. Conformality for d = 1 is meaningless, so we assume C1+ε in order to be able to rely on the Bounded Distortion for Iteration lemma.

We shall outline a theory of Gibbs measures from the point of view of multifractal spectra of dimensions (Section 9.2) and pointwise fluctuations due to the Law of Iterated Logarithm (Section 9.3).

For d = 2 we shall apply this theory to study the boundary Fr Ω of a simply connected domain Ω, and in particular a simply connected immediate basin of attraction to a sink for a rational mapping of the Riemann sphere.