This chapter relies on ideas of the proof of the rigidity theorem drafted by D. Sullivan in the Proceedings of Berkeley's International Congress of Mathematicians in 1986: see [Sullivan 1986]. In Chapter 7, Example 7.1.10 shows that two expanding repellers can be Lipschitz conjugate, but not analytically (nor even differentially) conjugate.

So in Chapter 7 we provided an additional invariant, the scaling function for an expanding repeller in the line, taking ‘gaps’ into account, and proved that it determined the C1+ε-structure.

In this chapter, following Sullivan, we distinguish a class of conformal expanding repellers (CERs) called non-linear, and prove that the class of equivalence of the geometric measure, and in particular the class of Lipschitz conjugacy, determines the conformal structure.

This is amazing: a holomorphic structure preserved by a map is determined by a measure.

Equivalent notions of linearity

Definition Consider a CER (X, f) for compact X ⊂ ℂ. Denote by Jf the Jacobian of f with respect to the Gibbs measure μX equivalent to a geometric measure mX on X. We call (X, f) linear if one of the following conditions holds:

1. (a) The Jacobian Jf, is locally constant.

2. (b) The function HD(X) log∣f′∣ is co-homologous to a locally constant function on X.

3. (c) The conformal structure on X admits a conformal affine refinement so that f is affine (that is, there exists an atlas {φt} that is a family of conformal injections ϕt: Ut → ℂ where ∪tUt ⊃ X such that all the maps ϕtϕs−1 and ϕtfϕs−1 are affine).

This book is an introduction to the theory of iteration of expanding and non-uniformly expanding holomorphic maps and topics in geometric measure theory of the underlying invariant fractal sets. Probability measures on these sets yield information on Hausdorff and other fractal dimensions and properties. The book starts with a comprehensive chapter on abstract ergodic theory, followed by chapters on uniform distance-expanding maps and thermo-dynamical formalism. This material is applicable in many branches of dynamical systems and related fields, far beyond the applications in this book.

Popular examples of the fractal sets to be investigated are Julia sets for rational functions on the Riemann sphere. The theory, which was initiated by Gaston Julia [1918] and Pierre Fatou [1919–1920], has become very popular since the publication of Benoit Mandelbrot's book [Mandelbrot 1982] with beautiful computer generated illustrations. Top mathematicians have since made spectacular progress in the field over the last 30 years.

Consider, for example, the map f(z)= z2 for complex numbers z. Then the unit circle S1 = {∣z∣ =1} is f-invariant, f(S1)= S1 = f−1(S1). For c ≈ 0, c ≠ 0 and fc(z)= z2 + c, there still exists an fc-invariant set J(fc) called the Julia set of fc, close to S1, homeomorphic to S1 via a homeomorphism h satisfying the equality f ∘ h = h ∘ fc. However, J(fc) has a fractal shape.

In this chapter we shall consider a compact metric space X with an open, distance-expanding map T on it, embedded isometrically into a smooth Riemannian manifold M. We shall assume that T extends to a neighbourhood U of X to a mapping f of class C1+ε for some 0 < ε ≤ 1 or smoother, including real-analytic. C1+ε and more general Cr+ε for r = 1, 2, … means that the r-th derivative is Hölder continuous with the exponent ε for ε < 1 and Lipschitz continuous for ε = 1. We shall also assume that there exists a constant λ > 1 such that for every x ϵ U and for every non-zero vector v tangent to M at x, ∥Df(v)∥ > λ∥v∥ holds, where ∥·∥ is the norm induced by the Riemannian metric. The pair (X, f) will be called an expanding repeller and f an expanding map. If f is of some class A, e.g. Cα or analytic, we shall say that the expanding repeller is of that class, or that this is an A-expanding repeller. In particular, if f is conformal we call (X, f) a conformal expanding repeller, abbreviated to CER. Finally, if we skip the assumption that T = f∣x is open on X, we shall call (X, f) an expanding set. Sometimes, to distinguish the domain of f, we shall write (X, f, U).

In Sections 6.2 and 6.3 we provide some introduction to conformal expanding repellers, studying the transfer operator, postponing the main study to Chapters 9 and 10, where we shall use tools of geometric measure theory.

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.