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).