10 - Sullivan's classification of conformal expanding repellers

Summary

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 C^1+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 {φ_t} that is a family of conformal injections ϕ_t: U_t → ℂ where ∪_tU_t ⊃ X such that all the maps ϕ_tϕ_s⁻¹ and ϕ_tfϕ_s⁻¹ are affine).