Are there critical points on the boundaries of singular domains?

Abstract

Generalizing a result of E. Ghys, we prove a general theorem that implies that if a rational functionf of the Riemann sphere of degree ≧2 leaves invariant a singular domainC (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that on $$\bar C$$ f is injective, then each boundary component ofC contains critical point off. The injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off(z)=z n +a, withneℕ,n≧2 andaeℂ. We also show that the singular disks, associated to periodic elliptic points off(z)=e az that satisfy a diophantine condition, are unbounded in ℂ. In the end of the paper, we give a survey of the theory of iteration of entire functions of ℂ.