On the instability of Herman rings
Denote by E the space of rational self-maps, of degree greater than one, of the Riemann sphere, furnished with the \(C^ 0\) topology. A Herman ring R of \(f\in E\) is a component of the complement of the Julia set of f, which is homeomorphic to an annulus and mapped to itself by some iterate \(f^ n\) of f, which is analytically conjugate to a rotation in R. The author shows that those f, which do not possess Herman rings, form an open dense subset of E. This extends the approach of the author, \textit{P. Sad} and \textit{D. Sullivan}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 193-217 (1983; Zbl 0524.58025), to the discussion of whether hyperbolic members are dense in E.
scientific article; zbMATH DE number 3957553