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On the instability of Herman rings. (English) Zbl 0595.30032

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, P. Sad and 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.
Reviewer: I.N.Baker

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics

Citations:

Zbl 0524.58025
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References:

[1] Ahlfors, L.: Lectures on quasiconformal mappings. Van Nostrand Co. 1966 · Zbl 0138.06002
[2] Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. Éc. Norm. Super.,16, 193-217 (1983) · Zbl 0524.58025
[3] Sullivan, D.: Quasi conformal homeomorphisms and dynamics I, II, III. (To appear in Ann. Math.)
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