We prove that has at least two disjoint and infinite critical orbits in the Julia set if it has a Herman ring.

This result is sharp in the following sense: there exists a cubic rational map having exactly two critical grand orbits but also having a Herman ring. In particular, has no Herman rings if it has at most one infinite critical orbit in the Julia set.

## Holomorphic Dynamics

These criterions derive some known results about the rational maps without Herman rings. References [Enhancements On Off] What's this? Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. Derrida , L. Itzykson , Fractal structure of zeros in hierarchical models , J. Devaney , Daniel M. Look , and David Uminsky , The escape trichotomy for singularly perturbed rational maps , Indiana Univ. Devaney and E. Theory Appl. Lyubich , Dynamics of rational transformations: topological picture , Uspekhi Mat. Nauk 41 , no.

- AMS :: Conformal Geometry and Dynamics of the American Mathematical Society.
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- Eine Frage von Glück oder Zufall: Roman (German Edition);

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