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.
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.
- Dewey and Other Naval Commanders.
- Eine Frage von Glück oder Zufall: Roman (German Edition);
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