Density of repelling fixed points in the Julia set of a rational or entire semigroup

Abstract

We briefly survey several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, in particular, focusing on those methods which can be extended to the case of semigroups. We then present an elementary proof that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points.

Keywords:

• complex dynamics
• Julia sets
• random dynamics
• dynamics of semigroups

AMS Subject Classification::

• 37F10
• 37F50
• 30D05

Acknowledgements

The author would like to thank both Hiroki Sumi and the referee for their careful reading and helpful comments regarding this manuscript.

Notes

1. It is important to note, however, that in [5] Bergweiler does provide elementary proofs of the special cases of the key results of the Ahlfors theory which are used in complex dynamics.