Residual Julia set for functions with the Ahlfors' Property

Abstract

We consider two classes of functions studied by Epstein [A.L. Epstein, Towers of finite type complex analytic maps, Ph.D. thesis, City University of New York, 1993] and by Herring [M.E. Herring, An extension of the Julia–Fatou theory of iteration, Ph.D. thesis, Imperial College, London, 1994], which have the Ahlfors' Property. We prove under some conditions on the Fatou and Julia sets that the singleton buried components are dense in the Julia set for these classes of functions.

Keywords::

• meromorphic functions
• Julia and Fatou sets
• Ahlfors' Property
• residual Julia set
• singleton components

Keywords::

• 37F10
• 30D05

Acknowledgements

We would like to thank the late Professor Baker for giving us the inspiration to produce this paper. We also thank Adam Epstein and especially the referee for the valuable suggestions to improve this paper. The authors were supported by CONACYT (Projects 128005 and 153850).