Meromorphic functions whose action on their Julia sets is non-ergodic

Abstract

We study transcendental meromorphic functions having two prepole asymptotic values and no critical values. We prove that these functions acting on their Julia sets are non-ergodic, which illustrates the antithesis of the Keen-Kotus result in [KK2] on the ergodicity of another subfamily of functions with two asymptotic values and no critical values.

After we circulated this preprint, Professor Janina Kotus told us that her former Ph.D. student, Bartłomiej Skorulski, has studied a more general family than the one in this paper. Therefore, we want to withdraw our paper. A reader who is interested in this interesting subject can refer to Skorulski's paper. We would like to thank Professor Janina Kotus for kindly pointing this to us