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

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.
Comment: After we circulated this preprint, Professor Janina Kotus told us that her former Ph.D. student, Bart{\l}omiej 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