Dynamics of transcendental meromorphic functions (()/()) having rational Schwarzian derivatives

Abstract

The dynamics of one parameter family of transcendental meromorphic functions , for λ > 0, a fix μ0>0 and , that have rational Schwarzian derivatives, is investigated in the present article. A computationally useful characterization of the Julia set of h λ(z) as complement of the basin of attraction of an attracting real fixed point of h λ(z) is proved and applied for computer generation of the images of the Julia sets of h λ(z). It is observed that for functions in our family bifurcations in the dynamics occur at three real parameter values, while for the family of functions λ tan z investigated in [Devaney, R. L., and Keen, L., 1989, Dynamics of meromorphic maps: Maps with polynomial schwarzian derivative. Annales Scientifiques de l'Écol1 Noemale Supérieure, 22(4), 55–79.], bifurcation in the dynamics occurs at just one real parameter value. Further, it is found that explosion in the Julia sets of h λ(z) occurs for certain ranges of parameter values. Our results found here are compared with recent results in [Devaney, R. L., and Keen, L., 1989, Dynamics of meromorphic maps: Maps with polynomial schwarzian derivative. Annales Scientifiques de l'Écol1 Normale Supérieure, 22(4), 55–79.; Devaney, R. L., and Tangerman, F., 1986, Dynamics of entire functions near the essential singularity. Ergodic Theory and Dynamical Systems, 6, 489–503; Kapoor, G. P., and Prasad, M. G. P., 1998, Dynamics of (e z − 1)/z: the Julia set and bifurcation. Ergodic Theory and Dynamical Systems, 18(6):1363–1383; Gwyneth Stallard, M., 1994, The Hausdorff dimension of julia sets of meromorphic functions. Journal of London Mathematical Society, 49(2), 281–295.].