Dynamics of transcendental meromorphic functions (([image omitted])/([image omitted]))[image omitted] having rational Schwarzian derivatives
Abstract:The dynamics of one parameter family of transcendental meromorphic functions [image omitted], for > 0, a fix 0>0 and [image omitted], that have rational Schwarzian derivatives, is investigated in the present article. A computationally useful characterization of the Julia set of h(z) as complement of the basin of attraction of an attracting real fixed point of h(z) is proved and applied for computer generation of the images of the Julia sets of h(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 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(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 (ez - 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.].
Document Type: Research Article
Affiliations: Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur - 208 016, India
Publication date: September 1, 2007