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Dynamics of transcendental meromorphic functions (([image omitted])/([image omitted]))[image omitted] having rational Schwarzian derivatives

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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

DOI: http://dx.doi.org/10.1080/14689360701208131

Affiliations: Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur - 208 016, India

Publication date: September 1, 2007

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