Dynamics of a family of transcendental meromorphic functions having rational Schwarzian derivative

In the present paper, a class F of critically finite transcendental meromorphic functions having rational Schwarzian derivative is introduced and the dynamics of functions in one parameter family K ≡ {fλ(z) = λf (z): f ∈ F, z ∈ Cˆ and λ>0} is investigated. It is found that there exist two parameter values λ∗ = φ(0) > 0 and λ∗∗ = φ( ˜x) > 0, where φ(x) = x f (x) and ˜x is the real root of φ (x) = 0, such that the Fatou sets of fλ(z) for λ = λ∗ and λ = λ∗∗ contain parabolic domains. A computationally useful characterization of the Julia set of the function fλ(z) as the complement of the basin of attraction of an attracting real fixed point of fλ(z) is established and applied for the generation of the images of the Julia sets of fλ(z). Further, it is observed that the Julia set of fλ ∈ K explodes to whole complex plane for λ > λ∗∗. Finally, our results found in the present paper are compared with the recent results on dynamics of one parameter families λ tan z, λ ∈ Cˆ \ {0} [R.L. Devaney, L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. 22 (4) (1989) 55–79; L. Keen, J. Kotus, Dynamics of the family λ tan(z), Conform. Geom. Dynam. 1 (1997) 28–57; G.M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. 49 (1994) 281–295] and λez−1 z , λ>0 [G.P. Kapoor, M. Guru Prem Prasad, Dynamics of (ez − 1)/z: The Julia set and bifurcation, Ergodic Theory Dynam. Systems 18 (1998) 1363–1383]. © 2006 Elsevier Inc. All rights reserved