Abstract
The dynamics of one parameter family of non-critically finite even transcendental meromorphic function xi(lambda)(z) = lambda sinh(2)z/z(4), lambda > 0 is investigated in the present paper. It is shown that bifurcations in the dynamics of the function xi(lambda)(x) for x is an element of R \ {0} occur at two critical parameter values lambda = x(1)(5)/sinh(2)x(1) (approximate to1.26333) and lambda = (x) over tilde (5)/sinh2 (x) over tilde (approximate to2.7.715), where x(1) and (x) over tilde are the unique positive real roots of the equations tanh x = 2x/3 and tanh x = 2x/5 respectively. For certain ranges of parameter values of lambda, it is proved that the Julia set of the function xi(lambda)(z) contains both real and imaginary axes. The images of the Julia sets of xi(lambda)(z) are computer generated by using the characterization of the Julia set of xi(lambda)(z) as the closure of the set of points whose orbits escape to infinity under iterations. Finally, our results are compared with the recent results on dynamics of (i) critically finite transcendental meromorphic functions lambda tan z having polynomial Schwarzian Derivative [10, 15, 19] and (ii) non-critically finite transcendental entire functions lambda e(z)-1/z [14].