Dynamics of a family of non-critically finite even transcendental meromorphic functions

The dynamics of one parameter family of non-critically finite even transcendental meromorphic function $\xi_{\lambda}(z) = \lambda \frac{\sinh^2 z}{z^4}, \lambda > 0$ is investigated in the present paper. It is shown that bifurcations in the dynamics of the function $\xi_{\lambda}(x)$ for $x \in \mathbb{R} \backslash {0}$ occur at two critical parameter values $\lambda = \frac{x_1^5}{\sinh^2 x_1}(\approx 1.26333)$ and $\lambda = \frac{\tilde{x}^5} {\sinh^2 \tilde{x}}(\approx 2.7.715)$, where $x_1$ and $\tilde{x}$ are the unique positive real roots of the equations $\tanh x = \frac{2 x}{3}$ and $\tanh x = \frac{2x}{5}$ 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 \frac{e^z-1}{z} $[14].