Dynamics of non-critically finite odd transcendental meromorphic function \(\lambda\,\frac{\sinh z}{z^2}\) (Q2572645)

Let \(f\) denote a transcendental meromorphic function. A singular value of \(f\) is defined to be either a critical value or an asymptotic value of \(f\). \(f\) is called critically finite if it has only finitely many critical values and asymptotic. Otherwise, \(f\) is called non-critically finite. The dynamics of these two types of functions are quite different. For instance, if \(f\) is critically finite, then \(f\) has no wandersing domain and Baker domain, while for a non-critically finite function, it may have wandering domian and Baker domain. In this paper, the dynamics of a one-parameter family \(T_\lambda=\{\lambda\sin hz/z^2,\lambda> 0\}\) has been studied. It is shown that the set of critical points of any \(f\in T_\lambda\) consists of infinitely many pure imaginary numbers and at most two real numbers. Also according to an extended Denjoy-Carleman-Ahlfors theorem, any \(f\in T_\lambda\) has at most two asymptotic values. Thus the set of singular values of any \(f\in T_\lambda\) is bounded and we say \(f\) is of bounded type. It follows that for any such a function \(f\), \(J(f)= \overline{(I(f))}\), where \(I(f)= \{z\in C\mid f^n(z)\to\infty\) as \(n\to\infty\) but \(f^n(z)\neq\infty\), where \(f^n\) is the \(n\)th iterate of \(f\}\) and is called the escaping points of \(f\). This fact, is useful to identify the Julia set \(J(f)\) and generate the computer graphics of \(J(f)\). Moreover, comparisons of dynamics between the family \(T_\lambda\) and family of non-critically finite transcendental entire functions \(E_\lambda= \{\lambda(e^z- 1)/z,\lambda> 0\}\) are made. Remarks: The techniques, agruements used in the present paper can be found in the paper ``Dynamics of a family of non-critical finite even transcendental meromorphic functions'' (written by the same author), Regular and Chaotic Dynamics 9, No. 2, 1--20 (2004). Also, in the same paper, more detailed comparisons of dynamics among several types of one-parameter familities of transcendental entire and meromorphic functions are listed.