A STUDY OF THE DYNAMICS OF THE FAMILY λ

Abstract

In this paper, the dynamical behavior of a family of non- critically finite
transcendental meromorphic function fλ (z) =λ m
m
x
x
2
sinh , λ >0 and m is an even
natural number is described. The Julia set of fλ (z) , as the closure of the set of
points with orbits escaping to infinity under iteration, is obtained. It is observed that,
bifurcation in the dynamics of fλ (z) occurs at two critical parameter values λ = λ1,
λ2, where λ1 =
1
2 1
1
sinh x
x
m
m+
and λ2 =
2
2 1
2
sinh x
x
m
m+
with x1 and x2 are the unique
positive real roots of the equations tanh x =
2m−1
mx and tanh x =
2m +1
mx respectively.