Certain dynamical aspects of a family fλ(z) =λ ez/z+1 for z ∈ C when λ < 0

Abstract

We study the change of dynamics of transcendental meromorphic functions f? = ? ez/z+1 for z ? C when ? varies on the negative real axis. It is shown that there is a ?^ such that the Fatou set of f? is empty for ? &lt; ?^ whereas the Fatou set is an invariant parabolic basin corresponding to a real rationally indifferent fixed point x^ if ? = ?^ . In fact, the Fatou set is an invariant attracting basin of a real negative fixed point ?? if ?^ &lt; ? &lt; 0. Also the dynamics of fn? for n ? 2 at the fixed points is investigated for different values of ?. As a generalization of f?, we observed some dynamical issues for the class of entire maps F?,a,m(z) = ?(z+a)m exp(z) where ?, a ? C and m ? N.