Julia set of \(\lambda \exp(z)/z\) with real parameters \(\lambda\) (Q1702457)

The author considers the dynamics of the map \(F_\lambda(z)=\lambda e^z/z\) for a real parameter \(\lambda\). It is shown that the Julia set \(J(F_\lambda)\) is different from the whole plane if and only if \(\lambda\in (0,4e^{-2}]\). If \(\lambda=4e^{-2}\), then \(F_\lambda\) has a parabolic point. If \(\lambda\in (0,4e^{-2})\), then \(F_\lambda\) has an attracting fixed point. It is further shown that then \(J(F_\lambda)\) has infinite area and the \(\omega\)-limit set \(\omega(z)\) equals \(\{0,\infty\}\) for almost all \(z\in J(F_\lambda)\).