The M-set of λexp(z)/z has infinite area

Abstract

It is known that the Fatou set of the map $exp\left(z\right)/z$ defined on the punctured plane ${\mathbb{C}}^{\ast }$ is empty. We consider the $M$-set of $\lambda exp\left(z\right)/z$ consisting of all parameters $\lambda$ for which the Fatou set of $\lambda exp\left(z\right)/z$ is empty. We prove that the $M$-set of $\lambda exp\left(z\right)/z$ has infinite area. In particular, the Hausdorff dimension of the $M$-set is 2. We also discuss the area of complement of the $M$-set.