The Hausdorff dimension of directional edge escaping points set

In this paper, we define the directional edge escaping points set of function iteration under a given plane partition and then prove that the upper bound of Hausdorff dimension of the directional edge escaping points set of \(S(z)=a e^{z}+b e^{-z}\), where \(a, b\in \mathbb{C}\) and \(|a|^{2}+|b|^{2}\neq 0\), is no more than 1.