No invariant line fields on escaping sets of the family \(\lambda e^{iz}+\gamma e^{-iz}\) (Q1948777)

The authors consider the family \(f_{\lambda,\gamma}(z) = \lambda e^{iz}+\gamma e^{-iz}\), where \(\lambda, \gamma \in \mathbb{C}^* = \mathbb{C}\setminus \{0\}\) which contains the sine family \(f_{\lambda} = \lambda \sin z\). For \(\lambda, \gamma \in \mathbb{C}^*\) the escaping set of \(f_{\lambda,\gamma}(z)\) is given by \(f_{\lambda,\gamma} = \{z \in \mathbb{C}: f^k_{\lambda,\gamma} (z) \to \infty\) as \(k \to \infty\}\), where \(f^k_{\lambda,\gamma}\) denotes the \(k\)-th iteration of \(f_{\lambda,\gamma}\). The set \(f_{\lambda,\gamma}\) is completely \(f_{\lambda,\gamma}\) -invariant. The authors give a direct proof of the main theorem of the article which says that for any \(\lambda, \gamma \in \mathbb{C}^*\) the escaping set \(I_{\lambda, \gamma}\) supports no \(f_{\lambda, \gamma}\)-invariant line fields.