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

Abstract

Consider the family $f_{\lambda, \gamma}(z) = \lambda e^{iz}+\gamma e^{-iz}$ where $\lambda$ and $\gamma$ are non-zero complex numbers. It contains the sine family $\lambda \sin z$ and is a natural extension of the sine family. We give a direct proof of that the escaping set $I_{\lambda, \gamma}$ of $f_{\lambda, \gamma}$ supports no $f_{\lambda,\gamma}$-invariant line fields.