This is a high-resolution version of Figure 2(c) from the paper "Hyperbolic entire functions with bounded Fatou components" by Bergweiler, Fagella and Rempe-Gillen (https://arxiv.org/abs/1404.0925). 
The Julia set is shown in dark tones of grey; the Julia set is locally connected.
the Fatou set is shown in white. Every Fatou component is a quasidisc.
The map is f(z)=λ(1−cos⁡(z)){\displaystyle f(z)=\lambda (1-\cos(z))}, with λ=4π3{\displaystyle \lambda ={\frac {4\pi }{3}}}.
The superattracting fixed point at z=0{\displaystyle z=0} is the only attracting cycle;
the second critical value λ{\displaystyle \lambda } is mapped to 0{\displaystyle 0} after two iterations.
Own work
L Rempe-Gillen