This is a high-resolution version of Figure 3 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 map is f(z)=3−cos⁡(arcosh2⁡3+π2)z2−arcosh2⁡33+1{\displaystyle f(z)={\frac {3-\cos {\sqrt {\left(\operatorname {arcosh} ^{2}3+\pi ^{2}\right)z^{2}-\operatorname {arcosh} ^{2}3}}}{3+1}}}. This function has three critical values and two fixed Fatou components, one of which is a bounded Jordan domain, while the other is unbounded with non-locally-connected boundary. The fixed points 0 and 1 are superattracting; the basin of 0 is depicted in white and the basin of 1 in light grey. Non-periodic critical points are marked by asterisks.
Own work
L Rempe-Gillen