Lodge's cubicThis map is the cubic map f(z) = 3z2/(2z3+1), which is discussed at length in Russell Lodge's thesis, "Boundary values of the Thurston pullback map" (Indiana University, 2012). The map had been considered previously by Buff-Epstein-Koch-Pilgrim.
The Julia set for this cubic is shown in the figure below, which was created from Mathematica.
The diagram for Lodge's cubic map
Input and output files