Critical Behavior of Complex Logarithmic Map

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It is exhibited, for a complex logarithmic map Z ↦ln Z + C , that there exists a k -period region near the point with Ω=1/ k on the boundary curve of the cycloid C =exp ( i2πΩ)- i2πΩ where a Hopf bifurcation takes place, and that a circle map is embedded into the curve. The Devil's staircase emerges just within the boundary and the Farey tree is constructed. The area of the k -period region in a periodic sequence 2→3→4→··· is predicted to be scaled by k -6 for large k , and which is also numerically ascertained.