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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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7. Smooth Julia Sets

Most Julia sets tend to be complicated fractal subsets of image, but there are three exceptions: D. H. Hamilton has shown that every Julia set which is a 1-dimensional topological manifold must be either a circle or closed line segment up to Möbius automorphism, or must have Hausdorff dimension strictly greater than 1. (See Hamilton [1995].) If we count the entire Riemann sphere as another smooth example, it follows that, up to automorphism, there are only three possible smooth subsets of image which can be Julia sets of rational functions ...

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