Certain Julia sets include smooth components

Abstract

The Julia set F* of the map z → ỹ(z) = z²-μ may be the boundary of an atom, of a molecule, or of a “devil’s polymer” in the z-plane. Denote the boundary of one of the atoms of F* by H. When μ ≠ 0 is the nucleus of a cardioid-shaped atom of the M-set, it is conjectured that the fractal dimension D of H is 1. Thus, H may be a be a rectifiable curve (of well defined length) or perhaps only a borderline fractal curve (of logarithmically diverging length). This paper comments on a clearer version of Figure 5 of M19831{C5} and develops a remark made there, but not very explicitly.