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

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§14. Repelling Cycles Are Dense in J

We saw in Lemma 4.6 that every repelling cycle is contained in the Julia set. A much sharper statement was proved in quite different ways by both Fatou and Julia, and both proofs are given below. Using our terminology, it reads as follows.

Theorem 14.1. The Julia set for any rational map of degree ≥ 2 is equal to the closure of its set of repelling periodic points.

Proof following Julia. According to Corollary 12.7 to the Rational Fixed Point Formula, every rational map f of degree 2 or more has either a repelling fixed point or a parabolic fixed point with λ = 1. In either case, by Lemmas 4.6 and 4.7 this fixed point belongs to the Julia set J(f).

Figure 31. A homoclinic orbit with zj zj – 1, limj→∞

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