### §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 z*_{j} *z*_{j – 1}, lim_{j→∞}