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

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§13. Most Periodic Orbits Repel

This section will prove the following theorem of Fatou.

Theorem 13.1. Let f: imageimage be a rational map of degree d ≥ 2. Then f has at most a finite number of cycles which are attracting or indifferent.

By a cycle we will mean simply a periodic orbit of f. Recall that a cycle is called attracting, indifferent, or repelling according to whether its multiplier λ satisfies |λ| < 1, |λ| = 1, or |λ| > 1. We will see in §14 that there always exist infinitely many repelling cycles. Shishikura [1987] has given the sharp upper ...

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