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

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§19. Hyperbolic and Subhyperbolic Maps

This section will describe some examples of locally connected Julia sets, using arguments due to Sullivan, Thurston, Douady, and Hubbard.

Definition. A rational map f will be called dynamically hyperbolic if f is expanding on its Julia set J in the following sense: There exists a conformal metric μ defined on some neighborhood of J, such that the derivative Dfz at every point z image J satisfies the inequality

Image

for every nonzero vector v in the tangent space Tz. (Notation as in the proof of Theorem 2.11.) Since ...

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