### §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 *Df*_{z} at every point *z* *J* satisfies the inequality

for every nonzero vector *v* in the tangent space *T*_{z}. (Notation as in the proof of Theorem 2.11.) Since ...