By Heinz Lüneburg

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4 Hyperbolicity In this section we give several equivalent definitions of hyperbolic rational maps, displaying some of the properties that make these dynamical systems especially well-behaved. 9(Ergodic or attracting) to show the Julia set of a hyperbolic map has measure zero. 13 (Characterizations of hyperbolicity) Let f be a rational m a p of degree greater than one. T h e n the following conditions are equivalent. 1. T h e postcritical set P ( f ) is disjoint from the Julia set J ( f ) . 2. There are no critical points or parabolic cycles i n the Julia set.

2. A parabolic basin: there is a parabolic periodic point w E a U and f " p ( z ) + w for all z i n U . 3. A Siegel disk: the component U is a disk o n which fp acts by a n irrational rotation. 4. A Herman ring: the component U is a n annulus, and again f P acts as a n irrational rotation. Remarks. The classification of periodic components of the Fatou set is contained in the work of Fatou and Julia. The existence of rotation domains was only established later by work of Siegel and Herman, while the proof that every component of the Fatou set is Preperiodic was obtained by Sullivan [Su131.

Let y be a path of length d ( x 1 , x 2 )joining xl to 22. 22, the injectivity radius r ( x ) is bounded below along y in terms of d ( x l ,x 2 ) and r ( x l ) . By the preceding result, we obtain an upper bound on llNf (x)11 along y. The integral of this bound controls 1 Df ( x l )- D f (x2)),and thereby the ratio 1) f ' ( x l ) ) ) /f )' )( ~ 2 ) ) ) . Chapter 3 Dynamics of rational maps This chapter reviews well-known features of the topological dynamics of rational maps, and develops general principles to study their measurable dynamics as well.