By Alexander Ziwet, Louis Allen Hopkins

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15 The attractor A of a hyperbolic rational map consists of a finite set of cycles which can be located by iterating the critical points of f . 5. 5 Invariant line fields and complex tori The measurable dynamics of a rational map can be extended by considering the action of f on various bundles over the sphere. For the theory of quasiconformal rigidity, the action of f on the space of unoriented tangent lines plays an essential role. 10). All known examples of rational maps supporting invariant line fields on their Julia sets come from a simple construction using complex tori.

Is a sequence of disjoint open sets i n the plane, such that 1. En is a finite union of disjoint unnested annuli of finite moduli; 2. a n y component A of En+1is nested inside some component B of En; and 3. for any sequence of nested annuli A,, where An is a component of E n , we have C mod(An) = oo. Let F, be the union of the bounded components of @ - En, and let F = Fn. T h e n F is a totally disconnected set of absolute area zero. 8. ABSOLUTE AREA 21 ZERO The set F consists of those points which are nested inside infinitely many components of U E n .

Given a holomorphic family of rational maps f A , we say the corresponding Julia sets Jx c move holomorphically if there is a holomorphic motion q5x : J, e +e such that +x(J,) = Jx and for all z in J,. Thus g5x provides a conjugacy between f , and f x on their respective Julia sets. The motion q4A is unique if it exists, by density of periodic cycles in J,. The Julia sets move holomorphically at x if they move holomorphically on some neighborhood U of x in X. A periodic point z of f , of period n is persistently indiflerent if there is a neighborhood U of x and a holomorphic map w : U --+ such that w ( x ) = z , f,"(w(A)) = w ( A ) , and I (f,")'(w(A))I = 1 for all A in U .

### Analytic geometry and principles of algebra by Alexander Ziwet, Louis Allen Hopkins

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