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Then, if one changes the complex structure of M on the set UF-n (X) , such that F becomes a holomorphic conjugation with an isomorphism from M to L, this gives the desired rational function R with a Hennan ring. Figure 19 : Construction of a Hennan ring by Shishikura. 4 The Classification Theorem We now state the classification theorem of Sullivan [82,83]. 1 Let R be a rational function and F a periodic component of :FR of period n and let r = Uk:6 Rk( F) be the associated cye/e. Then r is one of the following: NEWTON'S METHOD AND COMPLEX DYNAMICAL SYSTEMS 39 • An immediate basin A*( "Y) of a superattractive cycle "Y.
3 If Fjwith R( Fj ) = FJ· is a singular domain. then RIFJ is holomorphically conjugate to € I--t e 2 ",a€ for some irrational number a. Remarks • The irrational number a is called the rotation number of RIFj. It is a topological invariant. • Clearly, the same holds if Fj is a singular domain of some period k ~ 2. Now we return to rationally indifferent fixed points. Let Zo be a rationally indifferent fixed point of R, then (cf. 3 ) R is topologically conjugate to € I--t >-€( 1 + e"), where>- = R'(zo) .
Z) be a complex analytic mapping from A x V to C with 'I'/~ (0) = 0 and 1'1'/). ,z) E A xV. SetU). = DIO = {z E C Ilzl < lO}forall >. ). Then f). : U{ ---t U). is a polynomial-like mapping of degree 2 for 1>'1 ~ 4. 7 Let f = (f). 6. Then X: ML ---t M is an homeomorphism. In that case one says that f is a Mandelbrot-like family. Let us come back to Newton's method N). for>. close to >'0. For >'0 Newton's method has an superattractive cycle of period 2. Setting f). 02}, U). 3} and U~ = U). 7 are true.