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The iteration of cubic polynomials Part II: patterns and parapatterns

1992; Mittag-Leffler Institute; Volume: 169; Linguagem: Inglês

10.1007/bf02392761

1871-2509

Bodil Branner, John H. Hubbard,

Functional Equations Stability Results

CI~AZI'ERB. BRANNER AND J. H. HUBBARD critical point to2 escaping to infinity more slowly than to~ or not at all, i.e. hp(to2) 0, and fractal if he (to2)=0.They are difficult to "know" in any very precise sense.Forgetting about their embedding in C and remembering only their structure as Riemann surfaces, they are also fairly complicated, depending on a complex number ~ E C-/), a real number h<log ]~1 and many (sometimes infinitely many) combinatorial data.But for all that they are "knowable", as follows.The first step is to construct an analytic mapping q0~, defined in a neighborhood of infinity and conjugating P to z,--~z 3.There are only two such q0e, and if P is monic there is a unique one which is tangent to the identity at o~.Now for purely topological reasons, q0e can be extended to Ue (wO, and qge (Ue (to l))=C-Dr where D r is the disc of radius r=e her'~ Thus, as an abstract Riemann surface, Ue(tOl) is simply the complement of a disc.The point P(to0 is in Ue(w~), and the number q~e(P(wl)) is a dynamical invariant of the polynomial.Call it ~3, because we will see in a moment that it has a distinguished cube root.Observe that Ue(tol) is a domain in C bounded by a curve homeomorphic to a figure eight, with double point the critical point to~; and that this curve contains one extra inverse image of P(to0, which we will call the co-critical point to~'.The mapping qoe can be extended to a neighborhood of Wl', and q~e (~01')=r is the distinguished cube root mentioned above.From the complex number r we can reconstruct Ue(to~) as C-DIe I with the points jr and j2r identified, where j and j2 are the non-real cube roots of 1.Now consider P-l(up(a~l)); if hp(w2)<~hl,(tOl), the mapping P:P-l(Ue(tol))--~ Ue(to ~) is a ramified triple covering, ramified at the single point tot.There are exactly two such coverings, classified by the component of C-Ue(to 1) containing the critical value P(t09.If he(~o2)<~he(tol), the next inverse image P-2(Ue(to O) is again specified by the component of C-P-~(Ue(to~)) containing P(to2); of course the enumeration of the possible components depends on our previous choice.This construction of successive inverse images of Ue(to l) continues until an inverse image contains P(t02), and can be continued for ever if to2 EKe.The successive specification of the component of C-P-n(Ue (to 1)) containing P(t02) is called the combinatorial information.With ~ E C-/) and h<log fixed and he (to2) h only depends on the combina-Fatou-Julia problem on wandering domains.