Result. Math. 32 (1997) 115-l20 0378-6218/97/020115-6 $ 1.50+0.20/0 o Birkhauser Verlag, Basel, 1997 I Results in Mathematics

Accessible Fixed Points on the Boundary of Stable Domains

Walter Schmidt

Abstract

Let U be an invariant stable domain of a rational function Rand Zo E au a weakly repelling fixed point of R. Assuming a "local surjective condition", which is obviously satisfied in the case of a completely invariant domain U, we show that Zo is an accessible boundary point of U. This generalizes theorems of several authors.

1991 Mathematics SUbject Classification: 30D05

Keywords: Julia Set, Rational Functions, Accessible Fixpoint

1. Introduction and results

We assume the reader to be familiar with the basic notation in Rational Iteration. For a detailed treatise of this subject we refer to the books [1], [2] or [9].

Let :J denote the Julia set of a rational function R. A fixed point Zo of R is called weakly repelling when it is repelling (i.e. IR'(zo)1 > 1) or when R'(zo) = 1. In this case we have Zo E :J. Let us assume Zo E au ~ :J for some stable domain U, i.e. a connected component of the Fatou set q; \:J. It is of interest whether Zo is accessible from U. Several authors made contributions to this problem, cf. [2], [3], [4], [5], [7].

The crucial hypothesis needed in all theorems concerning this problem is described with the following notion of "local surjectivity".

Definition. For Zo E au and Wo = R(zo) E a(R(U)) we say R is locally surjective (at Zo with respect to the stable domain U), if, for any sufficiently small neighborhood N of zo, we have R(N n U) = R(N) n R(U).

We always have the inclusion R(N n U) ~ R(N) n R(U), and for a noncritical point Zo local surjectivity means that no other preimage U1 i= U of R(U) has Zo as a boundary point. Of course, if U is a completely invariant stable domain R is locally surjective at any point Zo E au.

On the other hand there are examples where R is not locally surjective, at least for some nonfixed zoo Take the rational function

64 R(z) = (z + 3)(z _ 3)2 - 3,

which has a superattracting fixed point at -1. The corresponding B6ttcherdomain U is not completely invariant and we have Wo = -V17 E au. The symmetry of :J with respect to 1R implies that all three preimages wo, Zo ;:::: 3.56155 + 2.77212i and Zo of Wo lie on the boundary of

116 Schmidt

U. It follows easily that R is not locally surjective at Zo with respect to U, since R : U --7 U is a proper mapping of degree two.

In this example we had Zo =J woo It is an open problem whether the condition in the definition above may be violated at a fixed point zoo In this case we would face difficult topological situations, since there would have to exist, for each n E IN, an n-th preimage Un =J U of R(U) with Zo E aUn.

The main result of this paper is

Theorem 1. Assume R to be locally surjective at the weakly repelling fixed point Zo E aU on the boundary of an invariant stable domain U. Then Zo is an accessible boundary point of U.

The proof will show the existence of an n E IN and an arc, : [0, 1) --7 U with ,(1- ) = Zo and Rnh) 2 " so that Zo is in fact accessible along a periodic arc.

With the introduction of the notion of external rays Douady and Hubbard [3] established Theorem 1 in the special case R = P a polynomial and a simply connected outer domain U = A( (0). Later Eremenko and Levin [4] used the Denjoy-Carleman-Ahlfors-Theorem for subharmonic functions to drop the connectivity-assumption.

Independently Carleson and Gamelin [2] and Petersen [5] proved the Theorem above for general rational functions, but they had to assume the simple connectivity of U again. Finally Przytycki [7J used the notion of coding trees to prove the theorem in the form stated above. It is the aim of this paper to give a much easier proof, which is a refinement of the one in [2].

One problem is not solved satisfactorily: Most researchers believe that the hypothesis on the local surjectivity should be redundant. But no corresponding result is known so far. Here we present a partial result in this direction.

Theorem 2. Ltf R be real with all critical values being real. Then R is locally surjective at any fixed point Zo with respect to any stable domain U where Zo E aU.

Of course, the class of rational functions treated in this theorem is a rather simple one. For example, Cremer points and rotation domains are excluded. On the other hand this class is closed nnder iteration, so we can pa.ss over to an appropriate iterate of R for technical simplifica.tion. This also implies, that the theorem holds for periodic points zoo

Note that the assertion is not true when Zo E au is not a fixed point of R, a counterexample is provided with the example above. However, a simple modification of the proof of Theorem 2 will show the local surjectivity when Wo = R( zo) ¢ lR. Acknowledgments. The results of this paper are part of my doctoral thesis prepared at the University of Dortmund [8]. I am very grateful to Prof. Dr. Norbert Steinmetz who taught me most of what I have learned in complex analysis.

Schmidt 117

2. The proofs

For the proof of Theorem 1 we need the following

Lemma. Let the domain U have a regular boundary for the Dirichlet Problem and R : U ---> U a proper map. Denote by g( z , () Green's Function of U with pole at ( E U. Then there exist a compact subset f( <:;; U and constants c, C > 0, such that the estimate

cg(z,()::; g(R(z),()::; C g(z,()

holds for z E U \]( .

Proof of the Lemma. Let R : U -+ U be proper of degree d and (1, ... , (d E U the preimages of (, counted by multiplicity. As R(z) -+ 8U for z ---> 8U by the hypothesis of the Lemma, the extended Maximum Principle yields

d

g(R(z),() = Lg(z,(j). j=l

Choose]( <:;; U compact, containing (,(b ... ,(d as interior points. Since g(z,() and g(z,(j) are continuous and positive on 8](, we have for some Cj,Cj > 0 the inequality

for z E /)](, j = 1, ... , d. By the Maximum Principle this estimate holds for z E U \ ](, as U has d d

a regular boundary. Obviously we can choose c = L Cj and C = L Cj to establish the assertion j=l j=l

of the Lemma.

Next we use an estimate of Tsuji for the harmonic measure. Let U be a domain having regular boundary for the Dirichlet Problem with 0 E 8U and r > 0 such that U ~ {I zl < r}. For the connected component Uo of U n {Izl < r} which contains a given point Wo E U with Iwol < r/4 we have BUo = ro u 0 0 with ro = BUo n {Izl < r} ::f. 0 and 0 0 = BUo n {Izl = r} ::f. 0.

Consider the harmonic measure wo(z) = w(z, 0 0 , Uo), i.e. the solution of the Dirichlet Problem in Uo with prescribed boundary values h = 0 on ro and h = 1 on 0 0 ,

Given this configuration we have the following

Proposition (Tsuji).

wo(wo)::; 15exp (-~ l g8~~g)) 21wol

where g80(g) is the linear measure of Uo n {Izl = g}.

Proof. This is a simple consequence of [10, Theorem III.67] with K = !. It follows easily using Harnack's inequality for positive harmonic functions, as Tsuji indicates in his remark to the corollary of his theorem cited above.

118 Schmidt

Proof of Theorem 1. First we note that U is regular for the Dirichlet Problem. This follows from the fact that J is uniformly perfect (see e.g.[2]) and a Theorem of Pommerenke [6, Remark 1].

Let us first prove the theorem in the case of a repelling fixed point zoo

We may assume Zo = 0 and U ~ Q::. Fix any C E U and denote by g(z) = g(z,() the Green's Function of U with pole at C. Write A = R'(O), IAI > 1. For r > 0 we set W = Un {Izl < r} and for sufficiently small r we have the following:

(i) U!lc {Izl < r}, so we can apply Tsuji's proposition.

(ii) In {Izl < r} there exists a contracting inverse function 8 = R- I with 8(0) = O. Since R is locally surjective we see 8 ( ur) ~ Ur.

(iii) g( z) < 1 in Ur , since we know that U is regular for the Dirichlet Problem.

(iv) In ur the estimate cg(z) ::::: g(R(z» ::::: Cg(z) of the Lemma holds for some constants c,C> o.

Following [2] we choose any Wo E U with Iwol < !i and set Wj = sj(wo). Let [Tj be the component of ur containing Wj. Since 0 is an attracting fixed point for 8 with multiplier A-I we see IWjl < AI.AI-j for some A > o.

We claim Uk = Uk+n for some integers k, n 2': 1. Assuming this, the proof is finished as follows. There is some arc joining Wk and Wk+n in Uk = Uk+n ~ {Izl < r}. The union of the images of this arc under iterates of sn forms a curve I in [1' terminating at Zo = O. Obviously it satisfies RnC/) "2 I and we are done.

We proceed by contradiction and assume the UJ to be pairwise distinct. Define Wj, Bj for UJ , Wj

analogous to wo,Bo for Uo, Wo in the configuration of Tsuji's proposition above. Then (iii) and the Maximum Principle imply g(z) < Wj( z) for z E Uj, j = 0,1,2, ....

Using this for Wj E Uj, we have by Tsuji's estimate

Now fix some N E IN. Since the domains Uj are disjoint, we have

2N

L Bj(Il)::::: 21r j=N+I

for AIAI-N ::::: Il ::::: ~, and by the Cauchy-Schwarz inequality

Hence for sufficiently large N

2N r/2 2N r/2 r/2

L J IlB~~Il) 2': L J 1l~~Il) 2': J J=N+I2IwJI J=N+I2A1>'I-N 2AI>'I-N

Schmidt 119

for some constant B > 0 depending only on A, r, A.

Thus we can find j, N < j ::; 2N, with

and combining these estimates we obtain for this j

g( Wj) < 15 exp (-~ T g ~~g )) 21w)1

::; 15 exp(-~BN2)

::; 1.5exp(-~Bj2/4).

On the other hand, induction on the est imate (iv) implies g( Wj) ~ cj g( wo) for all j ~ 0 ~ a contradiction for sufficiently large N, which completes the proof in the case of a repelling fixed point Zoo

Essentially this proof carries through for parabolic fixed points Zo with RI( zo) == 1. Since the result is trivial when U is a corresponding parabolic domain , we may assume that ur lies in a repelling petal of Zo == O. Here we have IWjl ::; Aj-I/m, where m is the number of petals at O. If the components Uj were pairwise disjoint, we would have

2N r/2 r/2

-'ft I J g~~g) ~ J J- + 21w,1 lAN-lim

for sufficiently large N and a suitable constant B > O.

Like above

(j)-1fBiI2

g(Wj) < 1.5exp(-~BNlogN)::; 1.5 "2

for some j and again this contradicts the estimate of the Lemma for large N.

Proof of Theorem 2. Passing over to an appropriate iterate of R we may assume V == R( U) to be invariant.

We first. consider the case Zo E ill, t.he upper halfplane. There exist.s some neighborhood N of Zo where the inverse function S == R- I with S( zo) == Zo is declared and such that N, R( N) and R2(N) do not meet JR.

Given any z E un N, we will show z E V which implies U == V and hence the local surjectivity of R. Set WI == R(z) E V, W2 == R(wJl E V, so that WI == S(W2). Since R is real, its Fatou set is symmetric with respect to JR and we can find an arc /1 joining W2 and WI and lying in ill n V. The segment /2 with endpoints W2 and WI does not. meet JR either, therefore /1 - /2 is a closed curve in ill .

Under the hypothesis of the theorem /1 -/2 is homotopic to a point in the domain if: punctured at the (real) critical values of R. By the monodromy theorem we find that the analytic continuation of S along /1 yields the same result as along /1. Of course, continuing along /2 results in the

120 Schmidt

value S( WI) = z, since S is well defined on 12. On the other hand 11 does not meet J and so all lifts of 11 under R do not either. It follows that z E U and WI E V indeed are in the same stable domain U = V.

It remains to prove the theorem in the case Zo E JR. Without restriction we have RI( zo) > 0, since otherwise we could consider the iterate R2. Carrying out the construction above we find WI

and W2 lying in the same halfplane and hence the argument given there applies.

References

[1] Bearden, A. F., Iteration of Rational Functions, Springer 1991

[2] Carleson, L., Gamelin, T. W., Complex Dynamics, Springer 1992

[3] Douady, A., Hubbard, J. H., Etudes dynamique des polyn6mes complexes, I & II, Publica­tions MatMmatiques d'Orsay (1984/85)

[4] Eremenko, A. E., Levin, G. M., Periodic Points of Polynomials, Ukrainian Mathematical Journal, 41 (1989),1258-1262

[5] Petersen, C. 1., On the Pommerenke-Levin- Yoccoz Inequality, Ergodic Theory & Dynamical Systems 13 (1993), 785-806

[6] Pommerenke, C., Uniformly Perfect Sets and the Poincare Metric, Archiv Mathematik 32 (1979), 192-199

[7] Przytycki, F., Accessibility of Typical Points for Invariant Measures of Positive Lyapunov Exponents for Iterations of Holomorphic Maps, Fundamenta Mathematicae 144 (1994), 259-278

[8] Schmidt, W. , Fixpunkte im Rand stabiler Gebiete, Dissertation Universitiit Dortmund (1996)

[9] Steinmetz, N., Rational Iteration, de Gruyter 1993

[10] Tsuji, M., Potential Theory in Modern Function Theory, Maruzen 1959

Eingegangen am 3. Februar 1997

Lehrstuhl IX fUr Mathematik U niversitiit Dortmund 44221 Dortmund Germany