Semiconjugacies between the Julia sets of geometrically finite rational maps Tomoki Kawahira * University of Tokyo October 22, 2003
Abstract
A rational map f is called geometrically finite if every critical point contained in its Julia set is eventually periodic. If a perturbation of f into another geometrically finite rational map is horocyclic and preserves the critical orbit relations with respect to the Julia set of f , then we can con- struct a semiconjugacy or a topological conjugacy between their dynamics on the Julia sets.
1
Introduction
Let f : Ĉ → Ĉ be a rational map of degree d ≥ 2. We call such a map geomet- rically finite if all critical points contained in the Julia set J(f) are eventually periodic.  A geometrically finite rational map can have (super)attracting and parabolic basins, but no Siegel disks or Herman rings. In particular, if a rational map is (sub)hyperbolic or parabolic, then it is geometrically finite. In this paper, we discuss perturbations of a geometrically finite rational map f within Rat d , the space of all rational maps of degree d . The topology of this space is defined by uniform convergence on the sphere with respect to the spherical distance d σ ( ·, ·).  Our aim is to study the dynamical stability of f on its Julia set; that is, structural stability of f restricted on the Julia set. Perturbations of f. Let us consider a family of rational maps of degree d ≥ 2, { f ε ∈ Rat d :  ε ∈ [0, 1]} with the following conditions: ● fo = f; and ● sup x ∈ ˆ d σ ( f ε ( x ), f ( x )) → 0 as ε ＼ 0. We represent this family in the convergence form, f ε → f , and call it a perturba- tion of f. For this perturbation f ε → f , let us consider whether the dynamics on J ( f ) is perturbed continuously to that on J ( f ε ). More precisely, we consider the existence of a map h ε : J(f ε ) → J(f) for each ε ∈ [0, 1] such that ● h ε is a homeomorphism with h ε ○ f ε = f ○ h ε on J(f ε ); and ● h − 1 : J(f) → J(f c ) tends to id : J(f) → J(f) as c → 0. Such an h ε with the first condition is called a (topological) conjugacy between f ε and f on their respective Julia sets. In addition, for the first condition, if h ε is not a homeomorphism but merely continuous and surjective, then such an h ε is called a semiconjugacy between f ε and f on their respective Julia sets. By the Mañé-Sad-Sullivan theory[15], if f has a connected neighborhood U ⊂ Rat d where each f ε ∈ U has the same number of attracting cycles as f , then for each f ε ∈ U there exists a unique quasiconformal conjugacy h ε : J(f ε ) → J(f) as above. This means any small perturbations of f have desired conjugacies. For example, hyperbolic rational maps have this property. On the other hand, when f is geometrically finite f can have parabolic cycles: As we will describe, those parabolic cycles may change into attracting cycles under some perturbations.  Thus the number of attracting cycles may change and we cannot apply the Mañé-Sad-Sullivan theory. Moreover, by a perturbation of parabolic cycles into attracting cycles, the topology of J(f) may change and we cannot even hope that J(f) and J(f) are homeomorphic in general. However, in our main theorem (Theorem 1.1), we will give a sufficient condi- tion for perturbations f ε → f to be accompanied by such conjugacies as above or best possible semiconjugacies between the dynamics on their Julia sets. Parabolic points.  Let f : Ĉ → Ĉ be a rational map of degree d ≥ 2, and let a be a periodic point of f with period l and multiplier ( f t ) '(a) =: λ. We say a is a parabolic (periodic) point if λ is a root of unity. Now let us suppose that a is a parabolic point and λ is a primitive q -th root of unity. Taking a local coordinate near a which maps a to 0, we obtain (1.0) with A p+1 ≠ 0 and p ≥ 1. (Moreover, we can normalize A p+1 to be 1 by using a linear transformation.)  It is known that p is a multiple of q which does not depend on the choice of local coordinates. We call p = p ( a ) the petal number of a. We also say that a has p petals. Note that a is a fixed point of f 1 q of multiplicity p + 1. By a perturbation of f into f ε , a splits into p + 1 fixed points of f ℓ q counting with multiplicity. This may cause drastic change of the dynamics, so we have to control the perturbation in order to change the original dynamics tamely. Horocyclic perturbations.   After C. McMullen, we say a perturbation f c → f is horocyclic if each parabolic point a of f as above satisfies the following: (a) There are fixed points a c of f t with multipliers ( f t ) ′ (a c ) = λ c satisfying αε → α and λε → λ; (b) There is a neighborhood D of a with local coordinates φε , φ : D → C such that: 1. α ε ∈ D and φε(αε) = φ(α) = 0; 2. φε → φ uniformly on D ; and 3.  If we represent the actions of f l q and f l q on D by φ c and φ respectively, we obtain the local representation of the perturbation as: feq(z) = λξz + zp+1 + O(zp+2) → feq(z) = z + zp+1 + O(zp+2).  (1.1) (c) If we set exp( L c + iθ c ) := λ q , which tends to 1 as ε → 0, then θ z = o(| L c |) as Le, θe → 0. Form (1.1) implies that the symmetry of the local dynamics near a is preserved by the perturbation. In particular, φ , φ ε are not necessarily conformal, can be just homeomorphisms from D to their images. By condition (c), a avoids being perturbed into an irrationally indifferent periodic point. See §2 for more details. Horocyclic perturbation was originally defined as horocyclic convergence of rational maps, to study the continuity of the Hausdorff dimensions of the Julia sets of geometrically finite rational maps[12, §7-9]. J-critical relations.  A geometrically finite rational map may have critical points in its Julia set.  Here we introduce a condition which controls the per- turbations of the orbits of such critical points. Let c 1 , . . . , c N be all critical points of f contained in J ( f ), where N is counted without multiplicity. A J-critical relation of f is a set of non-negative integers (i,j,m,n) such that fm(ci) = fn(cj). Let deg(f, x) denote the local degree of f at x . We say a perturbation f ε → f preserves the J-critical relations of f if: ● For all i = 1,...,N, the maps f c have critical points c i ( ε ) (may be in the Fatou set) satisfying c i ( ε ) → c i and deg( f ε , c i ( ε )) = deg( f , c i ) as ε → 0; and ● For each J -critical relation ( i, j, m, n ) of f, f c satisfies f ε π ( c i ( c )) = f n ( c j ( c )). If f is geometrically finite, then the maps f ε are also geometrically finite. If f is hyperbolic or parabolic, then C ( f ) ∩ J ( f ) = ∅ and any small perturbation of f automatically preserves its J -critical relations. Our main result is: Theorem 1.1  Let f : and f ε → f  a horocyclic perturbation which preserves the J-critical relations of f For each ε which is sufficiently small, there exists a unique semiconjugacy h ε : J(f ε ) → J(f) with the following properties: 1. If card( h ε 1 (y)) ≥ 2 for some y ∈ J(f), then there exists an n such that f n (y) is a parabolic point of f  and card(h ε 1 (y)) = deg(f n ,y) · p (f n (y)). 2. h c can be arbitrarily close to the identity on J(fc).  That is, if we fix an arbitrarily small r > 0, then for all sufficiently small e, h ε satisfies sup {dσ(hε(x), x) : x ∈ J(fε)} < r. Property 1 implies that the injectivity of h ε may break on the backward orbits of parabolic points of f .  Since such points are countable, we say that h c is almost bijective.  However, even though f has parabolic points, h ε can give a topological conjugacy. The precise condition for this is described in Corollary 7.1. In addition, Property 2 implies: Corollary 1.1 For f ε → f as above, J(f ε ) converges to J(f) in the Hausdorff topology. For a given geometrically finite rational map, the existence of such perturba- tions is guaranteed by [10]. Example 1.  Let us consider perturbations of a geometrically finite map f(z) = z(1+z)™ with m ≥ 2. Now −1 is a preparabolic critical point and 0 is a parabolic fixed point with one petal. Here are two typical perturbations: ● f ε ( z ) = λ ε z (1 + z ) m with real λ ε \ ε \ ε 1 ● fε(z) = λε z(1 + z) m with real λε̸/ 1 For both cases, 0 is split into a pair of attracting and repelling fixed points, 0 and −1 + 1/ ♕ λ e . For the first case, 0 is the repelling one, and for the second case, the attracting one. In Figure 1, curves roughly show the shape of the Julia sets for m = 3. These split fixed points and their first preimages are shown by heavy dots. Both two perturbations are horocyclic and preserving the J -critical relations of f . For the first case, we obtain h ε as a topological conjugacy. For the second case, h ε is a semiconjugacy which pinches the backward images of −1 + 1/ ♕ λ ε onto those of 0. The injectivity is broken only at these points. Remark on the Goldberg-Milnor conjecture. Theorem 1.1 gives a partial and affirmative answer to the following Goldberg-Milnor conjecture[6]:  For a polynomial f  which has a parabolic cycle, there exists a small perturbation of f such that ● the immediate basin of the parabolic cycle is converted to basins of some attracting cycles; and ● the perturbed polynomial on its Julia set is topologically conjugate to the original polynomial f  on J(f). Some horocyclic perturbations of a geometrically finite polynomial explicitly give such perturbations. For example, the first perturbation in Example 1 gives an affirmative answer to this conjecture for f(z) = z(1 + z)m. In general, any geometrically finite rational map has such a perturbation. See [10]. For other partial solutions of this conjecture, see [3] and [7]. Example 2. Let us consider a Blaschke product f(z) = (3z² + 1)/(3 + z²) with a parabolic fixed point at z = 1, which has 2 petals. The critical points of f are 0 and ∞. The Julia set is the unit circle and the Fatou set is the parabolic basin of z = 1. Let us consider perturbations of f of the form
For ε ≪ 1, f , are also Blaschke products and the Julia sets are contained in the unit circle. By this perturbation, the parabolic point z = 1 of f splits into the following three fixed points (counting with multiplicity): zo = 1 with multiplier λε, z 1 = (−λε + 2√−1 + λε)/(−2 + λε) and z 2 = (−λε − 2√−1 + λε)/(−2 + λε) with the same multipliers −1 + 2/λε. Now consider the case of real λε with (a)λε ＼ 1 or (b)λε ／ 1. For each cases, one can check that f ε → f is a horocyclic perturbation. When (a), z 0 = 1 is repelling and z 1 , z 2 are attracting. The Julia set of f c is also the unit circle. By Theorem 1.1, there is a conjugacy between f c and f on the unit circle. When (b), z 0 = 1 is attracting and z 1 , z 2 are repelling. The Julia set of f c is a Cantor set contained in the unit circle. By Theorem 1.1, there is a semiconjugacy between f ε and f on their respective Julia sets.  Note that the semiconjugacy maps a Cantor set onto the unit circle. Sketch of the proof of the main theorem.   Let us roughly sketch the proof of Theorem 1.1; the construction of the semiconjugacy between f ε and f on their respective Julia sets. Let f be a geometrically finite rational map and let f ε → f be a horocyclic perturbation which preserves the J -critical relations of f.  We investigate the properties of such a perturbation in §2. In §3, we prepare the ingredients for the semiconjugacy. For f , we construct a compact set Ω such that J(f) ⊂ Ω ⊂ f(Ω). Correspondingly, for each fixed f ε , we construct a compact set Ω ε such that J( f ε ) ⊂ Ω ε ⊂ f ε (Ω ε ). We also construct a certain surjective map h 0 (= h 0 c ) : Ω c → Ω as the "0-th" step to the semiconjugacy. Then in §4, we inductively construct a sequence of "lifts" {hn(= hn,ε) : fçn(Ωε) → f−n(Ω)}∞ satisfying f o h n+1 = h n o f ε . In §5, we investigate the expanding property of f ; in other words, the contracting property of f-1. By using this property, in §6, we show that { h n } converges uniformly to a surjective map h c on J ( f c ) if c ≪ 1.
In §7, we check that h ε satisfies the properties in Theorem 1.1. To simplify the argument, from §3 to §7, we suppose that J(f) ≠  C.  The case of J(f) =  C is treated in §8.
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Notes.
1. For the basic properties of the Julia sets and parabolic points, refer to [1], [2] and [14], etc. 2. If f is hyperbolic, we obtain h c as a topological conjugacy. In particular, by uniqueness, h ε coincides with the quasiconformal conjugacy obtained by using λ-Lemma in [15]. In general, for a perturbation f ε → f as Theorem 1.1, if each f ε for ε ∈ (0, 1) is hyperbolic, then each h ε is characterized as a uniform limit of quasiconformal conjugacies. 3. If a rational map f has no Siegel disks or Herman rings and f ε → f horo- cyclically, it is known that J(f ε ) → J(f) in the Hausdorff topology[8],[12, Theorem 9.1]. Corollary 1.1 gives another proof of this fact in a special case by using the existence of the semiconjugacy. 4. Theorem 1.1 is an improvement of an author's result on horocyclic pertur- bation of parabolic rational maps in [9] or [8]. Notation. Here we list some notation used throughout this paper. ● σ := 2|dz|/(1 + |z|2) is the spherical metric on the Riemann sphere C. ● d σ ( ·, ·) : the spherical distance measured in σ . ●  Bσ(x,r) := {y ∈ ˆ C : dσ(x,y) < r} ● F(f) : the Fatou set of f ● C(f) : the set of all critical points of f . ● P(f) := {fn(c) : c ∈ C(f), n = 1,2, . . .}; the postcritical set of f. ● For any map f , f 0 denotes the identity map on the domain of f . ● n ≫ 0 means that n > 0 is sufficiently large. ● ε ≪ 1 means that ε > 0 is sufficiently small. 2
Horocyclic perturbations
Bifurcations of parabolic periodic points have a strong effect on the local dy- namics as well as the global dynamics. In this section, we describe a horocyclic perturbation f ε → f of a geometrically finite rational map f in further detail. In particular, we introduce the notion of planet and satellite for periodic points generated by perturbation of parabolic points. Roughly speaking, a planet is the central periodic point which determines the properties of the perturbed local dy- namics. Satellites accompany a planet. Moreover, we will show a key lemma on horocyclic perturbation (Lemma 2.1), and see the local dynamics near parabolic points change tamely under such perturbations.
2.1
Planets and satellites.
First we consider condition (b)-3 of horocyclic perturbation. Let a be a parabolic point of f as in the preceding section, which has a local representation as (1.0). As we will see afterward, condition (b)-3 is important to keep the original symmetry of the local dynamics for the petals of α . However, if we suppose only conditions (a), (b)-1 and (b)-2 for f c → f , we just obtain a local representation of the convergence near a as the following:
where 2 ≤ r ≤ p. In [12, §7], C. McMullen gave some conditions which insure form (2.1) becomes form (1.1) by taking suitable local coordinates. One of such conditions is: Proposition 2.1  For the form (2.1) above, if Ae,i/(Xe – 1) converges as ε → 0 for each r ≤ i ≤ p, then through a continuous change of coordinates near a, we obtain the normalized form of the convergence as (1.1). For example, if p = q then A c,i satisfies this condition. See [12, Proposition 7.2 and 7.3] for more details. Planets and satellites.  Next, we consider the effect of condition (c) of horo- cyclic perturbation.  Let f ε → f be a horocyclic perturbation.  Now λ ε = exp( L c + iθ c ), with the assumption that θ c = ο(| L c |) as L c , θ c → 0.  By this relation, L ε = 0 implies θε = 0. In other words, if | λξ | = 1 then a ε is persistently a parabolic point of f ε with the same multiplier λ as a . This means, perturbations of a into another kind of indifferent periodic point are prohibited. Let us look the relation θ ε = o ( | L c |) in the complex plane.  If we fix a pair of arbitrarily small closed disks on the both sides of the imaginary axis, so that they are tangent to the axis at the origin, then they contain L c + iθ c for all ε ≪ 1. Thus L c + iθ c cannot converge to 0 along the imaginary axis, but can converge along a curve tangent to the imaginary axis with order < 2. From (1.1), the solutions of the equation f ξ q ( z ) = z near the origin are z = 0 and z ≈ (1 − λ q ) 1 / P and they correspond to the symmetrically arrayed fixed points of f l q generated by the perturbation of a (See Figure 2). We classify them into two types: planet and satellite. First, we consider the case of multiple petals: That is, p ≥ 2. Then we have the following three cases corresponding to L ε = 0, < 0, or > 0:
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(1) a ε is persistently a parabolic point with p petals and the multiplier λ ε = λ ; (2) a ε is an attracting periodic point, and there are p symmetrically arrayed repelling periodic points near a c ; or (3) a ε is a repelling periodic point, and there are p symmetrically arrayed at- tracting periodic points near α ε . For cases (2) and (3), these symmetrically arrayed periodic points have the same period lq and the multipliers ≈ λ ε pq . Moreover, they are contained in an open ball centered at a ε with radius O ( | 1 − λ ξ | 1 / p ). We call them the satellites of a ε and a ε itself the planet. In particular, for case (2), we say that the parabolic point a is perturbed into an attracting planet a c . As we will see in the following sections, attracting planets are the cause of non-injectivity of the semiconjugacies.  For case (1), we also call a c the planet, although it has no satellite. Next, we consider the case of one petal. Now p = 1, then automatically q = 1 and λ = 1. If λ ε = λ(= 1), α ε is persistently a parabolic point with one petal. In this case, we also call a c the planet. If λ c̸= λ , α splits into a pair of repelling and attracting periodic points. Which one is suitable for the planet? To define the planet in this case, we need to consider the J -critical relations. Preparabolic critical orbits in J(f).  Let b be a preimage of a such that a = f i ( b ) = f i+l ( b ). If deg( f i , b ) = m , we can take a local coordinate near b such that ζ(b) = 0 and
with a suitable branch of f-å. This implies that there are mp petals attached to b as preimages of the petals of a. Let us suppose that a horocyclic perturbation f ε → f preserves the J -critical relations of f .  Then there exists b ε such that a ε = f ξ (b ε ) = f ξ + 1 (b ε ) and deg(f ξ ε , b ε ) = m. Taking a suitable local coordinate near b ε such that ζ(b ε ) = 0, we obtain the corresponding normalized form of f ε ; f ξ i o f ξ i q o f ể (ζ) = λ ξ ζ + ζ mp+1 + O (ζ mp+2). If λ q̸= 1 (that is, L e̸= 0) and p ≥ 2, there are symmetrically arrayed mp “satellites” near b ε as the preimages of the satellites of a ε . Recall that a ε may be attracting: this implies, b c may be in the Fatou set. Now let us return to the definition of the planet when a has one petal. In the case of λ ε = λ(= 1), it has been defined by α ε . In the case of λ ε̸= λ , a splits into a pair of repelling and attracting fixed points of f ξ , say α ξ and α ε respectively. If a has a critical point in its preimages, then either a ξ or α ε has a critical point in its preimages because the J -critical relations are preserved. In this case, we define the planet as one containing a critical point in its preimages, and the satellite bas the other one. In particular, if a ,  is the planet, we also say that a is perturbed into an attracting planet a − . If a has no critical point in its preimages, then we formally define the planet as a ξ and the satellite as a ε . Example. Let us consider perturbations of f(z) = z(1+z)m with m > 1 again. Recall that 0 is a parabolic fixed point with one petal. For both perturbations in Example 1, 0 is the planet and −1 + 1/ "V/X c is the satellite (See Figure 1). For the second perturbation, 0 is perturbed into an attracting planet. On the other hand, for a trivial perturbation f ε ( z ) = z (1 + λ ε z ) m with λ ε → 1, where f ε are conjugate to f by linear transformations, 0 is the planet with no satellite. Prerepelling critical orbits in J ( f ).   By geometric finiteness of f , some crit- ical orbits in J(f) land on repelling cycles. Since the J -critical relations are pre- served, such repelling cycles are perturbed into repelling cycles of f ε for ε ≪ 1. Let us consider local representations of the perturbations near such cycles. Let b be a repelling periodic point of f in P(f) ∩ J(f) , with multiplier λ and period l .  Then there exists a repelling periodic point b ε of f ε in P(f ε ) ∩ J(f ε ), with multiplier λ ε and period l , such that b ε → b and λ ε → λ . By using a fundamental fact about linearization near repelling fixed points, we can take suitable local coordinates ψε , ψ on a neighborhood of b such that ψε ( bε ) = ψ( b ) = 0 and
where ψε converges to ψ uniformly near b. See [14, 8.3 Remark].
2.2
Key lemma on horocyclic perturbation.
Here we show a key lemma on horocyclic perturbation, which describes the per- turbation of an orbit which accumulates on parabolic periodic points. We will
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see how horocyclic perturbations control the parabolic bifurcations. Let a 0 be a periodic point of f with period l . The cycle α of a 0 is defined by
When a 0 is parabolic (resp. attracting, etc.), we call α a parabolic (resp. attract- ing, etc.) cycle. Let us fix an x ∈ C whose orbit accumulates on a parabolic cycle α.  For an arbitrarily small δ > 0, set △ = △(δ) := ∪ α∈α Bσ (α,δ), and take N 0 = N 0 ( x, δ) ≫ 0 such that f n ( x ) are contained in ∆ for all n ≥ N 0 .  Now the key lemma is described as: Lemma 2.1  If the perturbation f ε → f is horocyclic, then there exists an N ≥ N 0 such that f n (x) are contained in ∆ for all n ≥ N and all ε ≪ 1. To simplify the proof of this lemma, we use "linearization" of parabolic bifur- cations due to C. McMullen[12]. We begin the proof with constructing a simpler representation of the Proof. perturbation. Linearizing parabolics.  Let us take an integer k so that fk(a) = a and ( fk ) ′ ( a ) = 1 for any a ∈ α , and replace f by fk . Then we may assume that α = { a } is a fixed point with multiplier 1 and that ∆ = B σ ( a, δ ). It is sufficient to prove the statement in this case. From the conditions of horocyclic perturbation, there exists a fixed point a e of f ε converging to a . We may assume ε ≪ 1 such that a ε is contained in △ and sufficiently close to a. Now we set λε = exp(Lε + iθε) := 1/f'(αε), which tends to 1 with θ ε = ο(| L ε |). By replacing △ = △( δ ) with smaller δ and the definition of horocyclic pertur- bation, we can take a normalized convergent form on ∆ as (1.1); fe(z) = λ ε ¹ z + z p +¹ + O(z p +²) → f(z) = z + z p +¹ + O(z p +²) where z(a ε ) = z(a) = 0 and p is the petal number of a.  Moreover, we take a simpler form of the convergence as follows. First, by using local coordinates such that z(a ε ) = z(a) = ∞, we obtain
as a normal form of the convergence.  Next, by using [12, Theorem 8.3] and additional linear conjugacies, we can show that there exist quasiconformal maps Φε,ο, φ 0 with φε,ο → φ 0 near infinity and φε,ο(∞) = φ 0 (∞) = ∞ such that
Where p -th roots are taken so that ( λε z ρ +1) 1 /p = λε z +O(1) and ( z ρ +1) 1 /p = z + O(1). Note that T ε and T are p -fold branched coverings of linear transformations T ε ( w ) = λ ε w + 1 and T ( w ) = w + 1 respectively (where w = z P ). We call this form (2.4) a linearized model of the perturbation f ε → f near a. Let φε (resp. φ) be the composition of local coordinates of aε (resp.  a) as (2.3) with φε,0 (resp. φ 0 ) as (2.4).  Then we obtain φε → φ, a uniformly convergent family of local coordinates near α , which satisfies φε (αε) = φ(α) = ∞ and conjugates f ε → f to T ε → T .  Finally, by replacing △ = △( δ ) with much smaller δ, we may assume that ∆ is the domains of φε and φ. Now let us show the lemma by using the linearized model as (2.4). Take a constant R ≫ 0 and a closed disk D := {| z | ≥ R} , such that D is contained in both φε(Δ) and φ(Δ). Then there exists an N 1 ≥ N 0 such that φ(f n (x)) ∈ D for all n ≥ N 1 . Moreover, by uniform convergence of f ε → f and φ ε → φ , we may assume that φε ( f ε N 1 ( x )) ∈ D . To prove the lemma, it is enough to show that there exists an N ≥ N 1 such that φε ( f ε ( x )) ∈ D for all n ≥ N . The proof breaks into the cases of p = 1 and p ≥ 2. Case 1: p = 1.   Now φε → φ conjugates fε → f to
on D , with φε ( αε ) = φ ( α ) = ∞. (See Figure 3. The four regions are centered at infinity.) When λε = 1, Tε is still parabolic and
for all k ≥ 0. This implies that f ∈ 1+k ( x ) never escapes from φε 1 ( D ) ⊂ △ for all k ≥ 0. Hence we take N 1 as N in this case. We henceforth assume that | λε |̸= 1. By the perturbation, a splits into a pair of attracting and repelling fixed points.  We may suppose that a c is the repelling one, and let b c denote the attracting one.  (Here we do not consider which the planet is.) Then |1/λε| = |ƒε(αε)| > 1, that is, Lε ／ 0. Moreover, in the linearized model (2.5), φε ( bε ) must be the attracting fixed point of T ε ; thus φε(bε) = (1 − λε)− 1 =: bε, and the multiplier of b ′ε is λε. Since the real part of T n ( z ) tends to infinity, there exists an integer N ≥ N 1 such that φ( f N ( x )) is in D ∩ {| arg z | < π/4}. By uniform convergence of f ε → f
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and φε → φ, we may also assume that y := φε( fε ζ ( x )) is in D ∩ {| arg z | < π / 4} for all ε ≪ 1(Figure 4). To see the dynamics of T ε in detail, we take a Möbius conjugacy of T ε by
which maps ∞ → ∞, b ξ → 0 and y → 1. This conjugates the action of T ε to w → λεω with |λε| < 1. Hence 1 = ψε(y) is attracted to 0 = ψε( ζ ) by the iteration of w → λεw. Now we claim: For any fixed ε ≪ 1, f r ( x ) is contained in △ for all n ≥ N , and converges to b ε as n → ∞. In other words, the whole orbit of 1 = ψε ( y ) is contained in ψε ( D ) where the conjugation between T ε and w → λε ω holds. Set B := C − D and B ′ := ψ ε ( B ). Then B ′ is defined by this inequality:
We will show that the orbit of 1, that is, {1 = ψε(y), λε, λε, . . .}, never enters B'. For all ε ≪ 1, the center b ε /( b ε − y ) of B ′ is approximately 1 − y( L ε + iθ ε ). On the other hand, for any k such that | k ( L c + iθ c ) | ≪ 1, λ k is approximately 1 + k ( L ε + iθε ).  Since |arg y | < π/4, the direction of first several points of the orbit {1, λε, λε, . . .} is opposite to the center of B ′ with respect to 1. This means, at least, the orbit does not go to B ′ immediately (Figure 5). Suppose that θε = 0. Then the orbit of 1 accumulates on 0 along the real axis, and it is disjoint from B'. Suppose that θ ε̸= 0. We may assume that θ ε > 0 because the signature of θ ε determines only the direction of the rotation by the action of w → λε w. Then the orbit of 1 returns near the positive real axis by nearly 2π/θ ε times iterations of w → λεω. Now we have to handle the case where the order of θε ＼ 0 is lower than that of L ε / 0: Then the orbit might touch B ′ . However, we will show that it cannot occur if ε ≪ 1. Now note that the following two facts:  when the orbit of 1 returns near the pos- itive real axis, the distance between 0 and the orbit is nearly l c := exp(2π L c/θ c ); on the other hand, by (2.6), B ′ is contained in a ball centered at 1 with radius O(|L ε + iθε |), that is, every point in B ′ tends to 1 as ε → 0. By these facts, if lim inf | L ε /θ ε |̸≠ 0, l ε does not tend to 1 and the orbit of 1 never touches B' (Figure 6). Otherwise we can take a decreasing sequence ε n ＼ 0 such that L ε n /θ ε n → 0. Now l ε n → 1 as n → ∞ . In this case, |1 − l ε n | ≈ 2π | L ε n | θ ε n for n ≫ 0 thus
This means, for any choice of { ε n } , every point in B ′ tends to 1 faster than l ε n does. Note that the order of convergence in (2.7) depends only on the order of L ε , θ ε → 0 (not on the choice of { ε n }).  Hence for ε ≪ 1, the orbit of 1 is attracted to 0 without entering B'.
on D . As in the case of p = 1, we may assume that
for an N ≥ N 1 , and
for all ε ≪ 1.
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Let us consider a semiconjugation of T ε by a branched covering w = π(z) = z ρ . Then the dynamics of T ε on D is reduced to the dynamics of T ε ( w ) = λ ε w + 1 on π(D) = {|w| ≥ R p }(Figure 7). Similarly, π(z) gives a semiconjugacy from T(z) on D to T(w) = w + 1 on π(D). By the same argument as the case of p = 1, when λε = 1, the orbit of π(y) tends to w = ∞ and never escapes from π( D ). Similarly, if |λε |̸= 1, the orbit of π(y) tends to an attracting fixed point, which is either w = ∞ or w = 1/(1 − λξ), and never escapes from π( D ). Thus the original orbit of φε ( f ε N ( x )) by T ε never escapes from D. ■ One can easily check that the same result holds if we replace x with Remark. a compact set in the parabolic basin of a .  We will use this in the proof of Proposition 3.2.
3
Construction of Ω and Ω
In this section, we prepare the ingredients for the construction of the semiconju- gacy; Ω, Ωε and h 0 : Ωε → Ω. To simplify the arguments, from this section to § 7, we assume that J(f)̸= C. The case of J(f) = C is treated in §8. Let us introduce some notation. Let A denote the finite set of all parabolic points of f . We define the sets of all preperiodic critical orbits in the Julia sets by
In addition, we set Z 1 := f-1 (Z) and Z 1 := f c-1 (Z c). Since f c → f preserves the J -critical relations of f , card( Z ε ) ≤ card( Z ) < ∞ in general. The equality holds precisely if none of the parabolic points of f is perturbed into an attracting planet.
16
3.1
Construction of Ω.
Here we construct a compact set Ω for f. Proposition 3.1  There exists a finitely connected compact set Ω ⊂ Ĉ with the following properties: 1. Ω∩(P(f)∪C(f)) = J(f)∩(P(f)∪C(f)). This set is the union of A and all critical orbits in J(f). 2. J(f) ⊂ Ω and f-1(Ω) ⊂ Int(Ω) ∪ A. To define the compact set Ω, we will construct two open sets F and V Proof. which consist of finitely many simply connected components. Let a be an attracting or parabolic periodic point of f and α the cycle of a. First, we construct F : If α is attracting, we take a small disk neighborhood F α for each a ∈ α such that f(F a ) ⊂ F f(a) . Here we can take { F a } to be pairwise disjoint. If α is parabolic, we take F a for each point a ∈ α to be a small "flower" (that is, a union of attracting petals for each attracting directions of a ) such that f(F a − {a}) ⊂ F f (a) . Here we can also take { F a} to be pairwise disjoint, and each ∂F a to be tangent to the repelling directions. Now we set
where α ranges over all attracting and parabolic cycles. Note that f(F − A) ⊂ F. Next, we construct V: Let C(f,α) denote the set of all critical points of f whose orbits accumulate on α but never land on it. Now let us set Fα := ∪ σεα Fα For each c ∈ C ( f,α), there exists a natural number N = N (c) such that f n (c) ∈ F α for all n ≥ N . Then we can take a family of open disks { V c } i=0 satisfying the following conditions (See Figure 8): ● V c is a small disk-neighborhood of f i (c); ● V ξ ∩ V ξ = ∅ for i̸= j; ● VcN C Fa; and ● f(V c ) ⊂ V c +1 for all i < N . Now we set
where α ranges over all attracting and parabolic cycles. Note that f(V) ⊂ V ∪ F. Using F and V , we define Ω as C − ( F ∪ V ). Then we can easily check that Ω satisfies the conditions in the statement. ■
3.2
Construction of Ω c and the “0-th” map h 0 .
Next we consider a horocyclic perturbation f ε → f preserving the J -critical relations of f.  For each f ε , we construct a compact set Ω ε corresponding to Ω = C − ( F ∪ V ), and the correspondence is represented by the map ho(= ho,c ) : Ως → Ω. Proposition 3.2 For each ε ≪ 1, there exists a compact set Ω ε ⊂ ⊂ and a continuous map h 0 (= h 0,ε) : Ωε → Ω with the following properties: 1. Ως∩(P(fς)∪C(fς)) = J(fς)∩(P(fς)∪C(fς)), and this set is the union of all parabolic points of f ε and all critical orbits in J(f ε ). 2. J(fe) ⊂ Ω e and fe-1(Ω e ) ⊂ Ω e 3. ho : Ω ε → Ω is surjective. 4.  If there exists y ∈ Ω such that card(h − 1 (y)) ≥ 2 then y is a parabolic point and card( h − 1 (y)) = p(y).  Moreover, y is perturbed into an attracting planet and h − 1 (y) is the set of p(y) repelling satellites of the attracting planet. 5. For each b ε ∈ Z ε , there exists a unique b ∈ Z 1 such that b ε → b, and ho(be) = b. Moreover, for any fixed r > 0, we can make h 0 satisfy sup {d σ (h 0 (x), x) : x ∈ Ω ε } ≤ r for all є ≪ 1. For example, suppose that f is hyperbolic; that is, both A and J(f) ∩ C(f) are empty. For ε ≪ 1, f c is a very small perturbation of f , thus every attracting cycle of f is perturbed into an attracting cycle of f c .  By uniform convergence of f ε → f , we obtain f ε ( F ) ⊂ F for all ε ≪ 1. Similarly, if ε ≪ 1, V satisfies f ε (V) ⊂ V ∪ F . Hence we can set Ω ε := Ω = C − (F ∪ V) and h 0 := id. For general geometrically finite rational maps, to construct Ω ε for f ε → f , we need to modify F ; in particular, certain parts of the flowers { F a } a ∈ A . We also need additional modification near the critical orbits in the Julia set. Let us fix an r > 0 and set B x := B σ ( x, r /2) for each x ∈ A ∪ Z 1 . We suppose that r is sufficiently small so that B x ∩ B x ′ = ∅ for different x, x ′ ∈ A ∪ Z ¹ and that B x ⊂ Int(Ω) for x ∈ Z 1 − A. Modification of Ω near the parabolics.  Fix a parabolic point of f , say a ∈ A. Set E α := Ω ∩ B α . We may assume that E α is a union of p ( a ) narrow cusps near the repelling directions. Lemma 3.1  For each ε ≪ 1, there exists a compact set E ′ and a map h α : E ′ → Ea with the following conditions: ● ∂E α ∩ ∂B α = ∂E α ∩ ∂B α , and h α is the identity on this set. ●  f − 1 (E ′ f (α)) ∩ B α ⊂ E ′ α . ●  B α − E'α ⊂ F(f ε ). ● h α : E ′ α → E α is continuous and surjective. ● If y ∈ E α and card( h − 1 (y)) ≥ 2, then y = a. In this case, a is perturbed into an attracting planet a ε and h − 1 (y) is the set of all repelling satellites of a ε . ● d σ (h α ( x ), x ) ≤ r for any x ∈ E ′α. Proof.   For simplicity, here we only treat the case where a is a fixed point with multiplier 1. The case of a with multiplier ≠ 1 or period ≠ 1 is similar. As f ε → f horocyclically, suppose that α is perturbed into the planet a ε , a fixed point of f ε Let us consider the local dynamics by f− 1 and f − 1 restricted near B a . We denote by g (resp. g ε ) the branch of f − 1 (resp. f − 1 ) near B α which fixes α (resp. α ε ). Then a is still a parabolic fixed point of g and a ε is a fixed point of g ε with multiplier 1/f i (α ε ). Note that g ε → g is a locally defined horocyclic perturbation, thus we can apply Lemma 2.1. Set p := p(a), the petal number of a . The construction of E ′ and h α breaks into the cases of p = 1 and p ≥ 2. Case 1 : p = 1. In this case, we may assume that α c is an attracting or parabolic fixed point of g c . (Here we need not distinguish planet from satellite.) Now ∂E α ∩∂B α is an arc. Let ε 1 and ε 2 be its end points. Since r is sufficiently small, we may assume that e 1 and e 2 are enough close to the attracting direction for g , and that their orbits by g accumulate on a within E g . Then we may apply the argument in Lemma 2.1 to the orbits of e 1 and e 2 by g ε . For ε ≪ 1, joining the orbits of e i ( i = 1,2) by g c contained in B a , we obtain a piecewise smooth Jordan arcs η i with the following properties: ● Joining from ei to a ε . ● gε(η i ) ⊂ η i ⊂ B α ∪ { e i } and fε(η i ) − B α ⊂ F α ● η ∩ η ₂ = {α ε}. In fact, joining e i and g ε ( e i ) by nearly straight curve and taking the union of their forward images by g ε , we obtain such a curve η i . We define E α as the closure of the region in B a enclosed by η 1 , η 2 and ∂ E α ∩ ∂ B α . Then we see that f − 1 (E' a ) ∩ B α ⊂ E' a . We claim that B α − E ′ α ⊂ F ( f ε ) for ε ≪ 1. Let us take an arbitrary x ∈ B α − E ′ α . If the orbit of x never escapes from B and is attracted to the parabolic or attracting point of f c in B a , then x ∈ F ( f c ). So we consider the case where the orbit of x escapes from B α . Then for some i > 0, f i ( x ) is contained in the compact set F a − B a ⊂ F(f). By the local dynamics in F α , there exists N ≫ 0 such that f N (F α − B α ) is contained in B α and is sufficiently near the attracting direction of a . By uni- form convergence of f ε → f , we may suppose the same holds for f ε N (F α − B a ). Furthermore, since f n ( F a − B a ) converges uniformly to a within B a as n tends to infinity, we may apply the argument in Lemma 2.1 to the forward images of f ξ ′ ( F α − B α ) by f ε ; thus f ε ′ ( F α − B α ) converges uniformly to the parabolic or attracting point of f ε within B α . This implies x ∈ F ( f ε ) Finally we define the map h α : E α → E α : Let us take a Riemann map R ε : Int( E ′ α) → D, here D is the unit disk. Since the boundary of E ′ a is a Jordan curve, R ε is extended to a homeomorphism R ε : E ′ α → D. Similarly, we take an extended Riemann map R : E α → D.  By choosing a suitable topological map H c : D → D, we obtain h a := R-1 o H c o R c such that: ● h α : E 'α → E α is a homeomorphism; ●  h α |(∂E' α ∩ ∂B α ) = id; and ● h α(αε) = α. Furthermore, since the radius of B a is r/2, we obtain d σ ( h α ( x ), x ) ≤ r for any x E E'a Case 2 : p ≥ 2.  Now E α is the union of p narrow cusps which intersect only at a . We distinguish these p cusps as { E 1 , . . . , E p }; that is, each E j is a union of {a} and one of the p connected components of E α − {a}. Let e 1j and e 2j be the end points of ∂E j ∩ ∂B α for j = 1, . . . ,p. As in the case of p = 1, let us apply the argument in Lemma 2.1. Then we can take g c -invariant path η ij which joins e ij and a parabolic or attracting point of g ε generated in B α by the perturbation of α . We define E ε ε ε as the compact set in B a enclosed by ηιj, η 2j , and ∂E j ∩ ∂B a .  Note that we obtain the following three cases: 1. The planet a c is a parabolic fixed point of f c , that is, the multiplier f ξ ( a c ) satisfies f ξ (α ε ) = 1. In this case, each E ξ joins E j ∩ ∂ B α to α ε and ∩ ξ = 1 E j ; = {ae}. 2. The planet a c is a repelling fixed point of f c , that is, the multiplier f ξ ( α c ) satisfies | f ′ ( α e ) | > 1. In this case, each E ′ j joins E j ∩ ∂ B α to a e and ∩ j = 1 E j ; = {αε} (Figure 10). 3. The planet a c is an attracting fixed point of f ε , that is, the multiplier f ξ ( α ε ) satisfies | f ζ ( a c ) | < 1. In this case, each E ζ joins E j ∩ ∂B α to one of the symmetrically arrayed repelling satellites of a c and ∩ j = 1 E j = ∅ (Figure 10). Now we set E ′ := U j − 0 E j . We can show B α − E ′ α ⊂ F ( f ε ) for ε ≪ 1 by the same argument as the case of p = 1. h α for p = 1, and define a continuous map h α : E ′ → E γ in the same way as h a has the following properties: ● h α |(∂E'α ∩ ∂B α ) = id; ● h α : E'α → E α is surjective; and ● if y ∈ E α and card( h − 1 (y)) ≥ 2, then y = a. Moreover, a is perturbed into the attracting planet a ε , and h − 1 ( y ) consist of p repelling satellites of a ε . In particular, we also obtain d σ (h α ( x ), x ) ≤ r for any x ∈ E ′α . Finally let us show the existence of Ωε. Proof(Proposition 3.2).   For each fixed ε ≪ 1, set
By the construction of E α , one can easily check that J(f ε ) ⊂ Ω ε and f ε 1 (Ω ε ) ⊆ Ω ε . To check that Ω ε ∩ (P(f ε ) ∪ C(f ε )) = J(f ε ) ∩ (P(f ε ) ∪ C(f ε )), it is sufficient to show that the critical orbits in the Fatou set never land on Ω e . Let us take c ε ∈ C ( f ε ) ∩ F ( f ε ). Then there exists c ∈ C ( f ) such that c ε → c (ε → 0). If c ∈ J(f), by geometric finiteness of f, the orbit of c lands on a parabolic or repelling cycle, say α .  Since the J-critical relations of f are preserved, c ε also lands on a cycle. By our assumption that c ε ∈ F ( f ε ), α must be parabolic and the orbit of c ε must land on an attracting cycle which is generated by the perturbation of α. Thus the orbit of c ε never lands on Ω ε by the definition of Uaea Ea If c ∈ F(f), the orbit of c accumulates on a parabolic or attracting cycle. By the construction of Ω, c is not contained in Ω. Similarly, by the definition of Ω ε , we may assume that c ε ξ Ω ε . Let us suppose that f ε (c ε ) ∈ Ω ε for some n . Then c ε ξ ε n (Ω ε ) ⊆ Ω ε and it is a contradiction. Thus f ε (c ε )̸̸ξ Ω ε for all n . Finally we define h 0 : Ω ε → Ω. Since f ε → f preserves the J -critical relations of f , we may assume that for any b ∈ Z 1 − A, B b contains only one point of Z ξ , say b ε , such that b ε → b. Recall that B b ⊂ Int(Ω), by the assumption for r . Let h b : B b → B be an arbitrary topological map which satisfies h b (b c ) = b and h b | ∂B b = id. Then we obtain d σ (h b (x), x) ≤ r for x ∈ B b . Let us define h 0 : Ω ε → Ω by on E' a for a ∈ A, ho = ha on B b for b ∈ Z¹ − A, and ho = hь ho = id otherwise.
4
Construction of hn
For Ω ε and Ω constructed in §3, we set Ω ξ := f ξ n (Ω ε )  and  Ω n := f − n (Ω)   (n = 0, 1, 2, . . .). In addition, we set U c := Int(Ω n ) and U n := Int(Ω n ).  By the construction of these sets, f ε : Ω n +1 → Ω n and f : Ω n +1 → Ω n are branched covering maps, where the critical values are contained in Z ε and Z respectively. Note that {Ω n } and {Ωn} form the decreasing sequences as below:
In this section, we inductively construct a sequence of lifts of h 0 : Ω ξ → Ω 0 ,
satisfying f 0 h n+1 = h n 0 f ε . Proposition 4.1  For an n ≥ 0, assume that there exists hn(= hn,c) : Ω ε → Ω n satisfying the following properties: (1, n) hn is continuous and surjective. (2,n) hn maps Un onto Un homeomorphically. Moreover, if there exists y ∈ Ωn such that card( h − 1 (y)) ≥ 2 then f n (y) is a parabolic point of f perturbed into an attracting planet and card( h π ¹( y )) = deg( f n , y ) · p( f n (y )). (3,n) For any b ε ∈ Z ε , there exists a unique b ∈ Z 1 such that h n (b ε ) = b. Under these assumptions, there exists h n+1(= h n+1,ε) : Ω n +1 → Ω n+1  satisfying f 0 hn+1 = hn 0 fe and properties (1, n + 1),  (2, n + 1) and (3, n + 1). Recall that the map h 0 : Ω ξ → Ω 0 has properties (1, 0), (2, 0), and (3, 0). Thus this proposition gives us desired { h n : Ω n → Ω n } n =1· Proof.   The proof breaks into 3 steps. Step 1: Interior correspondence. The first step is to try to construct a homeomorphism between U n+1 and U n+1 . To begin with, we construct h n+1 such that the following diagram commutes:
Here f|(U n+1 – Z 1 ) and f ε |(U n+1 – Z ε ) are d-sheeted covering maps.  Moreover, by properties (2, n ) and (3, n ), h n | ( U ε − Z ε ) is a homeomorphism.  We will construct prospective h n+1 in the diagram by lifting this h n |(U n − Z c ). Note that U n and U n for n ≥ 1 are either connected or finitely many connected components. (For example, suppose that J(f) is a Cantor set.) Hence we construct h n+1 on each connected component of U n+1 – Z l Let Q ξ be a connected component of U ε +1 − Z ε , and take a base point x ξ ∈ Q ε . Set Q ε := f ε (Q ε ), a connected component of U ε r, and set x 0 := f ε (x δ ) ∈ Q ε . Moreover, set Q := h n (Q ε ) and y 0 := h n (x 0) ∈ Q . Let y 1 ∈ Un+1 be the closest point to x 1 in f −1 (y 0 ).  Such y 1 is uniquely determined, since critical values in the Fatou sets stay a bounded distance away from Q ε and Q . Let Q 1 denote a connected component of f − 1 (Q) containing y 1 . We will lift h n to h n+1 such that the following diagram commutes:
Take a point x 1 ∈ Q 1 and a curve η ε : [0,1] → Q 1 such that η ε (0) = x 1 and ηε(1) = x 1 . Then the curve h n (fε(ηε)) has the initial point y 0 . We lift this curve to η : [0,1] → Q 1 with the initial point y 1 , and define h n+1(x 1 ) as its end point η(1). Since h n Q ε is a homeomorphism and the J -critical relations of f are preserved, for the fundamental groups π 1 (Q ε , x 1 ) and π 1 (Q 1 , y 1 ),
is a group isomorphism. Hence the above definition of h n+1( x 1 ) gives the home- omorphism h n+1 : (Q ε , x φ ) → (Q 1 , y φ ) as a lift of h n : (Q ε , x 0 ) → (Q , y 0 ) (See [11, Ch.III). Now we have a homeomorphism h n+1 : U ε +1 − Z ε → U n+1 − Z 1 . For x ∈ U ε ∩ Z ε , let us set h n+1(x) := h n(x). Then we obtain a homeomorphism h n+1 : U n+1 → U n+1 as a natural lift of h n : U n → U n .
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Step 2:  Boundary correspondence.   The second step is to extend h n+1 de- fined on U ε +1 to the boundary ∂U ε +1 = ∂Ω ε +1 , in a natural way. Here we should be careful about the boundary correspondence near the preimages of a parabolic point which is perturbed into an attracting planet. Note that the injectivity of h n has already been broken at some of these points. To construct h n +1 | ∂Ω ε +1 , it suffices to construct h n +1 | ∂Ω ε for each Ω ε in Step 1. For x 1 ∈ Q ε and x 1 ∈ ∂Q ε , take a curve ηε : [0, 1] → Q ε ∪ { x 1 } with ηε (0) = x 1 and ηε (1) = x 1 . Now the value of h n +1 at x 1 is defined by
One can easily check that this value does not depend on the choice of η ε . By this definition, if a ∈ ∂Q 1 is a parabolic point with p ≥ 2 petals and is perturbed into an attracting planet, then h − 1 ( a ) is p distinct points in ∂Q 1 corresponding to p distinct accesses to a in E α . The case of k -th preimages of a with k ≤ n+1 is similar. Moreover, note that h n+1(x 1 ) = h n(x 1 ) if x 1 ∈ ∂Q i ∩ Z i . Step 3:  Checking the properties.   Now we have already defined a continuous map h n +1 : Ω ε → Ω. For the last step, we check that h n +1 has properties (1, n +1), (2, n + 1) and (3, n + 1). Note that h n+1 | Q c is a homeomorphism and h n+1 | Q c is continuous.  Thus bijectivity of h n+1 may break only at the boundary points. For a boundary point y 1 of Q 1 , take a curve η : [0, 1] → Q 1 ∪ { y 1 } such that η(0) = y 1 and η(1) = y 1 . Then the limit of h − 1 ( η ( t )) as t → 1 determines an element of h − 1 ( y 1 ) which is contained in the boundary of Q c . Hence h n+1 | ∂Q c is surjective and we obtain property (1, n + 1). Next, suppose that q := card( h − 1 ( y 1 )) ≥ 2. Note that η determines an access to y 1 within Q 1 and an element of h n +1 (y 1 ). Thus q ≥ 2 means that there are two or more distinct accesses to y 1 (more precisely, there are two or more distinct prime ends of Q 1 at y 1 ). By the definition of Ω n+1 , f n+1 ( y 1 ) must be a parabolic point with p ≥ 1 petals such that q = p · deg( f n +1 , y 1 ) ≥ 2. By the definition of Ω n +1 , such a must be perturbed into an attracting planet, since otherwise all possible η determines the same element of h n +1 ( y 1 ). Thus we obtain property (2, n + 1). Finally, we obtain property (3, n + 1) by the fact that h n+1( x 1 ) = h n( x 1 ) if x¹ E Z¹ ■
5
Contracting property of f-1
This By the construction above, h n is one of the branches of f-n o h 0 o f n . implies, to obtain the convergence of { h n } on J ( f ), it is necessary to use some kind of contracting property of the branches of f−1 (in other words, some kind of expanding property of f ) near the Julia set. In this section, to obtain such a
25
property of f , we follow [16, Step 2-5] with brief sketches of the proofs. The idea is originally due to A. Douady and J. H. Hubbard[4, Exposé No.X].
5.1
Branched covering of Ω
There exists a function v : Ω → N such that v(x) is the multiple of v(y) · deg(f,y) for each y ∈ f−1( x ). For example,
satisfies this condition.  Here we take v as the function which takes minimal possible values. Note that Z = {x E Ω : v(x) ≥ 2}. Let O be an open δ -neighborhood of Ω with δ ≪ 1. Then O contains a neighborhood of each a ∈ A. For x ∈ O − Ω, set v(x) = 1. Let us take an N -sheeted branched covering q : O* → O such that: ● O* is connected; ● there are N/v(x) points over x ∈ O; and ● for any y ∈ q-1(x), deg(q,y) = v(x). Now set U := Int(Ω), U* := q-1(U) and Ω* := q-1(Ω). For U* let us take the universal covering π : 𝔻 → U*, where 𝔻 is the unit disk. Then we obtain a branched covering p := q o π : D → U. Let Γ be the fundamental group of U* and Λ(Γ) the limit set of Γ . By lifting paths in Ω* terminating at boundary points, we can continuously extend π to the ideal boundary, π | (∂D − Λ ( Γ )) → ∂Ω*. Thus we obtain a branched covering p : D − Λ(Γ) → Ω. For a parabolic point a of f with multiple petals, every component Remark. of E a — { a } defines a different access to a . For such accesses, corresponding ideal boundary points of ∂D − Λ( Γ ) over a are distinct.
5.2
Lifting f-1
Next, we lift f-1 to the branched covering D – Λ(Γ) of Ω. Proposition 5.1  There is a holomorphic map g : D → D such that f opog = p. Moreover, g can be extended to g : D − Λ(Γ) → D − Λ(Γ) continuously. Sketch of the proof.   For x ∈ Ω, we take a small disk neighborhood B x .  Let G be one of the components of q-1( B x ), and H that of ( f o q )-1( B x ).  Then there exists a unique y such that {y} = f-1(x) ∩ q(H). By taking suitable local coordinates, q|G → B x and ( f o q )|H → B x are represented as z → z v(x) and z → zv(y)deg(f.y) respectively. Thus we can define the unique map ggh : G → H which has the form z + →  z v(x)/(v(y) deg(f,y)) as a branch of (f o q)−1 o q. Let us fix x 0 ∈ Ω − Z and ˜ ī 0 ∈ p− 1 ( x 0 ). Let η be a curve η : [0, 1] → Ω* with η(0) = π( ῖ 0) and η((0,1)) ⊂ U*, and η ′ be the unique lifting of η by π with ñ(0) = ñ 0 . Now we consider analytic continuation of the function elements {gcH} along ñ.  Let gcoHo be a function element at π(ño).  Since D − Λ(Γ) is simply connected, the analytic continuation of gc o H o along ñ determines a unique function element at ñ (1). Next, by ranging over all possible η , we obtain g : D – A( Γ ) → D – A( Γ ). It is clear that g | D is holomorphic. ■
5.3
The metric p
Proposition 5.2  There exists a piecewise continuous metric ρ with the following properties: ● ρ is defined on U — Z and small disk neighborhoods for each parabolic point of f ●  For every C 1  curve η ⊂ f − 1 (Ω) = Ω 1 , length ρ (f ◦ η) > length ρ (η). So f is expanding for ρ in the sense of this inequality. Sketch of the proof.   Let ρ 0 = υ 0 ( z ) dz | be a metric of U − Z induced from the Poincaré metric of D by the branched covering p : D → U.  Note that uo(z) < |z — b|−1+1/v(b) near b ∈ Z . Thus any rectifiable curve η : [0,1] → U passing through Z has finite length with respect to ρ 0 . However, any curve in f-1(Ω) terminating at A has infinite length with respect to ρ 0 . So we try to modify ρ 0 so that such a curve has finite length. For a sufficiently small δ > 0 and for each a ∈ A , set D a := B σ (a,δ) and D := UgEADa.  Note that Ω ∩ D is a finite union of narrow cusps near the repelling directions. Thus on each D α , we can take a suitable local coordinate ζα such that f is strictly expanding from the metric |dζα | to the metric |dζf(α)| on any compact subset of f-1(Ω ∩ D f(α)) ∩ D α − {α}. Furthermore, we take a sufficiently large M > 0 so that for any a ∈ A , f is expanding from ρ 0 to M | dζα | on a relatively compact set f-1(Ω ∩ D α − Z) − D. Set u α(z)|dz| := |dζα|. Then
27
we define the metric ρ = u(z)|dz| on U ∪ D − Z by u(z) := min { u 0(z), M u α(z)} for z ∈ D α , and by u(z) := u 0(z) otherwise. By construction, it is not difficult to show for z ∈ f− 1 (Ω − Z) − A. This implies length ρ (f ◦ η) > length ρ (η). for every C 1 curve η ⊂ f − 1 (Ω). ■
5.4
Continuous modulus
Let ˜ ρ be the lifting of ρ on p-¹(U — Z). Since f-¹(Ω) = Ω¹ has one or more connected components, p − 1 (Ω 1 ) is either connected or has countably many con- nected components. Take one of the components of p-1(Ω 1 ), say Q, and take x, y ∈ Q. We define the distance by
where ỹ ranges over all rectifiable curves such that ñ : [0, 1] → p-1(Ω1), ñ(0) = x, and ñ(1) = y. Note that such ñ has finite length with respect to ñ . Now (Q, d ñ ) is a complete metric space. For different components Q and Q ′ of p − 1 (Ω 1 ), we formally define d p (x, y) := ∞ if x ∈ Q and y ∈ Q ′ . For g , a lifting of f-1, we define a function τ g : R+ → R+ by τg(s) := sup {d p (g(x), g(y)) : x, y ∈ p − 1 (Ω 1 ), d p (x, y) ≤ s }. Furthermore, we define τ : R + → R + by τ(s) := sup {τg(s) : g a lifting of f-1}. Then we obtain: Proposition 5.3 τ has the following properties: (i) τ is an increasing and right-continuous function; (ii) s > τ(s) for any s; (iii) the function s → s − τ(s) is also increasing; and (iv) For any x, y ∈ p-1(Ω¹) and any lifting g of f-1, d p (g(x), g(y))  ≥  τ(d p (x, y)). Sketch of the proof.   If we replace τ by τ g , then (i), (ii) and (iv) are almost clear by definition. (iii) follows from the fact that τ g (s 1 + s 2 ) ≤ τ g (s 1 ) + τ g (s 2 ). A calculation shows that there exist d distinct liftings of f-1 , say g1 , . . . , gd , such that any τ g coincide with one of τ g 1 , . . . , τ g d . Thus τ(s) = sup {τ g i (s) : 1 ≤ i ≤ d }, and satisfies properties (i)-(iv). ■
6
Convergence of h n
In this section, we give the proof of the convergence of the sequence { h n : Ω n → Ω n } n =0 Here the expanding property of f with respect to ρ plays an important role. For instance, we can easily show the convergence when f is hyperbolic: Proposition 6.1 Suppose that f is hyperbolic.  For ε ≪ 1, the sequence h n converges uniformly to the limit h ε on J ( f ε ) which satisfies f o h ε = h ε o f ε . Proof.  Since f has no parabolic point nor critical point in J ( f ), the metric ρ in Proposition 5.2 is the Poincaré metric on U.  Now Ω 1 ⊂ U thus there is a constant C such that f*ρ/ρ ≥ C > 1 on Ω¹. Note that the constant M := sup {dρ(hρ(x),hρ(x)) : x ∈ Ω į } is finite since h 0 (Ω ε ) ⊂ U. For any x ∈ Ω ε , we obtain Cdρ(h1(x), h2(x)) ≤dρ(f(h1(x)), f(h2(x))) = dρ(h0(fε(x)), h1(fε(x))) <M, thus d ρ (h 1 (x), h 2 (x)) ≤ M/C. Similarly, for any x ∈ J (f c ), we obtain dρ(hn(x), hn+1(x)) ≤ M/C n → 0   (n → ∞). (Recall that J(f ε ) ⊂ Ω n and thus h n | J(f ε ) are defined for any n ≥ 0.) Hence h n converges uniformly and rapidly to the limit h ε on J ( f ε ). The relation f o h ε = h ε ◦ f ε follows from f ◦ h n +1 = h n ◦ f ε . ■ Let us consider the general case. When f has parabolic points, it is not uni- formly expanding on Ω 1 . However, since it is uniformly expanding on each com- pact subset of Ω 1 with respect to the metric ρ, h n converges slowly to the limit: Proposition 6.2  For ε ≪ 1, the sequence h n converges uniformly to the limit h c on J(f ε ) which satisfies f o h ε = h ε o f ε . Moreover, h ε can be arbitrarily close to the identity map:  That is, for arbitrarily small r > 0, if ε ≪ 1, h ε satisfies sup {dσ(hε(x), x) : x ∈ J(fε)} < r. Let us fix an arbitrary L > 0. Then we may assume that Proof. dρ(hρ(x), h 1 (x)) < L − τ(L) for any x ∈ J ( f ε ). In fact, by the construction of h 0 and h 1 , if ε ≪ 1 , d ρ ( h 0 ( x ), h 1 ( x )) can be arbitrarily small for any x ∈ J(f ε ). We claim that dρ(ho(x), hn(x)) < L for any n ≥ 1 and any x ∈ J(fε). If n = 1, dρ(hρ(x), h 1 (x)) < L−τ(L) < L. For n = k, let us assume that dρ(hρ(x), h k(x)) < L for any x ∈ J(f ε ). We first show that dρ(h 1 (x), hk+1 (x)) < τ(L). By assumption, we can take a rectifiable curve η : [0, 1] → Ω ¹ such that ● η(0) = h 0(f ε (x)) and η(1) = h κ (f ε (x)); ● η ∩ Z = ∅; and ●  L > length ρ (η). Fix zo ∈ p-1(ho(fe(x))), and let ñ be the lifting of η by p whose initial point is z 0 . Then the end point over h k ( f ε ( x )) is uniquely determined, say z 1 , and
By using the function τ , τ(L) > τ(d ρ ( z 0 , z 1 )) ≥ d ρ (g(z 0 ), g( z 1 )), where g is a lifting of f-1 such that p o g(zo) = h1(x). Then we can take a curve ñ' : [0, 1] → D − Λ(Γ) such that ● ñ'(0) = g(zo) and ñ'(1) = g(zi); ● ñ' ∩ p-1(Z) = ∅; and ● τ(L) > length ρ (ñ'). Hence
> dρ(ρ(ρ( g( zo )), p( g( z1 ))) = dρ( h1 ( x ), hk+1 ( x )). Then for n = k + 1 and for any x ∈ J(f ε ), dρ(hρ(x), hk+1(x)) ≤ dρ(hρ(x), h₁(x)) + dρ(h₁(x), hk+1(x)) < L − τ(L) + τ(L) = L. Thus we have shown the claim by induction on n . Let us show the convergence. By the same argument as above, for sufficiently large integer l, m,
Because we can take arbitrary x ∈ J ( f ε ), h n converges uniformly on J ( f ε ) with respect to the distance d ρ . Since the topology of Ω n defined by d ρ is equivalent to the topology defined by the spherical distance d σ , h n also converges uniformly on J(f c ) with respect to d σ . By continuity of each h n , the limit h c is also continuous. The relation f 0 h ε = h ε 0 f ε follows from f 0 h n+1 = h n 0 f ε . Finally we show the last part of the statement. Let us fix any r > 0 and suppose that ε ≪ 1. Then we can take h 0 such that d o (x,h 0 (x)) < r/2 for any x ∈ J ( f ε ). On the other hand, by the claim above, we obtain dρ (hρ(x), hε(x)) ≤ L for arbitrarily small L . Since we may also suppose that L is sufficiently small such that d σ (h 0 (x), h ε (x)) < r /2 for any x ∈ J (f ε ), we obtain dσ(x,hε(x)) ≤ dσ(x,hο(x)) + dσ(hο(x),hε(x)) < r. ■
7
Almost bijectivity and uniqueness of h ε
In this section, we prove that the continuous map h c in Proposition 6.2 maps J(f c ) onto J(f) "almost bijectively"; that is, there are at most countably many points in J ( f ) where h ε is not one-to-one. Furthermore we prove the uniqueness of such an h ε . First we show: Proposition 7.1 h ε maps J(f ε ) to J(f). Proof. Let X denote the set of all repelling periodic points of f ε . Since h ε ◦ f ε ＝ f n o h ε for any n , h ε maps X to a set of periodic points of f in Ω , which must be a subset of J ( f ). Since h ε is continuous and J ( f ε ) = X, h ε maps J ( f ε ) into J ( f ). ■ Next, we complete the proof of Theorem 1.1 under the assumption that J(f) ≠ C. For fixed ε, let A_ = A_,ε ⊂ A be the set of all parabolic points of f which are perturbed into attracting planets of f ε . Proposition 7.2  If ϵ ≪ 1, h ϵ : J ( f c ) → J ( f ) has the following properties: ● (Surjectivity) h ε is surjective. ● (Almost injectivity) If h ε ( x ) = h ε ( x ′ ) for distinct x, x ′ ∈ J ( f ε ), then there exists an integer N such that f N ( x ) and f N ( x ′ ) are repelling satellites of an attracting planet a ε generated by the perturbation of a point in A_. ● (Uniqueness) h ε is the unique semiconjugacy between f ε and f on their respective Julia sets which satisfies properties 1 and 2 in Theorem 1.1. By the almost injectivity above, we obtain the precise condition for h ε to be a topological conjugacy. Corollary 7.1 h ε is a topological conjugacy if and only if A _ = 0; that is, none of the parabolic points of f is perturbed into an attracting planet. Proof of Proposition 7.2:  Surjectivity.    Fix any y ∈ J(f).  By surjectivity of h n , there is a sequence x n ∈ Ω ε ⊂ Ω ε such that h n ( x n ) = y. Since Ω ε is compact, { x n } has an accumulation point x ∈ Ω ε and we can choose a subsequence x n k so that x nk → x (k → ∞). Now we claim that x ∈ J(f c ). If x ∈ F(f c ), f n (x) is attracted to an attracting or parabolic cycle as n → ∞. Thus there exists an N and a small disk neighborhood D such that f n ( D ) is outside of Ω c for all n ≥ N . On the other hand, for all k ≫ 0, we have n k ≥ N, x n k ∈ D, and f n k ( x n k ) ∈ Ω e . This is a contradiction. Since h n → h c uniformly and the family { h n } is clearly equicontinuous, the inequality implies y = h ε ( x ). Thus h ε is surjective. Preliminary to the almost injectivity and uniqueness.  Since f is geo- metrically finite and the assumption that J(f) ≠ C, f has at least one critical point in the Fatou set, and so does f ε . Now we take suitable conjugations of f ε → f 0 = f by rotations of C so that ∞ ∈ C (f ε ) ∩ F (f ε ). By the construction of Ωε, there exist R ≫ 0 such that D(R) := ˆ C − {| z | ≤ R} is a disk neighborhood of ∞ which is not contained in Ω c for all 0 ≤ ε ≪ 1. Then Ω c and J( f c ) are bounded sets in the complex plane. For δ > 0 and x ∈ C, we set
which is an open Euclidean ball. Now we fix δ to be sufficiently small so that the set
is a disjoint union of balls satisfying the following conditions: dρ(y, hε(x)) ≤ dρ(hnκ(xnk), hnk(x)) + dρ(hnk(x), hε(x)) ● if an x ∈ A ∪ Z 1 is periodic, then there exists a local chart on B(x, δ) as (2.2) or (2.4); and ● for x ∈ Z 1 − A, P(f) ∩ B(x, δ) = {x}. Set š := d(P(f), J(f) – B), where d( , · ) is the distance between sets measured by Euclidean distance. Since f is geometrically finite, every critical orbit either accumulates on an attracting or parabolic cycle, or is already contained in Z 1 . Hence we obtain 0 < š ≤ δ. Now we claim that d(P(fc), J(fc) - B) > š/2 for all ε ≪ 1.  It suffices to restrict our attention to the perturbation of the critical orbits accumulating on A or Z 1 . First, take a parabolic cycle α ⊂ A and a critical orbit accumulating to α. By horocyclicity of f ε → f , we may apply Lemma 2.1. That is, for ε ≪ 1, the corresponding perturbed critical orbit of f e is contained in UgeaB(a, δ) ⊂ B except finitely many points in the orbit. Since f ε → f uniformly, such finitely many points are very close to the original ones. On the other hand, J( f ε ) is very close to J(f) with respect to the Hausdorff topology, since h ε maps J(f ε ) onto J(f) and r-neighborhood of J(f) with respect to the spherical distance contains hç¬(J(f)) = J(fc). (Recall that r is fixed and arbitrarily small for ε ≪ 1.) Thus such finitely many points stay away from J( f c ) – B for ε ≪ 1, and the distance can be at least š/2. Next, take b ∈ Z 1 . Since f ε → f preserves the J -critical relations of f , for all ε ≪ 1, we may suppose that there exists a unique b ε ∈ f ε –1 (P(f ε )) such that | b − b c | < δ/2.  For such b c , d(b c , J(f c ) − B ) ≥ δ/2 ≥ δ/2.  Thus we conclude the claim. Replacing f ε (resp. f ) by its suitable iteration, we may consider the extreme case where every point in h − 1 (A) ∪ Z ε (resp. A ∪ Z ) is a fixed point of f ε (resp. f), and the multipliers of all parabolic points are 1. Then Z c and Z are the sets of all critical values of f c and f on their respective Julia sets. Set Γ_ = Γ_,ε := h ō 1 ( A_ ), the set of all repelling satellites generated by the perturbation of parabolic points in A_. Note that now every element in A_ or Γ_ is a fixed point of f or f ε respectively. Also, note that Γ_ and Z ε are disjoint. Almost injectivity.   Now let us start the discussion on the almost injectivity of h ε . We suppose that h ε ( x ) = h ε ( x ′ ) for distinct x, x ′ ∈ J ( f ε ). Set x n := f ε ( x ) and x'n := f ε (x'). Then h ε (x n) = h ε (x'n) because f n o h ε = h ε o f ε . Recall that dσ(x, hε(x)) < r for any x ∈ J(fε). Thus we obtain dσ(x n, x'n) ≤ dσ(x n, hε(x n)) + dσ(hε(x'n), x'n) < 2r and it implies | x n − x ′ n | = O(r).  Indeed, since the Julia set is contained in Ĉ − D(R), there exists a constant M ≈ 1 + R 2 such that | x n − x ′ − | ≤ M r for sufficiently small r . Now we set
Then we may suppose that r is sufficiently small such that ñ ≤ Mr < š/2 for ε ≪ 1. Note that ř ≤ Mr ≪ δ/2 also holds. For the orbit of the x and x ′ , we consider the following three cases: 1. Both x n and x ′ land on Γ_. 2. x n lands on Γ_ but x n never lands on Γ_ . 3. Both x n and x ′ never land on Γ_. Case 1:   Suppose that x n lands on h ε 1 ( a ) for some a ∈ A − when n = N . Here h ε 1 ( a ) ⊂ Γ_ is a set of repelling fixed points contained in B σ ( a, r ). By the facts that B σ (a, r) ⊂ B (a, Mr) ⊂ B (a, δ/2) and ñ < δ/2, x'n must be contained in B(α,δ) for all n ≥ N . If x'N̸f h − 1(α), by the local dynamics of f ε on B ( a, δ ) in the form (2.4), x ′ n goes out of B ( a, δ ). Thus x ′ N ∈ h ε 1 ( a ); that is, x n and x ′ n simultaneously land on repelling satellites in h − 1 (a), when n = N . Hence we need to show that the other cases cannot occur. Case 2:  We suppose again that x n lands on h ε 1 ( a ) for some a ∈ A_ when n = N. By the same argument as Case 1, x ′ n must be contained in B( a, δ ) for all n ≥ N . However, x ′ n̸f h − 1 (α) ⊂ Γ− , and thus by the local dynamics of f ε on B(a, δ) in the form (2.4), x'n goes out of B(a, δ). This is a contradiction. Case 3:   Furthermore we need to consider the following three cases: I. x n lands on Z ε but x ′ never lands on Z ε . II. Both x n and x ′ n land on Z ε . III. Both x n and x n never land on Z ε . Case 3-I:   Suppose that x n lands on h ε 1 ( b ) for some b ∈ Z when n = N . Here hç 1 ( b ) ⊂ Z c is a repelling or parabolic fixed point of f c contained in B σ ( b, r ). By the same argument as above, x ′ n must be contained in B( b, δ ) for n ≥ N . Now x', never lands on Z ε . This implies, by the local dynamics of f ε on B(b,δ) in the form (2.2) or (2.4), x n goes out of B(b, δ). This is also a contradiction. Case 3-II:   Since ř < δ /2 and all elements of Z ε remain at leaset δ apart, the orbits of x and x ′ have merged before landing on Z ε : That is, there exist two integers N 1 and N 2 with N 1 < N 2 such that ● XN1 F X'N1 and XN1+1 = X'N1+1, and
● x N 2 = x ′ N 2 ∈ Z 2 and x N 2 +1 = x ′ N 2 +1 ∈ Z ε .
Set w := x N 1 +1 = x ′ N 1 +1 ·  Since w is not contained in Z c , which is the set of critical values, the inverse image f ε 1 ( w ) consists of d distinct points. (Recall that d is the degree of f .) Similarly, by the construction of h ε , h ε ( w ) =: z is not contained in Z and f-1(z) also consists of d distinct points. Moreover, since h e is surjective, h ε 1 (f−1 (z)) must consist of at least d points. Note that f ε 1 (w) ⊂ h ε 1 (f − 1 (z)). Since h ε (x N 1 ) = h ε (x ′ N 1 ) for distinct x N 1 , x ′ N 1 ∈ f − 1 (w), there exists an x ′′ ∈ h − 1 (f − 1 (z)) − f − 1 (w) which satisfies f ε (x ′′ )̸= w and hε( fε( x ′′ )) = z. Setting w′ := fε( x ′′ ), we obtain hε( w ) = hε( w′ ) for w̸= w′ . Let us replace x and x ′ by w and w ′ respectively. This reduces Case 3-II with x N 2 ∈ Z e to Case 3-I or 3-II with x N 2 – N 1 –1 ∈ Z 1 . However, as we have seen, Case 3-I implies a contradiction. In Case 3-II, we can repeat the argument above. Hence we eventually consider the case where h ε ( x ) = h ε ( x ′ ) for x̸= x ′ with x ∈ Z ε . Suppose that f ε ( x ) = h ε 1 ( b ) for some b ∈ Z.  Then f ε ( x ) is contained in B o (b, r) and is a repelling or parabolic fixed point. On the other hand, since the elements of Z ε remain separated, x̸= x ′ implies x ′̸= Z ε l , and thus f ε ( x ′ )̸= Z ε . By the local dynamics of f ε on B ( b, δ ) in the form (2.2) or (2.4), f ε ( x ′ ) is not a fixed point and goes out of B(b, δ). This is a contradiction. Case 3-III:  If either x n or x'n lands in B , it goes out of B by finitely many iterations of f ε . Now we take a subsequence { n k } of { n } so that each x n k is never contained in B ; that is, x nk ∈ J(f ε ) − B. Recall that d(P(f ε ), J(f ε ) − B) > š/2. For any s satisfying ñ < s < š/2 and for any k ≫ 0, there exists a branch g nk of f ξ − nk on B ( x nk , s ) which is univalent and gnk ( x nk ) = x . Set V nk := gnk ( B ( x nk , ñ )). Then V nk contains x and x ′ . By applying the Koebe distortion theorem to g nk on B( x n k , s), we obtain
diam V nk = O(|g'nk(xnk)|) = O(1/|(f'nk)'(x)|).
If | P ( f ε ) | < 3, f ε is conjugate to z → z ≠ d , and thus it is hyperbolic. On the Julia set, |(f ε π κ ) ′ ( x ) | → ∞ as k → ∞ hence lim(diam V nk ) = 0. It contradicts x ≠ x'. If | P ( f ε ) | ≥ 3, let ρε be the Poincaré metric of ˆ C − P ( f ε ). By [13, Theorem 3.6], since x n $ P(f ε ) for any n , we obtain
Now recall again that d(P(fc), J(fc) − B) > š/2. Since x nk and x stay away from P(fc), ρε(fcrk(x)) and ρε(x) are bounded. Hence |(fcrk)'(x)| → ∞ as k → ∞ , then lim(diam V n k ) = 0. It contradicts x ≠ x ′ again.
35
From Proposition 6.2, Proposition 7.1 and the proof of the almost Uniqueness. bijectivity above, it is easy to check that h ε satisfies properties in Theorem 1.1. In particular, we obtain property 1 in Theorem 1.1 from the almost injectivity discussed above and property (2, n ) of h n in Proposition 4.1. Let h ′ ε be another semiconjugacy between f ε and f on their respective Julia sets with properties 1 and 2 in Theorem 1.1. Take a repelling periodic point x of f ε which has period more than one. By our assumption that h ε 1 ( A ) ∪ Z ε is a set of fixed points, x does not belong to Γ − ∪ Ζ ε . By surjectivity of h ε , there exists an x' E J(f ε ) such that h ε ( x ) = h ε ( x ′ ). It is easy to see that h ε ( x ) and x ′ are also repelling periodic points with the same period as x . Set x n := f ε ( x ) and x ′ n := f ε ( x ′ ). Then h ε ( x n ) = h ε ( x ′ n ) because h ε and h ε are semiconjugacies.  Moreover, we obtain dσ ( x n, x ′ n ) < 2r from property 2 in Theorem 1.1. Thus | x n − x ′ n | ≤ Mr < δ/2 for all n and we may suppose that x n belongs to Γ − ∪ Ζ ε as well as x n . Now we can apply the same argument as Case 3-III of the proof of the almost injectivity, and we conclude that x = x ′ . This means that h ε = h ′ , on the dense subset of J(f ε ), because repelling periodic points are dense in the Julia set. Since h ε and h ε are continuous, h ε must coincide with h ε on J ( f ε ). ■
Geometrically finite maps with the empty Fa-
8
tou set
In this section, we prove Theorem 1.1 for a geometrically finite rational map f with J(f) = C by using the same idea as in the case of J(f) ≠ C. Now f has no parabolic or (super)attracting periodic point.  Moreover, by the geometric finiteness, every critical point of f is preperiodic; that is, f is postcritically finite.  Then we can consider the orbifold O f with base space C which is parabolic or hyperbolic type[13, §A]. This Of has an orbifold metric ρ = ρ(z)|dz| which is induced from the Euclidean or hyperbolic metric of the universal covering. In both cases, there exists a constant C > 1 such that
(See the argument in [13, Theorem A.6]). Note that ρ has singularity at b ∈ P(f) as |d(z — b)¹/v(b)|. Let us consider a horocyclic perturbation f ε → f preserving the J -critical relations of f . Since f has no parabolic point, horocyclicity is trivial. By the J -critical relations of f , f c is also postcritically finite. Since f has no attracting
36
or superattracting periodic point, f c has no superattracting periodic point: This implies J(f ε ) is also the whole sphere (See [13, Theorem A.6] again). Now let us begin the construction of h ε . Proof of Theorem 1.1 in the case of J(f) = C.  First, set Ω := C and Ω ε := C. We take h 0 : Ω ε → Ω as a homeomorphism which satisfies condition 5 of Proposition 3.2. For any fixed r > 0, if ε ≪ 1, such h 0 satisfies d σ (h 0 (x), x) < r for all x E C. Next, we lift h 0 to the family of homeomorphism { h n : ˆ C → ˆ C } n =1 as in Proposition 4.1. We can show that h n converges to the limit h e in the same way as Proposition 6.1. In fact, we may replace the Poincaré metric in the proof of Proposition 6.1 with the orbifold metric ρ of O f . Furthermore, we can also lift h = 1 to the uniformly convergent sequence of homeomorphisms { h = 1 }. The limit must be surjective and thus h ε : C → C is a homeomorphism. Finally, we show the uniqueness in the same way as Proposition 7.2: Let h ξ be another conjugacy with property 2 in Theorem 1.1, and x be a repelling periodic point of f ε which does not belong to P ( f ). Since h ε is a homeomorphism, there exists a unique x ′ such that h ε ( x ) = h ε ( x ′ ). Set x n := f ε ( x ) and x ′ n := f ε ( x ′ ). By using the uniformly expanding property of f e with respect to the orbifold metric ρε of O fε, dρε ( x, x ′ ) is bounded by dρε ( x n, x ′ n )/C ε  with C ε > 1. This implies x = x ′ . Thus h c = h c on a dense subset of the sphere, which is a set of repelling periodic points.  By continuity of h ε and h ε , we obtain h ε = h ε on the whole sphere. ■ Remark.   If the orbifold O f does not have signature (2,2,2,2) , by Thurston's theorem([5], [13, Theorem B.2]), h ε is a Möbius transformation which conjugates f ε to f . Here we gave a general construction of the conjugacy h ε including such a particular case of signature (2,2,2). Acknowledgements.   I would like to thank H. Shiga, M. Shishikura, H. Sumi, M. Taniguchi and the referee for their helpful advice and encouragement. This paper was revised during my stay in SUNY at Stony Brook. I also would like to thank M. Lyubich and S. Sutherland for their hospitality and helpful comments.