Note on dynamically stable perturbations ofparabolics
Tomoki Kawahira ∗Nagoya University
kawahira@math.nagoya-u.ac.jp

Abstract
In this note, we sketch some results on almost-dynamics-preserving perturbations of rational maps with parabolic cycles.

1

Introduction with rabbits

Well known “Douady’s rabbit” has a friend called “fat rabbit” at the root of
1/3-limb of the Mandelbrot set. However the term “fat” does not sound good, so
we tentatively call him “chubby rabbit”.

Figure 1: “plump”, “chubby”, and “overweight”.
“Chubby rabbit” has a parabolic ﬁxed point with 3 petals and multiplier. Actually there is an overweight rabbit in the main cardioid, which has aneattracting ﬁxed point with multiplier of the form re2πi/3 (0 < r < 1). On the other
hand, there is a little bit thinner rabbit than “chubby” near “Douady’s”, which
2πi/3

∗

The author gave a talk on 20 February 2003 at RIMS, Kyoto. This note is written for
RIMS Kokyuroku.

1has a repelling ﬁxed point with multiplier Re2πi/3 (R > 1). So we tentatively
call them “overweight rabbit” and “plump rabbit”. (I don’t know whether theseterms are proper or not, though...)The change from “chubby” to “overweight” or “plump” is parameterized by
the multiplier of (α-)ﬁxed point. They look similar when r and R are very close
to 1 (Figure 1). Actually “overweight” and “plump” converge to “chubby” asr, R → 1 in the Hausdorﬀ topology. Moreover, the dynamics on the Julia sets arealmost the same as we can see by observing the combinatorics of landing external
rays. (By theorems in §2, we can also see that the dynamics inside the Julia sets
are almost the same.)
In the general case, changes from “parabolic” to “hyperbolic” (=“attracting
or repelling”), or opposite directions, are not easy as above. The diﬃculty comesfrom well-known parabolic implosion, but here we omit to deal with it. Our mainquestion is:
For a rational map with a parabolic cycle, can we give a way toperturb its parabolic cycle into another kind of cycle without changing
most part of the dynamics?
In this note, we will give a quick survey on this problem.

1.1

Preliminary

Here we list some deﬁnitions and notation.
Classes of rational maps. Let f : Ĉ → Ĉ be a rational map of degree d ≥ 2.
Here we recall some famous classes:• f is geometrically ﬁnite if every critical point in the Julia set J(f ) is preperiodic. If f is geometrically ﬁnite, the Fatou set F (f ) consists of attractingor parabolic basins.
• f is called subhyperbolic if f is geometrically ﬁnite without parabolic cycles.
• f is called semihyperbolic if f has no recurrent critical points nor parabolic
cycles in its Julia set.
In §3, we will deal with one more class of rational maps, called weakly hyperbolic.
Perturbation. A perturbation of a rational map (resp. polynomial) f is afamily of rational maps (resp. polynomials) {f :  ∈ [0, 0 ]} with some 0 > 0satisfying f0 = f ; deg f = d; and d ˆ (f , f ) → 0 as  → 0. For simplicity, we
represent such a family in convergence form, f → f .
2

Notation.
• For a parabolic or attracting periodic point α, A(α) denotes its immediatebasin.
• P (f ) denotes the postcritical set of f .
• C(f ) denotes the critical set of f .

2

Polynomial case: Theorems of P. Haı̈ssinsky

In the case of polynomial, there are some results by Peter Haı̈ssinsky. Here wesketch his sequential work related to our question. In this section we assume thatf is a polynomial of degree d ≥ 2.

2.1

Parabolic to repelling

The ﬁrst theorem is on a perturbation of direction “parabolic → repelling”.
Theorem 2.1 (Haı̈ssinsky, [5]) If f is geometrically ﬁnite with connected Julia
set, then there exists a polynomial perturbation f → f accompanied by conjugacies between the actions of the Julia sets. Moreover, f are all subhyperbolic.
This theorem is extended later in §3.
Sketch of the proof. The last sentence implies that every parabolic cyclein J(f ) is perturbed into a repelling cycle. We explicitly construct a rational
perturbation F = f + R, where R is a rational function which takes value
zero at all parabolic cycles and at ﬁnite critical orbits on J(f ). (Here we allow
deg F ≥ deg f .) Then F has cycles exactly the same places as the original
parabolic cycles, but their multipliers are changed slightly by R. Here we take aproper R to make them repelling. Moreover, to preserve the local degree of criticalorbits on the Julia sets, we take R to have enough tangency at those points. If  1, we can take a nice topological-disk neighborhood U of J(f ) such that
{U, F−1 (U ), F } is an analytic family of polynomial-like map. By straightening,
we obtain a subhyperbolic perturbation f → f . Now it is known that theconnected Julia sets of geometrically ﬁnite polynomials are locally connected.
Thus every external ray land on the Julia sets. To check the dynamical stabilityon the Julia sets, we check the stability of the ray equivalence, which is deﬁned
by shared landing points of external rays. 3Goldberg-Milnor conjecture. Theorem 2.1 gives an aﬃrmative answer tothe following Goldberg-Milnor conjecture in the case of geometrically ﬁnite polynomials: 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 someattracting cycles; and• the perturbed polynomial on its Julia set is topologically conjugate to theoriginal polynomial f on J(f ).
Conversely, is it possible to create parabolics from hyperbolics(=attracting or
repelling)? The following results give us some partial answers.

2.2

Repelling to parabolic

Next we consider the opposite direction: “repelling → parabolic”. The second
theorem is:
Theorem 2.2 (Haı̈ssinsky, [4]) Suppose f has an attracting ﬁxed point α anda repelling ﬁxed point β on ∂A(α). We also add the following condition:(B) β is accessible from A(α) and β ∈/ P (f )Then there exists a polynomial g of degree d and a homeomorphism h : C → C
such that• h ◦ f (z) = g ◦ h(z) for any z ∈ Ĉ − A(α);• h(β) is a parabolic ﬁxed point and h(A(α)) = A(h(β)); and• h|J(f ) gives a topological conjugacy between the actions on the Julia sets.
We can remove condition (B) when f is geometrically ﬁnite. Moreover, we
can modify the statement by replacing the term “ﬁxed point” with “cycle”.
This theorem says that we can convert an attracting basin into the parabolic
basin in our particular situation. The conjugacy breaks only on the immediatebasin A(α), where we operate tricky surgery by means of µ-conformal map. µconformal map is not a quasiconformal map, though it is exponentially close toquasiconformal in some sense.
Let µ : C → D be a measurable function which satisﬁes
Area{z ∈ C : |µ(z)| > 1 − } ≤ Ce−η/
for some C ≥ 0 and η > 0. Such a µ is called to be in the David class of functions
on C. Note that µ∞ can be 1, but it is quite close to the situation µ∞ < 1(that is,Area{z ∈ C : |µ(z)| > 1 − } = 0
4for all   1), which will give us a quasiconformal map by solving the Beltrami
equation ∂z̄ φ = µ(z)∂z φ.
Now the main tool is:
Theorem 2.3 (David, [3]) For µ in the David class, the Beltrami equation∂z̄ φ = µ(z)∂z φ has a unique solution ﬁxing 0, 1 and ∞.
We call this solution a µ-conformal map. In the proof, we will partially replace
the Möbius-hyperbolic-like dynamics near α and β by Möbius-parabolic-like one
and will obtain a new topological dynamics F : C → C. Then F admits aninvariant µ which is in the David class, and by solving the Beltrami equation wewill get the desired polynomial g.
Sketch of the proof. For simplicity we assume that A(α) contains a single
critical point only. Then the dynamics in A(α) is quasiconformally conjugate tothat of a Blaschke product B : D → D of the form B(z) = (z 2 + b)/(1 + bz 2 )with 0 < b < 1/3. Let Ψ : A(α) → D be the conjugacy. By comparing with the
dynamics of Bpar (z) = (z 2 + 1/3)/(1 + z 2 /3), we will ﬁnd invariant “sectors” S
and Spar which have similar dynamical behavior on their boundaries (Figure 2).
Indeed, there is a piecewise quasiconformal homeomorphism ψ : D → D which
satisﬁes ψ(S) = Spar and ψ ◦ B = Bpar ◦ ψ on D − S.

Figure 2: Dynamics of B and Bpar on D. The thickest curves show the boundaryof invariant “sectors” S and Spar . Thinner curves show their ﬁrst and second
preimages.
Now we deﬁne the topological endomorphism F : C → C by F := P on
C − A(α); and by F := (ψ ◦ Ψ)−1 ◦ Bpar ◦ (ψ ◦ Ψ) on A(α). (Here we replace thehyperbolic dynamics by parabolic one.) Let σ0 be the standard complex structureof C, and let σ1 := (ψ ◦Ψ)∗ σ0 on A(α). We deﬁne a new almost complex structure5

σ by σ := (F n )∗ (σ1 ) on P −n (A(α)); and by σ := σ0 elsewhere. Then we haveF ∗ σ = σ.
We have to use the property β ∈/ P (f ) in (B). Actually, even if some criticalorbits land on β but the other critical points do not accumulate on β, we canshow that the Beltrami diﬀerential µσ induced by σ belongs to the David class(by taking suitable ψ above). By Theorem 2.3, there exists a µσ -conformal φ withφ∗ σ0 = σ a.e., and the polynomial g := φ ◦ F ◦ φ−1 has the desired properties. 

2.3

Attracting to parabolic

Next direction is “attracting → parabolic”. Before stating the third theorem, let
us start with an easy example.
Pinching to be “Chubby”. We assume from now on that p and q are relatively prime positive integers. (That is, (p, q) = 1 where we allow p = q = 1.)
Set ω := e2πip/q , and consider a family of quadratic polynomialf (z) = (1 − )ωz + z 2 : 0 ≤  < 1 .
For ﬁxed 0 < 0 < 1, the dynamics of f = f0 near the origin is conformallyconjugate to T : w → (1 − 0 )ωw. Let Φ denote this local conjugacy. We extend
it analytically to Φ : A(0) → C by using the relation w = Φ(z) = T −n ◦ Φ ◦ f n (z).
Now there are q symmetrically arrayed rays joining 0 and ∞ whose union is T invariant on w-plane. By pulling them back by Φ, we can ﬁnd q arcs I1 , . . . , Iq
joining 0 and a repelling cycle γ1 , . . . , γq , which are permuted by f (That is,f (Ii ) = Ij iﬀ j ≡ i + p modulo q) and disjoint from P (f ) (Figure 3).

I2
I3

I1

Figure 3: Left, the Julia set for an f with p/q = 1/3. Right, the Julia set forf0 . Shadows distinguish the regions which never intersect by the iteration of f3
or f03 .

6
Set I := j Ij . By comparing with the parabolic dynamics of g = f0 , one caneasily see that I of f plays the role of the parabolic ﬁxed point of g, topologically.
A priori, we can get the dynamics of g by pinching the grand orbit of I. What thethird theorem states is that we can ﬁnd a family of quasiconformal deformations{f } of f which realizes the pinching as above, and we can get g as its limit.
Theorem 2.4 (Haı̈ssinsky, [6]) Suppose f is semihyperbolic with connected Julia set and an attracting ﬁxed point α. Then the following holds:
1. For any p and q as above, there exist q arcs I1 , . . . , Iq joining α and arepelling cycle of period q permuted by f as the example above.
2. There exists a polynomial g with a parabolic ﬁxed point β of multiplier e2πip/q
which satisﬁes the following:• There exist quasiconformal deformations {f : 0 <  ≤ 1} of f (= f1 )such that f → g is a perturbation.
• Let H denote the quasiconformal conjugacy from f to f . Then H
converges uniformly as  → 0 to the limit h which semiconjugates f tog.
• For y ∈ Ĉ, card(h−1 (y)) ≥ 2 iﬀ 
y eventually lands on β. In particular,such an h−1 (y) is either I = Ij or a connected component of its
preimages.
We will see some similar results later in §3.
Idea of the proof. Let us start with a model of pinching and a caricatureof quasiconformal deformation. By quasiconformal deformation, we may assume
that the multiplier of α is ω/2. By taking a linearizing coordinate z → w plus acovering w → ζ = wq , the action of f on A(α) is semiconjugated to ζ → λζ on
C, where λ = (ω/2)q > 0.
Now we put an almost complex structure (=a ﬁeld of inﬁnitesimal ellipses)on C as in Figure 4. By taking the angle  closer to 0 and making the constantC
1, we will have a family of almost complex structures σ and associating
Beltrami diﬀerentials µ which are invariant under the action of ζ → λζ. By
solving the Beltrami equation ∂ζ̄ φ = µ (ζ)∂ζ φ for each , the solution φ ﬁxing
1, λ, and ∞ gives a deformation of the quotient torus C∗ /λ to another torus
with lower modulus. Then quasiconformal map φ converges compact uniformly
to φ0 : C∗ −R− → C as  → 0, and conjugate the action of ζ → λζ to ζ → ζ +1−λ.
We can pull-back σ to A(α) by (z → w → ζ)∗ and denote it by σ . By puttingn ∗ (f ) σ on f −n (A(α)), and the standard complex structure elsewhere, we havean f -invariant almost complex structure and we will get a family of polynomialsf in the same way as the previous theorem. This may realize the pinching “fromhyperbolic to parabolic”, however, in our pulled-back situation, we are not sure
7

Figure 4: A caricature of the pinching. On the right half plane µ (z) = 0. Forπ|< | arg z| < π − , |µ (z)| increases with dilatation 1+|µ= 1 + tan2 (| arg z| − π2 ).
21−|µ |

= 1+tan2 ( π2 −Finally |µ (z)| becomes a constant 0 < C < 1 elsewhere, with 1+C
1−C). Moreover, we take arg µ = 0 and µ is constant along any radial lines fromthe origin. Hence µ is invariant under ζ → λζ.
that the limit of f as  → 0 exists and is g as we desired. To check them, we makesome eﬀort to show that the integrating map φ of σ is equicontinuous and any
subsequential limits coincide. In this technical part we use the semihyperbolicity
of f . (We also use the weak hyperbolicity of f . See §3.)

2.4

Explicit construction of pinching: Tessellation

Here we present an explicit way of constructing pinching semiconjugacy. The
idea is tessellation of ﬁlled Julia sets [11]. For simplicity we explain an example
just by ﬁgures.
Let us consider a family of quadratic polynomials,

F = fc (z) = z 2 + c : −3/4 < c < 0 .
Figure 5 is the pictures of tessellation for two quadratics, one is taken from F andthe other is f−3/4 . The construction of tiles are obviously based on linearizing
coordinates.
Each tile has an “address”, consists of angle θ ∈ Q/Z, level n ∈ Z, andsignature + or −. Addresses are organized so that the tile of address (θ, n, +) is
mapped to the tile of (2θ, n + 1, +) for example. It matches to the combinatorics
of external rays, and we can precisely describe the dynamics inside the Julia set.
Now it is not diﬃcult to construct a pinching by pasting tile-to-tile homeomorphisms which preserve addresses. Then some of arcs (as {Ij } in the previous
example f (z) = (1 − )ωz + z 2 ) are naturally pinched by continuous extentionsof the tile-to-tile homeomorphisms.
8

Figure 5: Tessellation.

9

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Figure 6: “Checkerboard” and “Zebra”, showing the structure of the addressesof tiles. “Checkerboard”, with some external rays drawn in, shows the relation
between the external angles and the angles of tiles. The invariant regions coloredin white and gray correspond to tiles of signature + and − respectively. “Zebra”
shows the levels of tiles. Levels get higher near the preimages of the attracting
ﬁxed point.
10

3

Rational case: Geometrically finite maps, etc.

Here we deal with the case of rational maps. Some results are natural generalization of theorems in §2.

3.1

G. Cui’s plumping deformation.

G. Cui made intriguing applications of well-known Thurston rigidity to geometrically ﬁnite branched coverings. Here we roughly sketch some part of his work
relating “parabolic → hyperbolic” perturbations. See [1] for his original, or [14]for a survey of his entire works.
Periodic star-like graphs. Before the main statement, let us consider a rational map f with an attracting cycle α of period l. For some integers n, m with
nm = l, suppose there are a repelling cycle β of period n and star-like graphs
I 1 , . . . , I n such that: each I k is centered at a repelling point in β; each I k has mfeet with their toes at α; and f (I k ) = I k+1 with superscripts modulo n. We callsuch an I k a repelling periodic star-like graph associated with α.
For example, readers may imagine the case of “plump rabbit”, with an invariant star-like graph connecting the central repelling ﬁxed point and the attracting
cycle, or the cases of its tuned quadratic polynomials (i.e. “plump rabbits” incopies of M in M ). One may call the graph I = Ij in Figure 3 an invariant
attracting star-like graph centered at 0.
Plumping to be “plump”. The theorem here deals with simple plumping,which replaces parabolic cycles by repelling periodic star-like graphs without
breaking the symmetry of petals, like the change of “chubby rabbit” into “plump”.
Now the statement is:
Theorem 3.1 (Cui, [1]) Let g be a geometrically ﬁnite rational map with parabolic
cycles. Then there exists a subhyperbolic rational map f satisfying the following:• There exist quasiconformal deformations {f : 0 <  ≤ 1} of f (= f1 ) suchthat f → g is a perturbation.
• Let H denote the quasiconformal conjugacy from f to f . Then H converges uniformly as  → 0 to the limit h which semiconjugates f to g.
• For y ∈ Ĉ, card(h−1 (y)) ≥ 2 iﬀ y eventually lands on a parabolic cycle. In
particular, such an h−1 (y) is either a repelling periodic star-like graph or aconnected component of its preimages.
Compare this with Theorem 2.4. (It was “overweight” to “chubby”.)
This theorem is stronger than Theorem 2.1, however the proof is quite diﬀerent
and complicated. It goes like this: Step 1, we construct a partially analytic
11branched covering F by replacing all parabolics with proper repelling star-likegraphs. This operation creates no Thurston obstruction. Step 2, by a result on
convergence of Thurston algorithm in [2], we can ﬁnd a subhyperbolic rational
map f which is conjugate to F . Step 3, we construct pinching deformations {f }as in Theorem 2.4 which has the limit rational map ĝ with recreated parabolics.
Step 4, under suitable normalization of f and f , we can show g = ĝ by a rigidity
result also due to [2].
Actually there is a much stronger theorem which makes Theorem 3.1 justa corollary. See Theorem D+F’+G’ in [14]. Instead of stating it in detail, we
consider an example.
Example (Example 3 of [14]). Set g(z) := z(1 − z 2 ), with a parabolic at
z = 0. The attracting (resp. repelling) directions lie on R (resp. iR). Let 1 bean invariant curve in the ﬁrst quadrant and L1 the region enclosed by 1 ∪ {0},called a sepal. For i = 2, 3, and 4, let i and Li be the symmetric image of 1 andL1 in the i-th quadrant. (See Figure 7.) In particular, L1 and L3 (resp. L2 andL4 ) are called right sepals (resp. left sepals).

L2

L1

L3

L4

Figure 7: Sepals and the ﬁlled Julia set of g.
General plumping is, roughly speaking, replacing a pair of right and left sepals
by an invariant arc joining two ﬁxed points. To describe possible ways of plumping, we consider plumping combinatorics τ as following: τ is an injective map
deﬁned on L2 , L4 , or L2  L4 ; and τ |Li (i = 2, 4) is just a symmetric reﬂection
which sends Li to L1 or L3 . Here is all the possible τ :(i) τ : (L2 , L4 ) → (L1 , L3 )(ii) τ : (L2 , L4 ) → (L3 , L2 )(iii) τ : L2 → L1 , τ : L4 → L3(iv) τ : L2 → L3 , τ : L4 → L1
12

There are corresponding plumpings of τ in Figure 8. Two τ ’s in (iii) or (iv)give topologically the same plumpings. For example, let us pick up a plumping

Figure 8: Plumpings of type (i)–(iv), from left to right. Black, white, and graydots show repelling, attracting, and parabolic ﬁxed points respectively.
combinatorics τ : L2 → L3 , that is, τ (L2 ) = L3 . Consider a Riemann map
φ : Ĉ − L2 ∪ τ (L2 ) → Ĉ − D which sends two prime ends at 0 to those of ±1.
Then there exists a gluing map T : Ĉ − D → C which identiﬁes the componentsof ∂D − {1, −1} such that T respects the holomorphic dynamics near 2 andτ (2 ) = 3 .
A new plumped map F : Ĉ → Ĉ will be deﬁned basically by (φ ◦ T ) ◦g ◦ (φ ◦ T )−1 . Indeed, it is naturally deﬁned as an analytic map except nearg −1 (0) − {0} = {1, −1}. Take a small neighborhood U of L2 ∪ τ (L2 ). Let U1 andU−1 be the component of g −1 (U ) around 1 and −1. Deﬁne F on φ(U±1 ) by asuitable topological map which sends ∂φ(U±1 ) to φ(∂U ). Then F is a partiallyanalytic branched covering.
Cui showed that there exists a polynomial f which is conjugate to F . Moregenerally,
Theorem 3.2 (Cui) For any plumping combinatorics τ of g above, there exists
a polynomial f which is conjugate to a partially analytic map deﬁned in a similar
way as F above. Moreover, there exists a quasiconformal deformation f (0 <
 ≤ 1) of f = f1 which gives a perturbation f → g, and is accompanied by(semi)conjugacies H → h as in Theorem 3.1.
See [14] for more precise statement and the proof, which deals with general
geometrically ﬁnite rational maps with parabolics.

3.2

Weakly hyperbolic maps

In the case of simple plumping, Tan Lei and Haı̈ssinsky generalized Step 3 ofthe proof of Theorem 3.1 to more bigger class of rational maps as following. A
rational map f is weakly hyperbolic if there exist δ ≥ 1 and r > 0 with thefollowing: For any z ∈ J(f ) − {parabolics}, there exists a subsequence {nk } of{n} such thatdeg{f nk : Wnk → B(f nk (z), r)} ≤ δ,
13

where B(x, r) is a spherical ball of radius r centered at x, and Wn is the component of f −n (B(f n (z), r)) containing z. It is known that semihyperbolic orgeometrically ﬁnite maps are weakly hyperbolic.
The statement is:
Theorem 3.3 (Haı̈ssinsky and Tan, [8]) Let f be a weakly hyperbolic rational map with attracting cycles. Let I be an f -invariant collection of periodic
repelling star-like graphs associated with the attracting cycles. Then there exists
a rational map g with parabolic cycles which satisﬁes the following:• There exist quasiconformal deformations {f : 0 <  ≤ 1} of f (= f1 ) suchthat f → g is a perturbation.
• Let H denote the quasiconformal conjugacy from f to f . Then H converges uniformly as  → 0 to the limit h which semiconjugates f to g.
• For y ∈ Ĉ, card(h−1 (y)) ≥ 2 iﬀ y eventually lands on a new parabolic cycle
created in g. In particular, such an h−1 (y) is either a repelling star-likegraph in I or a connected component of its preimages.
The proof goes in the same way as Theorem 2.4. The diﬃculty is also thesame, that is, we have to show the equicontinuity of H and the uniqueness ofevery subsequential limit. For the equicontinuity, they used a modulus-controllinginequality which is due to Cui. The uniqueness is followed by a rigidity result on
weakly hyperbolic maps which is due to Haı̈ssinsky[7].

3.3

Horocyclic perturbation of geometrically finite maps

What kind of perturbations give us dynamically stable perturbations? With theassumption of geometric ﬁniteness, here we give a suﬃcient condition for this:
Theorem 3.4 (K, [9]) Let f be a geometrically ﬁnite rational map. If a perturbation f → f is horocyclic and preserving J-critical relations of J(f ) (deﬁned
below), then for   1, there exists a unique semiconjugacy h : J(f ) → J(f )with the following properties:(1) h tends to the identity. That is, sup {d ˆ (h (x), x) : x ∈ J(f )} → 0 ( → 0);(2) If card(h−1 (y)) ≥ 2 for some y ∈ J(f ), then y eventually lands on a parabolic
cycle; and(3) h is a homeomorphism (thus is a conjugacy) between the Julia sets if andonly if none of parabolic cycles of f is perturbed into an attracting cycle.
To get a dynamically stable perturbation, we have to control two kinds ofbifurcation: One is the parabolic bifurcation of course, and the other is thebifurcation of ﬁnite critical orbits on the Julia sets.
14

Horocyclic perturbation. Suppose that f is a rational map with parabolic
cycles. We say a perturbation f → f is horocyclic if each parabolic point a of f ,with m petals and period l, satisﬁes the following:(a) There is a neighborhood D of a with local coordinates φ , φ : D → C (notnecessarily conformal) such that: φ(a) = 0; φ → φ uniformly on D; andthe perturbation is locally viewed as
m+1+ O(z m+2 )φ ◦ flm ◦ φ−1
 (z) = λ z + z−→ φ ◦ f lm ◦ φ−1 (z) = z + z m+1 + O(z m+2 ),where λ → 1.
(b) If we set exp(L + iθ ) := λ , which tends to 1 as  → 0, then θ2 = o(|L |)as L , θ → 0.
Condition (a) means that the perturbation preserves the symmetry of dynamics near parabolics. This is not so much an essential condition but it makes theargument simpler. Condition (b) is a more crucial condition: If f has no rotation
domains, then the Julia set varies continuously along horocyclic perturbations.
However, it is known that this continuity breaks when property (b) breaks.
Horocyclic perturbation was originally deﬁned as horocyclic convergence ofrational maps by C. McMullen[12, §7-9], to investigate the continuity of Hausdorﬀ
dimension of the Julia set.
J-critical relations. Let c1 , . . . , cN 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 nonnegative integers (i, j, m, n) such that f m (ci ) = f n (cj ).
We say a perturbation f → f preserves the J-critical relations of f if:• For all i = 1, . . . , N , the maps f have critical points ci () (may be in the
Fatou set) satisfying ci () → ci and deg(f , ci ()) = deg(f, ci ) as  → 0; and• For each J-critical relation (i, j, m, n) of f , f satisﬁes fm (ci ()) = fn (cj ()).
If f is geometrically ﬁnite, then each f is also geometrically ﬁnite.
Idea of the proof. To explain the idea, consider a perturbation of f (z) = z 2
into f (z) = z 2 +  with small  > 0. We can make a conjugation on the Julia set
in the following way: First take two attracting disks near 0 and ∞ of f . Thenthey are also attracting disks for f for   1, because of uniform convergencef → f . Let Ω be the compliment of these two disks, which is a closed annulus.
Set Ωn := f −n (Ω) and Ωn := f−n (Ω). Then Ωn+1 is contained in the interior of15Ωn , and the same is true for {Ωn }. Set h0 = id|Ω . By lifting hn to hn+1 by the
relation hn ◦ f = f ◦ hn+1 for n ≥ 0, we obtain the following diagrams:
..
.
Ω2



f 

..
.
h2

..
.

2

−−−→ Ω

f


..
.
hn+1

Ωn+1
−−−→ Ωn+1



f

f 


h

1
Ω1 −−−
→ Ω1



f
f 


h

n
Ωn −−−
→ Ωn
..
..
.
.

h

0
Ω0 −−−→ Ω0

Now one can show that hn converges to a unique limit h on the Julia set J(f ),by using the uniform expanding property of f near J(f ).
Even in the case of geometrically ﬁnite f , we can take such a nice compact
set Ω with J(f ) ⊂ Ωn+1 ⊂ Ωn . Indeed, we can ﬁnd such an Ω of the formĈ−{attracting disks and petals}∪{some ﬁnite disks near critical orbits in F (f )}.
We can ﬁnd a similar set Ω with J(f ) ⊂ Ωn+1
⊂ Ωn by modifying Ω nearthe parabolics. Then we can start lifting h0 : Ω → Ω, which is not id|Ω , but
arbitrarily close to id|Ω by horocyclicity of f → f . Since the J-critical relations
are preserved, the lifting is not so complicated. The convergence of hn is shownby means of a weakly expanding metric near J(f ), which is compatible with thespherical metric. For the construction of this metric we again use the geometricﬁniteness of f .
Remark. If J(f ) = Ĉ, then f is postcritically ﬁnite and thus has Thurston
rigidity. This implies that any perturbation f → f preserving the J-critical relations must be either a family of Möbius conjugation or a family of quasiconformal
deformation. The former happens when the associating orbifolds does not havetype (2, 2, 2, 2).
Existence of perturbation. For any geometrically ﬁnite rational map, there
exists such a perturbation as in Theorem 3.4.
Theorem 3.5 (K, [10]) Let f be a geometrically ﬁnite rational map with J(f ) =Ĉ. Then there exists a horocyclic perturbation f → f which preserves J critical
relations. Thus there exists a semiconjugacy as in Theorem 3.4. Moreover, one
can choose the direction of the perturbation such that the parabolic cycles of f are
perturbed into repelling, parabolic and attracting cycles of f in any combination.
Sketch of the proof. The proof uses quasiconformal perturbation developedby M. Shishikura[13]. First take a polynomial p(z) which vanishes at attracting
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cycles, parabolic cycles, and at the ﬁnite set (C(f ) ∪ P (f )) ∩ J(f ). (We may
assume that ∞ is not periodic.) In particular, we can take p with an expansionof the formp(z) = s(z − a) + (z − a)M +2 + · · · + (z − a)kabout every parabolic point a, where s = 1, 0, or −1 depending on what kind ofcycle we want, k = deg p  0, and M is the largest petal number of parabolics.
Let ρ : [0, ∞) → [0, 1] be a smooth non-decreasing function such that ρ(t) = 1
at t ∈ [0, 1] and ρ(t) = 0 at t ∈ [2, ∞). In particular, we can take such a ρ withbounded derivative. Set H (z) = z + h(z)ρ(1/k |z|). Then H : Ĉ → Ĉ is aquasiconformal map if   1, and satisﬁes H → id and its maximum dilatation
tends to 0 as  → 0.
Set g := f ◦ H . Note that the quasiregular map g is holomorphic exceptV := z : |z| > −1/k . In particular, if a is as above with period l and multiplierλ, then a is a periodic point of g with period l and multiplier (1 + s)l λ. Thus it
can be repelling, parabolic, or attracting depending on s = 1, 0 or −1. Moreover,g → f preserves the J-critical relations of f .
By taking a g -invariant region E and taking a Möbius conjugacy, we may
assume that g (V ) ⊂ E if   1. Indeed, we can construct such a family of sets
{E } by modifying a union of attracting disks and petals of f .
Let σ0 denote the standard complex structure on Ĉ. For   1, we put analmost complex structure σ deﬁned by (gn )∗ (σ0 ) on g−n (E ) and by σ0 elsewhere.
Then σ is g -invariant and we can ﬁnd a quasiconformal map Φ such that Φ∗ σ0 =σ a.e., and thus f := Φ ◦ g ◦ Φ−1 is a rational map of degree d. Since Φ → idby the deﬁnition of g and the continuous dependence of Φ on its Beltrami
diﬀerential, we obtain a rational perturbation f → f preserving the J-critical
relations of f . By investigating local dynamics near the parabolics, we can checkthat f → f is horocyclic. Now we can apply Theorem 3.4. 
Remarks. One can enjoy more various perturbations of parabolics by changing
the polynomial p(z) above. Note that this perturbation keeps superattracting
cycles, thus a polynomial is perturbed within the category of polynomials. In
particular, this gives a mild generalization of Theorem 2.1.

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