The fixed point of the parabolic renormalization operator
1. Introduction
In a recent paper [IS], H. Inou and M. Shishikura demonstrated that the successive parabolic renormalizations of the quadratic polynomial converge to an analytic map defined in a neighborhood of the origin, which satisfies the fixed point equation
Conjecturally, the solution of the above functional equation is unique in a suitably restricted class of maps.
In this paper we present a class of analytic maps which have a maximal analytic extension to a Jordan domain, satisfying the invariance property
The covering properties of a map admit an explicit topological model. We prove that the InouShishikura fixed point of is contained in , and conjecture that successive renormalizations of any map converge to .
The boundary of the maximal domain of analyticity of has a a highly degenerate geometry. It is this bad geometry that makes the study of so challenging. In contrast, consider the parabolic renormalization of critical circle maps with a parabolic fixed point on the circle, which both of the authors have studied [Lan, Ya, EY]. The corresponding renormalization fixed point also has a maximal analytic extension, whose covering properties are similar to those of . The geometry of its domain of analyticity, however, is rather tame, which permits both a numerical and an analytic study.
We present a numerical method for computing the Taylor’s expansion of with a high accuracy. Our approach also allows us to compute the domain , and the reader will see the first computergenerated images of it. Finally, we obtain a numerical estimate of the leading eigenvalue of .
2. Local dynamics of a parabolic germ
2.1. Fatou coordinates
We briefly review the local dynamics an analytic function in the vicinity of a parabolic fixed point at :
We consider first the case , that is, , and we write
(2.1) 
for some and . The integer can be recognized as the multiplicity of as the solution of .
A complex number of modulus one is called an attracting direction if the product , and a repelling direction if the same product is positive. The terminology has the following meaning:
Proposition 2.1.
Let be an orbit in which converges to the parabolic fixed point . Then the sequence of unit vectors converges as to one of the attracting directions.
We say in this case that the orbit converges to from the direction of .
If has a parabolic fixed point at , it admits a local inverse there, by which we mean a function , defined and analytic on a neighborhood of , so that for near enough to . The germ at of a local inverse is unique, but its domain of definition typically has to be chosen. A local inverse also has a parabolic fixed point at ; attracting directions for are repelling for the inverse and vice versa.
Definition 2.1.
Let be an attracting direction for . An attracting petal for (from the direction ) is a Jordan domain with closure in such that:


is injective on ;

;

for any , the orbit converges to from the direction , and the convergence of to is uniform on .

conversely, any orbit which converges to from the direction is eventually in .
Similarly, is a repelling petal for if it is an attracting petal for some local inverse of .
Judiciously chosen petals can be organized into a LeauFatou Flower at :
Theorem 2.2.
There exists a collection of attracting petals , and repelling petals such that the following holds. Any two repelling petals do not intersect, and every repelling petal intersects exactly two attracting petals. Similar properties hold for attracting petals. The union
forms an open simplyconnected neighborhood of .
The proof of this statement relies on some changes of coordinates. First of all: Every germ of the form (2.1) can be brought into the form
(2.2) 
in a suitable conformal local coordinate change at . In fact, a straightforward induction shows the following:
Proposition 2.3 (cf. [Mil1] Problem 10d,[Be]).
For every germ of the form (2.1) there exists a unique such that for every greater that there is a locally conformal change of coordinates , with , such that
Further, there exists a formal power series which formally conjugates
Thus, the number is a formal conjugacy invariant of , and specifies its formal conjugacy class uniquely.
For the next few paragraphs, we will take to have the special form (2.2). The attracting directions are then the th roots of . We will describe some ways of constructiong attracting petals for the attracting direction ; the adjustments necessary to deal with repelling petals are routine. The reader is reminded that (2.2) is not the general form of a mapping with a parabolic fixed point of order ; it has been cleaned up by making a preliminary analytic change of coordinates to eliminate some powers of in its Taylor series.
The behavior of orbits of such an near is greatly clarified by making the coordinate change
We are considering a particular attracting direction, and we take to be defined on the sector between the two adjacent repelling directions; it opens up this sector to the complex plane cut along the positive real axis. With its domain of definition restricted in this way, is bijective, and its inverse is given by
where the branch of the th root is the one cut along the positive real axis and taking the value at .
Then
We thus obtain:
where
Selecting a right halfplane for a sufficiently large , we have
The domain is then an attracting petal for the attracting direction . In the case of a simple parabolic point, what we have just shown simplifies to the assertion that any disk of sufficiently small radius tangent to the imaginary axis from the left at is an attracting petal.
The petals just discussed – pullbacks of halfplanes under – have boundaries tangent at the origin to the directions . For many purposes – such as the proof of Theorem 2.2 – we will need petals with strictly larger opening angle. There are many ways to construct such petals; here is one which is convenient for our purposes. Let , , and let
(2.3) 
(i.e., is the sector translated right by .) From
there exists a so that
(2.4) 
for . If is large enough, the domain does not intersect the disk , so (2.4) holds for . For such ’s, by elementary geometric considerations,
and for all . Further any contains a right halfplane and hence eventually contains any orbit converging to . Finally, it can be verified that the sequence of iterates converges uniformly to on . We omit this verification; it uses simplified versions of the ideas used in the proof of Lemma 2.16. Thus, sets of the form are attracting petals, symmetric about the attracting direction under consideration, with tengents at the origin in directions . It will be useful to have a general term for behavior for this: We will say that a petal with attracting or repelling direction is ample if
for some and sufficiently small .
The dynamics inside a petal is described by the following:
Proposition 2.4.
Let be an attracting petal for . Then there exists a conformal change of coordinates defined on , conjugating to the unit translation .
Proof.
For a traditional proof, see e.g. [Mil1] §10. We cannot resist giving a proof based on quasiconformal surgery, which probably originated in the work of Voronin [Vor]. For definiteness, we discuss the case of an attracting petal with attracting direction , and let
be as above. Also as above, we select a right halfplane . The main step will be to prove the existence of for the special petal , which we provisionally denote by . The case of a general petal will then follow by an easy extension argument.
As we know, so let us denote the closed strip
Setting , let be any diffeomorphism
which on the boundary of the strip conjugates to :
We will further require that the first partial derivatives of and be uniformly bounded in . Verifying the existence a diffeomorphism with these properties is an elementary exercise which we leave to the reader.
The diffeomorphism defines a new complex structure on , which we extend to the left halfplane by
Gluing together with the standard complex structure and the halfplane with structure via the homeomorphism (which is now analytic), and using the Measurable Riemann Mapping Theorem, we obtain a new Riemann surface . By the Uniformization Theorem, is conformally isomorphic either to or to the disk. By construction, is quasiconformally isomorphic to and therefore cannot be conformally isomorphic to the disk. We can specify a conformal isomorphism uniquely by imposing normalization conditions and .
The pair of maps and induces a conformal automorphism of , which we denote by . Then is a conformal automorphism of with no fixed point. It is a standard fact that the only such automorphisms are translations, and our choice of normalization for implies that
(2.5) 
But on , so we get
Moreover, the restriction of to is analytic in the standard sense. Thus, we set on and obtain
as desired. Since is a conformal isomorphism from to , the map is univalent on .
This proves the existence of on the particular petal . We provisionally denote the above , which is defined on , by . We define
If , then for sufficiently large . If , then is an open set containing and contained in , so is open. Since is mapped into itself by , and since
takes the same value for all for which . We denote this common value by , thus obtaining a function defined on all of and extending defined on . Tautologically,
If , then on a neighborhood of , so on this neighborhood, which shows that the extended is analytic, but not necessarily univalent, on all of .
Now let be a general attracting petal with the same attracting direction . By definition of petal, , so we can restrict to , thus obtaining an analytic function satisfying
It remains to show that the restriction of to is univalent. To see this, let , be points of with . For sufficiently large , and are both in , so
But, by construction, is univalent, so . The argument so far works for any pair , in with . Now, however, we use that facts that and are both in the petal , which is mapped into itself by and on which is univalent. Hence, from it follows that , proving univalence of on .
∎
We note for future reference a simple result which was proved in the course of the preceding argument:
Proposition 2.5.
Let be an attracting direction for , be an attracting petal from the direction , a univalent analytic function defined on and satisfying the function equation
Then has a unique extension to satisfying this equation.
We define attracting Fatou coordinate (for the attracting petal with attracting direction ) to be a function defined, analytic and univalent on and satisfying
As we have seen, such a function extends uniquely, via the above functional equation, to all of , and the extension restricts to an attracting Fatou coordinate on any other petal with attracting direction . It is clear that, if is an attracting Fatou coordinate then so is for any constant . We will see shortly that any two attracting Fatou coordinates differ only in this way.
Any attracting Fatou coordinate can be written in the form , where satisfies the functional equation
on an appropriate invariant domain “near infinity”. We will refer to such ’s as Fatou coordinates at infinity.
A repelling Fatou coordinate for means an attracting Fatou coordinate for an analytic local inverse of . If
which can be brought back into the standard form by conjugating with ( odd) or ( even). The above considerations then apply to define , repelling petals, etc. We note that:

a repelling Fatou coordinate satisfies the same functional equation
as does an attracting one, but the domains of definition are different, and

the image of a repelling petal by a repelling Fatou coordinate is mapped into itself by the unit left translation ; the image of an ample repelling petal under a repelling Fatou coordinate contains a left halfplane.
Again, it is useful to consider also repelling Fatou coordinates at infinity: If is a repelling Fatou coordinate, the corresponding one at infinity is
(but the appropriate branches of are different from the ones in the attracting case).
Our next step is to prove a crude asymptotic formula for a Fatou coordinate at infinity. It is advantageous here to deviate from what we have been doing. We consider a mapping of the form
i.e., we do not assume we have made a preliminary change of variable to eliminate, e.g., the terms for between and . We introduce as before; this time, the behavior of near infinity is
the series converges for sufficiently large . Let be an attracting Fatou coordinate at infinity. By what we have already proved: For any , there is an so that extends analytically to a univalent function on the set .
Proposition 2.6.
uniformly as in any sector with . The same limits hold for , but with in the opposite sector
This proposition is a lessprecise version of Lemma A.2.4 of [Sh], and the argument we give is the first part of Shishikura’s proof of that lemma. Shishikura carries the analysis further and is able to identify, in favorable cases, the first correction to the indicated asymptotic behaviors. We do not give his full argument here, as we will prove Theorem 2.15, which gives more precise information about the asymptotic behavior of Fatou coordinates.
Proof.
Fix with , and let . Take so that is defined and univalent in
and also so that on . For , denote by the distance from to the boundary of . We will investigate limits as in the strictly smaller sector ; then there is a constant so that, asymptotically, . In the following, we will frequently assume silently that is “large enough”. We will also use to denote a generic “universal” constant; different instances of need not denote the same constant.
For the first step, we use the Koebe Distortion Theorem: If – so the disk of radius 2 about is in – the mapping
is analytic and univalent on and has unit derivative at the origin. A simple rescaling of the Koebe Theorem to adapt it to the disk of radius 2 gives a universal constant so that
We insert into this estimate, use to ensure that , and use also the functional equation
to get
Applying the Cauchy estimates gives a bound
(with a different ).
Next we apply Taylor’s Formula with Integral Remainder to write
Again, we set and use to get
Since the estimate holds for all appearing in the integral on the right, we get
We have already remarked that as in the sector so
This establishes the asserted convergence of ; the assertion about follows by integration.
∎
Equipped with this information about the asymptotic behavior of Fatou coordinates, we can now show that the image of a Fatou coordinate is large enough. As usual, it suffices – up to insertion of some minus signs – to consider the attracting case. Let be an attracting Fatou coordinate, and let . Then, for sufficiently large , extends analytically to a univalent function on
Proposition 2.7.
Let . Then, for sufficiently large ,
Proof.
Let ; we want to investigate solutions to the equation which we rewrite as
The idea is to apply the Contraction Mapping Principle to , using the fact that which is small for large. To do this, we need to find a domain mapped into itself by and on which is contractive. Suppose we can find a so that

for

Then, for ,
so the disk of radius about will be mapped to itself, and will have a unique fixed point in this disk.
We implement this strategy as follows: First of all we arrange, by making larger if necessary, that on . We write
and we note that, by elementary geometry, implies . If we further take , then the disk of radius about is contained in , for any . Recall that we have already arranged that on . Finally, we apply Proposition 2.6 to see that, by taking large enough we can arrange that
All the element for the above contraction argument are now in place, and we can conclude that, for every , there is a unique with in . This proves the assertion .
∎
It follows from this proposition that:
Proposition 2.8.
The image under of any ample petal of contains a right halfplane.
A similar assumption holds for ample repelling petals, but with the image under covering a left halfplane.
Let be an attracting petal. We define a relation on by if and are on the same orbit, that is, if either or (with .) It is easy to check that this is an equivalence relation.
Consider the quotient . The canonical projection is locally injective, and it is straightforward to verify that there is a unique way to give a Riemann surface structure in such a way as to make analytic and therefore a local conformal isomorphism.
Let be another attracting petal contained in . Since the orbit of every point eventually lands in , the inclusion induces a conformal homeomorphism
Now if is any attracting petal with the same attracting direction as , the intersection is also a petal. Hence
Thus, the quotient does not depend on the choice of the petal but only on the choice of the attracting direction corresponding to . We will write
We will omit from the notation when the choice of the attracting direction is clear from the context (for instance, when there is only one attracting direction).
Let be an attracting Fatou coordinate defined on some petal . It is easy to verify, using the injectivity of on , that two points and are equivalent if and only if . Hence, defines by passage to quotients an injective mapping from to . If we take to be an ample petal, then it follows from Proposition 2.8 that takes on all values in . Thus:
Proposition 2.9.
The map is a conformal isomorphism from the Riemann surface to the .
In light of the preceding proposition, we will call the attracting cylinder corresponding to the direction .
The repelling cylinder for is the attracting cylinder for a local inverse of , fixing the origin.
If is an attracting petal, we will call the halfopen domain
a fundamental attracting crescent, the name reflecting its shape. A fundamental repelling crescent means a fundamental attracting crescent for a local inverse of .
Proposition 2.10.
For any attracting petal in the direction , the fundamental crescent projects bijectively onto the attracting cylinder . More concretely: Every point of lies on a forward orbit starting in , and distinct points of have disjoint forward orbits.
Proof.
From the requirement that converges uniformly to 0 on – condition (4) in of our definition of petal – it follows that no point of admits an arbitrarily long backward orbit in . Thus, every point of lies on the forward orbit starting outside ; the first point on this orbit inside is in . Let , be points of whose forward orbits intersect. From the injectivity of on , we have – possibly after interchanging and – for same . If , then , contradicting , so the only possibility left is , i.e., . ∎
We note the following standard fact:
Proposition 2.11.
Let be an injective holomorphic map. Then either or for a nonzero constant .
Proof.
Such an is in particular an analytic function with isolated singularities at and (and nowhere else.) By injectivity, neither singularity can be essential, so extends to a meromorphic mapping of the Riemann sphere to itself, i.e. to a rational function. Injectivity on the sphere with two points deleted implies that this rational function has degree one, i.e., is a Möbius transformation. In particular, the extended function maps the sphere bijectively to itself, so either , in which case , or . In the first case, is bounded at and has a removable singularity at , so, by Liouville’s Theorem, . In the case , applying the above to gives .
∎
A corollary of the above result is a uniqueness statement for Fatou coordinates:
Proposition 2.12.
Let be an attracting petal of , and let and be attracting Fatou coordinates on , i.e., univalent analytic functions satisfying . Then is constant on .
Proof.
By Proposition 2.9, and both induce conformal isomorphisms from to . It will be more convenient to work with with the punctured plane instead of . The function
induces – again by passage to quotients – a conformal isomorphism from to , so and both induce conformal isomorphism . Hence, the prescription
defines a conformal isomorphism . By Proposition 2.11, there are two possibilities:

there is a nonzero constant – which we write as – so that
or

there is a constant so that
In the first case,
But the expression on the left is continuous, and an integervalued continuous function on a connected set must be constant, so
which is what we wanted to prove.
In the second case, similarly,
and this contradicts
so this case is excluded. ∎
The critical values of Fatou coordinates are simply related to those of :
Proposition 2.13.
Let be an attracting Fatou coordinate defined on . Then critical values of all have the form
If is surjective, all numbers of this form are critical values.
Proof.
Iterating the functional equation for , differentiating, and applying the chain rule gives
For given and sufficiently large , is in an attracting petal, which implies . Thus: is a critical point of if and only if there exists such that is a critical point of . For such a , is a critical value of . Since
the assertion follows. ∎
For completeness, let us note how the situation changes if is a th root of unity with . A fixed petal for the iterate corresponds to a cycle of petals for . It thus follows that divides the number of attracting/repelling directions of as a fixed point of .
2.2. Asymptotic expansion of a Fatou coordinate at infinity
We will now specialize to the case and . By rescaling we can then bring the coefficient of to , and the normal form (2.2) becomes
(2.6) 
We will say in this case that is a simple parabolic fixed point of . There is one attracting direction () and one repelling direction ().
If the domain of definition is fixed, we let , as before, to denote the basin of the parabolic point at the origin. The immediate basin of , which we denote , is the connected component of which contains an attracting petal.
The change of variables moving the parabolic point to becomes simply
and we have
and is analytic at .
We showed earlier (Proposition 2.6) that any attracting Fatou coordinate at infinity for such an satisfies
We will prove shortly a much more precise result – an asymptotic expansion giving up to corrections of order for any . Before we do this, we investigate formal solutions to the functional equation
satisfied by both attracting and repelling Fatou coordinates.
Proposition 2.14.
There is a unique sequence of complex coefficients such that
(2.7) 
satisfies
(2.8) 
Furthermore, if we set
(2.9) 
then
(2.10) 
The logarithm appearing in (2.7) – and all other logarithms in this section – are to be understood as the principal branch, i.e., the branch with a cut along the negative real axis and real values on the positive axis. Because of the logarithmic term, as written is not exactly a formal power series in . To work around this, we rewrite the equation formally as
(2.11) 
Since is analytic at and takes the value there, is analytic at and vanishes there. Furthermore, the formal identity
holds literally on for sufficiently large , for any . It is equation (2.11) which we really solve.
The lefthand side of (2.11) is analytic at , and vanishes to second order there:
Further, is analytic at infinity and vanishes to order there, so
is indeed a formal power series in which begins with a term in . Furthermore, the coefficient of in the expression on the right in (2.11) can be written as
Thus, since the lefthand side of (2.11) is known, the ’s can be determined successively, and, by induction on , they are uniquely determined. The assertion about the order of the error term also follows, since
begins with a term in