Automorphisms of with an invariant nonrecurrent attracting Fatou component biholomorphic to
Abstract.
We prove the existence of automorphisms of having an invariant, nonrecurrent Fatou component biholomorphic to which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component.
Contents
Introduction
Let be a holomorphic endomorphism of or , . In the study of the dynamics of , that is of the behavior of its iterates, a natural dichotomy is given by the division of the space into the Fatou set and the Julia set. The Fatou set is the largest open set where the family of iterates is locally normal, that is the set formed by all points having an open neighborhood where the restriction of the iterates of the map forms a normal family. The Julia set is the complement of the Fatou set and is the part of the space where chaotic dynamics happens. A Fatou component is a connected component of the Fatou set.
A Fatou component for a map is called periodic if there exists , , such that (we call it invariant in case , i.e., ). A Fatou component is preperiodic if there exists such that is periodic. It is a wandering component otherwise. While in dimension one Sullivan’s NonWandering Domain Theorem states that there are no wandering Fatou components for rational maps of degree , in dimension two it has been recently proved by the second named author with M. Astorg, X. Buff, R. Dujardin and H. Peters, [1] that there exist polynomial endomorphisms of having wandering Fatou components.
In the classification of preperiodic Fatou components, one can essentially reduce to the case of invariant components. These are divided in nonrecurrent, if all the orbits escape to the boundary of the component, and recurrent, if there exists a point in the component whose limit belongs to the component itself.
We call attracting any invariant Fatou component for a map such that there exists with for all . Note that, in particular, is a fixed point for and is recurrent if , nonrecurrent if .
It follows from a result of H. Peters, L. R. Vivas and E. Fornæss Wold [11], that every attracting recurrent Fatou component of a holomorphic selfmap of is biholomorphic to and it is, in fact, the global basin of attraction of at , which is an attracting fixed point (namely, the eigenvalues of have modulus strictly less than ).
Recurrent invariant Fatou components for polynomial maps have been classified by J.E. Fornæss and N. Sibony in [7] (see also [16], where T. Ueda completed the classification by proving that a certain case cannot occur), while nonrecurrent components have been studied by B. J. Weickert in [18]. More recently, in [10], M. Lyubich and H. Peters classified nonrecurrent invariant Fatou components, under the additional hypothesis that the limit set is unique (see also [3]). In particular, as a consequence of [10, Theorem 6], they proved that, any nonrecurrent invariant attracting Fatou component of a polynomial automorphism of is biholomorphic to .
For nonpolynomial automorphisms of the situation is completely different. In fact this paper is devoted to prove the following result.
Theorem 0.1.
There exist holomorphic automorphisms of having an invariant, nonrecurrent, attracting Fatou component biholomorphic to .
Our result also shows that the socalled “Snail Lemma” no longer holds in two or more variables.
Let be an automorphism of and assume that the origin is an isolated fixed point. If the eigenvalues of have modulus strictly less than , then there exists an attracting, invariant, recurrent Fatou component containing which is biholomorphic to (see, e.g., [13]). If one eigenvalue is and the other has modulus less than , then T. Ueda [15] proved the existence of an immediate basin of attraction such that the union of all preimages forms an invariant open set biholomorphic to . A similar result has been proved by B. Weickert [17] (see also M. Hakim [9]) when both eigenvalues are equal to . It is not clear whether such attracting domains, which can be well characterized in terms of the behavior of the orbits, coincide with the Fatou components containing them. Using Hakim’s result, in [14], the third named author together with L. Vivas, constructed an automorphism of tangent to the identity at the origin, pointwise fixing an axis, and having an invariant attracting domain biholomorphic to . Although this domain is not known to be a Fatou component, one can see that the Fatou component containing such an invariant domain is not simply connected.
It is worth mentioning here that in general completely invariant domains are not Fatou components, and that, in a same Fatou component the behavior of the orbits (which is the way the attracting domains in the previous recalled papers have been defined) cannot be well characterized. Consider for instance the linear automorphism of given by is a recurrent, invariant, attracting Fatou component for . The set is however completely invariant, but it is not a Fatou component. Moreover, denote by and by , where and . Then, for every with , it follows that , but this is no longer true for points of the form or with . . Then
The previous discussion should make clear which difficulties one can encounter in constructing an automorphism of having an invariant, nonrecurrent, attracting Fatou component biholomorphic to . On the one hand, one has to find an automorphism having an attracting domain which is biholomorphic to , and on the other hand one has to prove that such an attracting domain is, indeed, a Fatou component.
We perform our construction starting from a simple case of a onedimensional resonant germ of , as introduced by the first named author with D. Zaitsev in [4] (see also [5]). We consider germs of holomorphic diffeomorphisms of at the origin of the form
(0.1) 
where is a germ of a holomorphic function in near such that , , , , is not a root of unity, and and . In [4] (see also Theorem 1.4) it has been proved that, provided that is sufficiently large (depending on ), there exists an invariant domain having the origin on the boundary such that for all . In this paper we show that is doubly connected and all orbits landing in can be characterized in terms of orbits behavior of . Moreover, we prove that there exist “Fatou coordinates” on which conjugate to a simple overshear of .
Thanks to a result of F. Forstnerič [8], there exist automorphisms of of the form (0.1) at the origin. For such an automorphism , one can define a completely invariant domain by setting . We prove that the domain can be characterized as the union of all points such that , where , .
Moreover the domain is non simply connected. The local Fatou coordinates on can be extended to a univalent map from to and we prove that the image of such a map is .
On the other hand, if the eigenvalue in (0.1) satisfies the socalled Brjuno condition (see Section 3 for details), which is satisfied by a set of full measure on the unit circle, we show that coincides with the Fatou component containing it. In order to prove this, we use the Brjuno condition to construct local coordinates in such a way that the axes are not contained in . Therefore, in these new coordinates, any two orbits in have finite Kobayashi distance with respect to a small ball at the origin minus the two axes. This enables us to use some estimates on the Kobayashi distance of the punctured disc, that, together with a careful description of the shape of , allows to prove that if there is an orbit in which is not contained in , then the Kobayashi distance in between this orbit and any orbit in is bounded from below by a fixed constants, which is not possible.
Acknowledgements. Part of this paper was written while the first and the third named authors were visiting the Center for Advanced Studies in Oslo for the 201617 CAS project Several Complex Variables and Complex Dynamics. They both thank CAS for the support and for the wonderful atmosphere experienced there.
The authors also thank Han Peters for some useful conversations.
Notations and conventions
We set up here some notations and conventions we are going to use throughout the paper.
We let , , be defined by
If is a map, we denote by the th iterate of , , defined by induction as , . Moreover, for and , we let
If are real positive functions of , we write
if there exist such that for all .
Moreover, we use the Landau little/big “O” notations, namely, we write
if there exists such that for all , while we write
if .
1. The local basin of attraction
Let be a germ of biholomorphism of , fixing the origin, of the form
(1.1) 
where, is a germ of a holomorphic function in near such that , , , , is not a root of unity, and .
Definition 1.1.
For and we let
Also, we let
Lemma 1.2.
Let be a germ of biholomorphism at of the form (1.1). Let
Fix and set
Then there exists such that for every ,
(1.2) 
and
(1.3) 
Proof.
Let and let . If , then , hence,
Therefore, for , recalling that , we have
Thus, if is sufficiently large, it follows that for every ,
(1.4) 
On the other hand, if is sufficiently large and ,
Thus, from the previous inequality and (1.4), we immediately obtain (1.2) for sufficiently large.
A similar argument holds for (1.3). ∎
In [4, Thm. 1.1] it has been proved that any small variation of admits a local basin of attraction. For the reader’s convenience and for the sake of completeness in the present work, we sketch the proof of such a result. To this aim, let us introduce some sets:
Definition 1.3.
For we let
For every , and , we let
We are now ready to state and prove the existence of local basins of attraction:
Theorem 1.4.
Let be a germ of biholomorphism at of the form (1.1). Let and let be the constants given in Lemma 1.2. Let be such that . Let , be such that . Finally, assume is a germ of biholomorphism at of the form
Then there exists such that for every the (nonempty) open set is a uniform local basin of attraction for . Namely, , and
uniformly in .
Sketch of the Proof.
For , we have
and
(1.5) 
Let . The function maps the set onto the set . Hence, if and only if .
Since implies that and , we have
(1.6) 
for all . Hence, if is sufficiently large so that , it follows that whenever , and .
Now we have to show that if then .
We prove that , a similar argument works for . From (1.1), we have
Since implies , by Lemma 1.2 (setting for simplicity),
for some .
Taking into account that , we can write
for some .
Hence, if we can prove that
which is equivalent to
(1.7) 
Now, if is large enough, bearing in mind that we assumed that , then for every such that ,
hence, (1.7) follows.
Summing up, we proved that there exists such that for any . Moreover, arguing by induction, from (1.6), one can easily prove that
for all , showing that converges uniformly to on .
Finally, note that is nonempty for all , since for all small enough. ∎
Definition 1.5.
Let be as in Theorem 1.4. We set
In the following, we shall use some properties of , that we prove here below. We start with a lemma, which allows us to characterize the preimages of .
Lemma 1.6.
Let and be as in Theorem 1.4. Let be such that and such that as . If there exists such that for all , then

and (in particular, ),

and ,

for every there exists such that for all .
In particular, eventually.
Proof.
The map has the form
(1.8) 
where and , .
Since
taking into account that , we have
For , and since , we have that is at most an . Hence,
(1.9) 
Fix . Let . We claim that there exists such that, for all we have
(1.10) 
and
(1.11) 
In order to prove (1.10) and (1.11), we first notice that, by (1.9), there exists such that for all ,
Then, starting from (1.9), one can prove easily (1.10) and (1.11) by induction on .
From (1.10) and (1.11), letting (so that ) we obtain immediately that
(1.12) 
In particular, this means that . Hence, , which implies at once that
(1.13) 
Hence statement (1) follows.
Now, for , we set
and
Arguing by induction, we have
(1.14) 
Now we concentrate on , as the argument for is similar. We have,
(1.15) 
Taking into account statement (1), we have
Therefore,
(1.16) 
Moreover, since eventually, and , it follows that there exists some constants such that
Hence, by (1.16) we have for sufficiently large
Since
Hence, from (1.15), there exists a constant such that
(1.17) 
A similar argument for , shows that
(1.18) 
By statement (1), it holds . Since and by (1.17) and (1.18), it follows that, in fact, and . Statement (2) is then proved.
Finally, by statement (2), there exist constants such that for all and . Fix . Hence, for every large enough,
Similarly, one can prove that . As a consequence, eventually for every . ∎
Remark 1.7.
We shall also need the following local result concerning the topology of :
Lemma 1.8.
Let and be as in Theorem 1.4. Then is a doubly connected domain (i.e., is connected and its fundamental group is ). Moreover, its fundamental group is generated by the loop for any fixed .
Proof.
Let be defined by
(1.19) 
Note that is a biholomorphism from onto . Since , it follows that is a biholomorphism. A straightforward computation shows that
(1.20) 
In particular, is a submersion over where each fiber is an annulus, centered at the origin, of internal radius and external radius .
Consider the map given by
(1.21) 
where, writing in polar coordinates, and ,
The map is a diffeomorphism onto its image, which is given by . Hence is doubly connected and clearly the loop , which is contained in , generates its fundamental group. ∎
Remark 1.9.
Let be a closed continuous curve, where is given by (1.20). The curve is contractible in if and only if the winding number around of the curve is zero. Indeed, by Lemma 1.8, is homotopic to for some , where . Hence, there exists a homotopy of paths between and . Clearly, is a homotopy between and and thus the winding number of is , and the statement is proved.
2. Local Fatou coordinates on
In this section we are going to introduce special coordinates on , which will be used later on in our construction. The first coordinate, was introduced in [5, Prop. 4.3]. Here we shall need more precise information, therefore we are going to prove the following result:
Proposition 2.1.
Let and be as in Theorem 1.4. Then there exists holomorphic such that
(2.1) 
The map is the uniform limit in of the sequence of functions , where