Renormalization in the GoldenMean SemiSiegel Hénon Family: Universality and Nonrigidity
Abstract.
It was recently shown in [GaYa2] that appropriately defined renormalizations of a sufficiently dissipative goldenmean semiSiegel Hénon map converge superexponentially fast to a onedimensional renormalization fixed point. In this paper, we show that the asymptotic twodimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of perioddoubling renormalizations in the Hénon family considered in [dCLM]. As an application of our result, we prove that the boundary of the goldenmean Siegel disk of a dissipative Hénon map is nonrigid.
1. Introduction
The archetypical class of examples in holomorphic dynamics is given by the quadratic family:
Despite its apparent simplicity, the dynamics of this family is incredibly rich, and exhibits many of the key features that are observed in the general case. In dynamics of several complex variables, the role of the quadratic family is assumed by its twodimensional extension:
called the (complex quadratic) Hénon family.
Since we have
a Hénon map is a polynomial automorphism of . Moreover, it is easy to see that has constant Jacobian:
Note that for , the map degenerates to the following embedding of :
Hence, the parameter determines how far is from being a degenerate onedimensional system. In this paper, we will always assume that is a dissipative map (i.e. ).
A Hénon map is determined uniquely by the multipliers and at a fixed point . In particular, we have
and
When convenient, we will write instead of to denote a Hénon map.
Suppose that one of the multipliers, say , is irrationally indifferent, so that
Then we have
In this case, the Hénon map is said to be semiSiegel if there exist neighborhoods of and of , and a biholomorphic change of coordinates
such that
where . A classic theorem of Siegel states, in particular, that is semiSiegel whenever is Diophantine. That is, for some constants and , we have
where are the continued fraction convergents of (see Section 2). In this case, the linearizing map can be biholomorphically extended to
so that its image is maximal (see [MNTU]). We call the Siegel cylinder of . In the interior of , the dynamics of is conjugate to rotation by in one direction, and compression by in the other direction. Clearly, the orbit of every point in converges to the analytic disk at height . We call the Siegel disk of .
The geometry of Siegel disks in one dimension is a challenging and important topic that has been studied by numerous authors; including Herman [He], McMullen [Mc], Petersen [P], Inou and Shishikura [ISh], Yampolsky [Ya], and others. In the twodimensional Hénon family, the corresponding problems have been wide open until a very recent work of Gaidashev, Radu, and Yampolsky [GaRYa], who proved:
Theorem 1.1 (Gaidashev, Radu, Yampolsky).
Let be the inverse golden mean, and let . Then there exists such that if , then the boundary of the Siegel disk of is a homeomorphic image of the circle. In fact, the linearizing map
extends continuously and injectively (but not smoothly) to the boundary.
In the author’s joint paper with Yampolsky [YaY], we have obtained the first geometric result about Siegel disks in the Hénon family:
Theorem 1.2 (Yampolsky, Y.).
Let and be as in Theorem 1.1. Then for the boundary of the Siegel disk of is not smooth.
The proofs of Theorem 1.1 and 1.2 are based on the renormalization theory developed by Gaidashev and Yampolsky in [GaYa]. Generally speaking, a renormalization of a dynamical system is defined as a rescaled first return map on an appropriately chosen subset of the phase space. In their paper, Gaidashev and Yampolsky considered the semiSiegel Hénon maps within the context of a Banach space of dynamical systems called almost commuting pairs. They then formulated renormalization as an operator from to itself. They were able to show that this operator is analytic, and that it has a hyperbolic fixed point . In [GaRYa], they went on to prove that the stable manifold of does indeed contain the almost commuting pairs that correspond to sufficiently dissipative semiSiegel Hénon maps of the goldenmean type.
It is important to note that the fixed point for is a degenerate onedimensional system. Hence, when the renormalization sequence of an almost commuting pair converges to , it loses its dependence on the second variable along the way. In fact, Gaidashev and Yampolsky showed that this must happen at a superexponential rate.
In this paper, we describe the behaviour of almost commuting pairs as they approach the space of degenerate onedimensional systems under renormalization. For this purpose, we adopt a new renormalization operator that we obtain by modifying the construction of . The main difference between these two operators is that while is based on a diagonal embedding of the pairs of onedimensional maps:
the operator is based on a Hénonlike embedding (see (6)). Although the former embedding has the benefit of being more symmetric, the latter embedding allows us to track twodimensional deviations from its image more precisely and more explicitly. However, it should be noted that and are still related closely enough that a number of proofs given in [GaRYa] can be directly transferred to our setting, mutatis mutandis (in particular, see Theorem 6.7 and 6.8).
The central result of this paper is that in the limit of renormalization, the almost commuting pairs take on a universal twodimensional shape as they flatten into degenerate onedimensional systems. This statement is formulated explicitly in Theorem 7.3. The proof relies on an analysis of the average Jacobian of almost commuting pairs on their invariant renormalization arcs. A similar approach was taken by de Carvalho, Lyubich, and Martens in [dCLM] to study the limits of perioddoubling renormalization in the Hénon family.
The universality phenomenon described in Theorem 7.3 has deep consequences on the geometry of the goldenmean Siegel disk of dissipative Hénon maps. In [dCLM], de Carvalho, Lyubich, and Martens used universality to show, in particular, that the invariant Cantor set for perioddoubling renormalization is nonrigid. In this paper, we are able to obtain the following analogous result:
Nonrigidity Theorem.
Let and be as in Theorem 1.1. If , and , then the two semiSiegel Hénon maps and cannot be conjugate on the boundary of their respective Siegel disks.
Nonrigidity is the first known property of Siegel disks of Hénon maps that is unique to higher dimensions. In the onedimensional case, McMullen showed that two quadraticlike maps with a Siegel disk of the same bounded type rotation number are conjugate on their Siegel boundary (see [Mc]).
2. Motivation
Consider the quadratic polynomial
that has a fixed Siegel disc with rotation number . Since must be irrational, it is represented by an infinite continued fraction:
The th partial convergent of is the rational number
The denominator is called the th closest return moment. The sequence satisfy the following inductive relation:
(1) 
We say that is of bounded type if ’s are uniformly bounded. The simplest example of a bounded type rotation number is the inverse golden mean:
The following theorem is due to Petersen [P].
Theorem 2.1.
Let
be a quadratic polynomial that has a fixed Siegel disk with bounded type rotation number . Then has its critical point on its Siegel boundary , and the restriction is quasisymmetrically conjugate to the rigid rotation of the circle by angle .
Assume that is of bounded type, so that the dynamics of on its Siegel boundary is characterized by Theorem 2.1. We are interested in studying the smallscale behavior of this dynamics.
Consider the orbit of the critical point under :
Denote by the closed arc containing the critical point whose end points are and , where and are the th and the st closest return moments respectively. The arc can be expressed as the union of two closed subarcs and , where has its end points at and . Observe

[label=()]

;

;

for and ; and

for and .
The subarc is called the th closest return arc.
The arcs form a nested neighborhood of in , and by Theorem 2.1, we see that
(2) 
Define the th prerenormalization of as the first return map on under iteration of . It is not hard to see that
Hence, we can consider as a pair of maps
(3) 
acting on the arc . Letting , we obtain a pair representation of :
Intuitively, captures the dynamics of on the Siegel boundary that occurs at the scale of .
Note that we can obtain the st prerenormalization by taking the first return map on under iteration of the th prerenormalization :
For the inverse golden mean , this corresponds to taking the following iterate of :
These observations suggest that the sequence of prerenormalizations of can be realized as the orbit of under the action of some prerenormalization operator defined on a space of pairs of maps.
By (2), we see that degenerates as to a pair of maps acting on a single point (namely, ). To obtain a more meaningful asymptotic behavior, we need to magnify the dynamics of and bring it to some fixed scale. The simplest way to do this is to conjugate
by a linear map that sends the critical value to . The resulting rescaled dynamical system
is called the th renormalization of . If we denote the rescaling operator on pairs by , we can define the renormalization operator as
Similar to , the renormalization operator acts on the space of certain pairs of maps. If belongs to this space, then it should satisfy the following properties.

[label=)]

The maps and each have a unique simple critical point at .

The scale of is normalized, so that the critical value is at .

The maps and extend to holomorphic maps on some neighborhoods and of in .

Where and are both defined, these maps should commute:
Observe that commutativity clearly holds for , since in this case, and represent different iterates of the same map .
The main goal of this paper is to extend the theory of renormalization to a higher dimensional setting. To this end, consider a quadratic Hénon map
that has a semiSiegel fixed point with multipliers and . Such Hénon maps are parameterized by their Jacobian:
As , the semiSiegel Hénon map degenerates to the following twodimensional embedding of the quadratic Siegel polynomial :
Hence, for , the dynamics of can be considered as a small perturbation of the dynamics of .
Let be the twodimensional Siegel disk of . A priori, we do not have an analog of Theorem 2.1 that characterizes the dynamics of on . However, we can still define the th prerenormalization of by taking the same iterates as in (3):
(4) 
In (4), the sets and are chosen to be some suitable domains in which intersect . By letting , we obtain a pair representation of :
Analogously to the onedimensional case, the sequence of prerenormalizations of can be realized as the orbit of under the action of some prerenormalization operator defined on a space of pairs of twodimensional maps. To transform into a proper renormalization operator , we need to compose with some suitable rescaling operator . However, this turns out to be more a intricate problem in twodimensions than in the onedimensional case. To ensure a tractable asymptotic behavior under iterations of , it is not only important to fix the scale of the dynamical systems, but we must also bring them back to Hénonlike form after each renormalization. To achieve this, we incorporate a nonlinear change of coordinates to the definition of . Further details are provided in Section 3.
Suppose that the renormalizations of are given by
where and are defined on some fixed neighborhoods and of in . Recall that and represent rescalings of the and iterates of respectively. If is sufficiently dissipative, so that for some , then by the chain rule, the Jacobians of and are on the order of and respectively. Hence, if the renormalization sequence converges to some limit , then we have
Thus, we see that the limit of the renormalizations of must be a degenerate onedimensional system.
3. Renormalization of Almost Commuting Pairs
In this section, we formalize the ideas discussed in Section 2. While previously, we considered any rotation number of bounded type, we will henceforth restrict to the case of the inverse goldenmean:
Onedimensional renormalization
For a domain , we denote by the Banach space of bounded analytic functions , equipped with the norm
Denote by the Banach space of bounded pairs of analytic functions from domains and respectively to , equipped with the norm
Henceforth, we assume that the domains and contain .
For a pair , define the rescaling map as
where
Definition 3.1.
We say that is a critical pair if

[label=()]

and have a simple unique critical point at , and

.
The space of critical pairs in is denoted by .
Definition 3.2.
We say that is a commuting pair if
It turns out that requiring strict commutativity is too restrictive in the category of analytic functions. Hence, we work with the following less restrictive condition.
Definition 3.3.
Note that if is a critical pair, then the firstorder commuting relation is automatically satisfied:
Proposition 3.4 (cf. [GaYa2]).
The spaces and have the structure of an immersed Banach submanifold of of codimension and respectively.
Definition 3.5.
Let . The prerenormalization of is defined as:
The renormalization of is defined as:
We say that is renormalizable if .
The following is shown in [GaYa2]:
Theorem 3.6 (1DRenormalization Hyperbolicity).
There exist topological disks and , and a commuting pair such that the following holds:

[label=()]

There exists a neighborhood of in the submanifold such that
is an analytic operator.

The pair is the unique fixed point of in . In particular, we have
where
is a universal scaling factor.

The differential is a compact linear operator. Moreover, has a single, simple eigenvalue with modulus greater than . The rest of its spectrum lies inside the open unit disk (and hence is compactly contained in by the spectral theory of compact operators).
Let
be the quadratic polynomial with a Siegel fixed point of multiplier , where is the inverse goldenmean rotation number. For sufficiently close to , we can identify the quadratic polynomial as a pair in as follows:
(5) 
where
The following is shown in [GaRYa]:
Theorem 3.7.
The oneparameter family intersects the stable manifold of the fixed point for the 1Drenormalization operator . Moreover, this intersection is transversal, and occurs at .
Twodimensional renormalization
For a domain , we denote by the Banach space of bounded analytic functions , equipped with the norm
Define
Denote by the Banach space of bounded pairs of analytic functions from domains and respectively to , equipped with the norm
Define
Henceforth, we assume that
where , , and are domains of containing . For a function
from or to , we define the projection map as
For a pair , we define the projection map as
and the rescaling map as
where
Definition 3.8.
For , we say that is an critical pair if

[label=()]

and have a simple unique critical point which is contained in a neighborhood of , and

.
The space of critical pairs in is denoted by .
Definition 3.9.
We say that is a commuting pair if
Definition 3.10.
We say that is an almost commuting pair if
The space of almost commuting pairs in is denoted by .
Notation 3.11.
We denote by the set of 1Drenormalizable pairs. That is, a pair is in if
is a welldefined element of .
We define an embedding of into as follows. For a pair , we let
where
(6) 
Observe
(7) 
Lemma 3.12.
Let and be domains in . For any , there exists a neighborhood of such that if , then the pair
is a welldefined element of .
For , we denote by the set of pairs of the form
such that the following conditions are satisfied.

[label=()]

We have and .
Lemma 3.13.
The set have the structure of an immersed Banach submanifold of .
We define the renormalization of in several steps. First, we define the prerenormalization of as
(8) 
Next, we denote
and consider the following nonlinear changes of coordinates:
(9) 
Define
Let
It is not hard to check that we have the following estimates:
(10) 
where is the embedding given in (6).
By (10) and the argument principle, it follows that if is sufficiently small, then the function has a simple unique critical point near . Set
(11) 
and define
Observe that we have
If is a commuting pair, then and would also commute. In this case, we would have
Thus, we see that the operator maps commuting pairs in into , where . To finish the definition of the renormalization operator, we need to compose with an appropriate projection operator, so that the image of noncommuting pairs map into as well.
Lemma 3.14.
There exists an analytic projection operator defined on such that for any , the following statements hold:

[label=()]

, where ,

, where and is the embedding given in (6), and

if is a commuting pair, then .
Proof.
By (10) and the argument principle, it follows that if is sufficiently small, then the function has a simple unique critical point near . Set
and define
Observe that we have
and similarly
Hence, we see that is in for some .
Write
To project into for some , we make the following modifications:
where
Observe that still belongs to . To determine the constants and , we compute:
and