Renormalization in the Golden-Mean Semi-Siegel Hénon Family

Renormalization in the Golden-Mean Semi-Siegel Hénon Family: Universality and Non-rigidity

Jonguk Yang
Abstract.

It was recently shown in [GaYa2] that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalizations in the Hénon family considered in [dCLM]. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Hénon map is non-rigid.

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 two-dimensional 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 one-dimensional system. In this paper, we will always assume that is a dissipative map (i.e. ).

Figure 1. A Hénon map . Note that vertical lines are scaled uniformly by , and then mapped to horizontal lines.

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 semi-Siegel 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 semi-Siegel 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 .

Figure 2. The Siegel cylinder and 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 two-dimensional 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 semi-Siegel 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 semi-Siegel Hénon maps of the golden-mean type.

It is important to note that the fixed point for is a degenerate one-dimensional 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 super-exponential rate.

In this paper, we describe the behaviour of almost commuting pairs as they approach the space of degenerate one-dimensional 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 one-dimensional maps:

the operator is based on a Hénon-like embedding (see (6)). Although the former embedding has the benefit of being more symmetric, the latter embedding allows us to track two-dimensional 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 two-dimensional shape as they flatten into degenerate one-dimensional 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 period-doubling renormalization in the Hénon family.

The universality phenomenon described in Theorem 7.3 has deep consequences on the geometry of the golden-mean 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 period-doubling renormalization is non-rigid. In this paper, we are able to obtain the following analogous result:

Non-rigidity Theorem.

Let and be as in Theorem 1.1. If , and , then the two semi-Siegel Hénon maps and cannot be conjugate on the boundary of their respective Siegel disks.

Non-rigidity is the first known property of Siegel disks of Hénon maps that is unique to higher dimensions. In the one-dimensional case, McMullen showed that two quadratic-like 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 quasi-symmetrically conjugate to the rigid rotation of the circle by angle .

Figure 3. The Siegel boundary of the golden-mean Siegel quadratic polynomial . The critical point is on .

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 small-scale 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

  1. [label=()]

  2. ;

  3. ;

  4. for and ; and

  5. 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 pre-renormalization 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 .

Figure 4. The first return map on under iteration of given by the pair .

Note that we can obtain the st pre-renormalization by taking the first return map on under iteration of the th pre-renormalization :

For the inverse golden mean , this corresponds to taking the following iterate of :

These observations suggest that the sequence of pre-renormalizations of can be realized as the orbit of under the action of some pre-renormalization 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.

  1. [label=)]

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

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

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

  5. 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 semi-Siegel fixed point with multipliers and . Such Hénon maps are parameterized by their Jacobian:

As , the semi-Siegel Hénon map degenerates to the following two-dimensional 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 two-dimensional 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 pre-renormalization 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 one-dimensional case, the sequence of pre-renormalizations of can be realized as the orbit of under the action of some pre-renormalization operator defined on a space of pairs of two-dimensional 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 two-dimensions than in the one-dimensional 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 Hénon-like form after each renormalization. To achieve this, we incorporate a non-linear 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 one-dimensional 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 golden-mean:

One-dimensional 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

  1. [label=()]

  2. and have a simple unique critical point at , and

  3. .

The space of critical pairs in is denoted by .

Figure 5. A critical pair .
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.

We say that is an almost commuting pair (cf. [Bur, Stir]) if

The space of almost commuting pairs in is denoted by .

Note that if is a critical pair, then the first-order 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 pre-renormalization of is defined as:

The renormalization of is defined as:

We say that is renormalizable if .

Figure 6. The 1D-renormalization .

The following is shown in [GaYa2]:

Theorem 3.6 (1D-Renormalization Hyperbolicity).

There exist topological disks and , and a commuting pair such that the following holds:

  1. [label=()]

  2. There exists a neighborhood of in the submanifold such that

    is an analytic operator.

  3. The pair is the unique fixed point of in . In particular, we have

    where

    is a universal scaling factor.

  4. 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 golden-mean 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 one-parameter family intersects the stable manifold of the fixed point for the 1D-renormalization operator . Moreover, this intersection is transversal, and occurs at .

Figure 7. The stable manifold of the fixed point for the 1D-renormalization operator . The family of quadratic polynomials intersect transversely at the golden-mean Siegel quadratic polynomial .

Two-dimensional 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

The following definitions are analogs of Definition 3.1, 3.2 and 3.3.

Definition 3.8.

For , we say that is an -critical pair if

  1. [label=()]

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

  3. .

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 1D-renormalizable pairs. That is, a pair is in if

is a well-defined 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 well-defined element of .

For , we denote by the set of pairs of the form

such that the following conditions are satisfied.

  1. [label=()]

  2. For

    the pair defined in Lemma 3.12 is a well-defined element of , where

  3. 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 pre-renormalization of as

(8)

Next, we denote

and consider the following non-linear changes of coordinates:

(9)

Define

Figure 8. The 2D-pre-renormalization .

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 non-commuting pairs map into as well.

Lemma 3.14.

There exists an analytic projection operator defined on such that for any , the following statements hold:

  1. [label=()]

  2. , where ,

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

  4. 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