User:Xavier Buff/Proposed/Siegel disks

• Prof. Xavier Buff, Laboratoire E. Picard, Université Paul Sabatier

• Arnaud Chéritat, Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France

• Prof. Lasse Rempe-Gillen, Department of Mathematical Sciences, University of Liverpool

Figure 1: The quadratic golden mean Siegel disk

A Siegel disk of a discrete holomorphic one-variable dynamical system is a maximal domain in which the system (or an iterate) is conjugate to an irrational rotation of a disk.

Example: Consider the map \(z\mapsto \rho z + z^2\) from \(\mathbb{C}\) to \(\mathbb{C}\ ,\) where \(\rho=e^{i2\pi\omega}\) is the complex number with modulus one and argument \(2\pi\omega\) and \(\omega=\frac{\sqrt{5}+1}{2}\) denotes the golden mean. It has a Siegel disk, shown in Figure 1. This particular example is known to be bounded by a fractal Jordan curve, whose Hausdorff dimension lies strictly between 1 and 2.

Problems regarding the existence and structure of Siegel disks appear to be among the most subtle in holomorphic dynamics. There has been consistent and at times spectacular progress in their study over time, but a number of important problems remain unsolved. Some of the key questions in the study of Siegel disks are:

• When do Siegel disks exist?
• What happens at the boundary of a Siegel disk? What is the topology of this boundary?
• What is the geometry of a Siegel disk?

The linearization problem

The question of the existence of Siegel disks is a subset of the linearization problem and closely related to small divisors. A short overview of these topics is given here, but for more detail the reader should refer to the separate subpage on linearization.

Setting and terminology

A discrete dynamical system is modeled by \(x_{n+1}=f(x_n)\) where \(x_n\) represents the state at time \(n\ .\) The objects \(x_n\) may be modeled by numbers, points in a manifolds, etc... We will call them points. The sequence \((x_n)\) is called the orbit. A change of variables (or change of coordinates, or conjugacy) consists in looking at \(y=\phi(x)\) instead of \(x\ .\) Then \(y_{n+1} = h(y_n)\) for some other map \(h\) that can be explicitly computed in terms of \(f\) and \(\phi\ :\) \(h=\phi\circ f\circ \phi^{-1}\ .\) For a well chosen change of variables, this new map may be easier to iterate. Mathematicians say that \(\phi\) conjugates \(f\) to \(h\ .\)

The composition \(f\circ\cdots\circ f\) with \(n>0\) occurrences of \(f\) is called the \(n\)-th iterate of \(f\) and is denoted by \(f^n\). In particular \(f^1=f\). By convention \(f^0\) denotes the identity map, which sends every \(x\) to \(x\). If \(\phi\) conjugates \(f\) to \(h\), then it conjugates \(f^n\) to \(h^n\), (at least for parts of orbits \(x_1,\ldots,x_n\) that are contained in the domain of \(\phi\); typically, near fixed points).

Statement of the linearization problem and first properties

The present encyclopedia entry focuses on holomorphic dynamical systems in one variable, i.e. \(f\) is a holomorphic function of one complex variable. For instance \(f:z\mapsto z^2+c\) from \(\mathbb{C}\) to \(\mathbb{C}\ ,\) where \(c\in\mathbb{C}\) is any given parameter. So a complex number will also be called a point when it represents an element of an orbit of such a system.

A periodic point is a point \(a\) that is eventually mapped back to itself after a finite number of iterates. The period of \(a\) is the minimal \(p>0\) such that \(f^p(a)=a\). A fixed point is a periodic point of period one, i.e. a point \(a\) such that \(f(a)=a\ .\) If \(f\) is differentiable, it is well approximated by its linear part, i.e. its differential at \(a\ .\) Linear maps are easy to iterate, and the linearization problem is the question whether or not a change of variable can be found for which f is conjugated to its linear part. A periodic point of period \(p\) is a fixed point of \(f^{kp}\) for all \(k>0\), and the linearization problem can also be formulated in this situation. It is not obvious that linearizability is independent of \(k\) but it will be the case in our context.

The multiplier and irrationally indifferent fixed points. If \(f\) is a holomorphic map and \(a\) is a fixed point, then the multiplier is the complex number \(\lambda=f'(a)\ .\) Usually, the multiplier of a point of period \(p\) refers to \((f^p)'(a)\), i.e. the multiplier associated to the minimal iterate that fixes it. The multiplier is invariant under holomorphic conjugacy. It is sometimes called eigenvalue, a terminology that comes from the case of higher dimensional phase spaces.

The linear part (a.k.a. the differential) of \(f^p\) at the point \(a\) takes the form \(z\mapsto \lambda z\ .\) If \(|\lambda| = 1\ ,\) then \(a\) is an indifferent fixed point and the linear part is a rotation. Write \(\lambda = \exp(i2\pi\theta)\) for some \(\theta\in\mathbb{R}\ .\) The number \(\theta\) is called the rotation number. If \(\theta\notin\mathbb{Q}\ ,\) then \(a\) is called irrationally indifferent.

Equivalence of topological and holomorphic linearizability. For holomorphic maps in one variable, it turns out that linearizability by a continuous conjugacy implies linearizability by a holomorphic conjugacy.

Stability implies linearizability. There is even a stronger result. A dynamical system is said to be stable near a fixed point if for any neighborhood \(V\) of the fixed point, there is a neighborhood \(U\subset V\) such that the sequence \((x_n)\) is defined for all \(n\geq 0\) and remains in \(V\) as soon as \(x_0\) is in \(U\ .\) This notion is usually referred to as Lyapunov stability. It is known Template:Citation needed that when \(f\) is a holomorphic map in one variable, a stable indifferent fixed point is linearizable.

As a consequence, in our setting, for a periodic point of period \(p\), if \(f^{kp}\) is linearizable for some \(k>0\) it will be linearizable for all \(k>0\).

For simplicity, in the sequel, we restrict the discussion to period one. The adaptation to higher period is usually straightforward, otherwise we give some precisions.

Brjuno's arithmetic condition

Brjuno's condition is an arithmetical condition that plays an important role in the question of when a holomorphic map is linearizable near an irrationally indifferent fixed point. To state this condition, let \(\theta\) be an irrational real number\[\theta\in\mathbb{R}\setminus\mathbb{Q}\ .\] Also let \(p_n/q_n\) be the sequence of continued fraction convergents of \(\theta\ .\) (For background on continued fraction expansions, see Linearization.)

Definition. The number \(\theta\) is said to satisfy Brjuno's condition (also called the Brjuno-Rüssmann condition) whenever \(\sum_{n=0}^{\infty} \frac{\log q_{n+1}}{q_n} < +\infty\ .\)

It can be checked that all Diophantine numbers satisfy Brjuno's condition. Therefore almost all real numbers do, in the sense that the set of real numbers that do not has Lebesgue measure equal to 0.

Theorem (sufficiency and optimality of Brjuno's condition). let \(\theta\) be irrational.

• If \(\theta\) satisfies Bjruno's condition, then all fixed points with multiplier \(e^{2i\pi\theta}\) are linearizable.
• If \(\theta\) does not, then there exists maps with a non linearizable fixed point with multiplier \(e^{2i\pi\theta}\ .\)

Historical notice. The first part of the previous theorem is due to Brjuno and Rüssmann in the 1960s (Brjuno, 1965). Yoccoz proved the second part in the late 1980s (see Yoccoz, 1995), and also gave an independent proof of the first. Historically, non-linearizable fixed points were proven to exist before linearizable ones, and it is considered as a major breaktrhough when Siegel managed to prove in the 1940s the linearizability for all maps when the rotation number is diophantine (Siegel, 1942).

In general, a fixed point with any non-Brjuno rotation number \(\theta\) may be linearizable. Indeed, the rotation \(z\mapsto e^{i2\pi\theta}z\) is linearizable by definition; the same is true for any map obtained from this rotation by a local holomorphic change of variable. However, when one restricts the class of functions under consideration, more can be said. Indeed, Yoccoz proved the following, more precise version of his theorem:

Theorem. If \(\theta\) does not satisfy Bjruno's condition, then the fixed point \(z=0\) of the polynomial \(e^{2i\pi\theta}z+z^2\) is not linearizable.

It completely settles the linearization question for fixed points polynomials of degree two (a.k.a. quadratic polynomials): given any quadratic polynomial \(P\) and a fixed point thereof, one can find a linear change of variable \(\phi(z)=az+b\) that sends the fixed point to the origin and conjugates \(P\) to the polynomial \(\lambda z+z^2\) where \(\lambda\) is the multiplier of the fixed point.

Following Yoccoz's proof, the necessity of Brjuno's condition for linearization was established for a large number of other families of holomorphic functions. (Compare Geyer, 2004.) In particular it holds for quadratic polynomials for periodic points of any period (it can also be deduced from Yoccoz's theorem using Douady-Hubbard renormalization). On the other hand, it is still an open question whether there exist cubic polynomials with non-Brjuno linearizable fixed points. It is expected that this is not the case; in fact, the following has been conjectured by Douady.

Conjecture (Douady). Let \(f\) be a rational function* of degree at least 2 and suppose that \(a\) is an irrationally indifferent periodic point of \(f\) of period $p$ whose rotation number does not satisfy Brjuno's condition. Then \(f^p\) is not linearizable near \(a\ .\)

(*) A rational function, also called a rational map, is a quotient of two polynomials\[f(z)=P(z)/Q(z)\ .\] All points \(z\in\mathbb{C}\ ,\) apart a finite number of them, have the same number of preimages by \(f\) (solutions \(w\) of \(f(w)=z\)). This number is called the degree of \(f\ .\) Polynomials are a particular case of rational maps and in this case the notion of degree coincides with the usual one.

Siegel disks

Now suppose that a holomorphic map \(f\) has an irrationally indifferent fixed point \(a\ ,\) and assume that \(a\) is linearizable. Then there is a maximal connected open set \(\Delta\) that contains \(a\) and on which \(f\) is conjugate to a rotation. This domain is called the Siegel disk of \(f\) at \(a\ .\) Any Siegel disk is simply connected, and thus homeomorphic to a disk.

In the case of a periodic point of period \(p\), a Siegel disk for \(f\) at \(a\) designates a Siegel disk for \(f^p\) at \(a\). If we want to distinguish between the period one case and the general case, we will use the terms invariant Siegel disk and periodic Siegel disk. For the purpose of the article, we shall consider only invariant Siegel disks and typically omit the adjective invariant for brevity; however, many of the results will hold also for periodic Siegel disks more generally. Note that iterates of a rational map are rational maps and iterates of a polynomial are polynomials.

Riemann maps are linearizers. Let \(\lambda=f'(a)\ .\) By Riemann's mapping or uniformization theorem, there exists a conformal isomorphism \(\phi\) from the Siegel disk to either \(\mathbb{D}\) or \(\mathbb{C}\ ,\) sending \(a\) to 0. (A conformal isomorphism is a holomorphic bijection between two open sets; the inverse of such a map is always itself holomorphic.) Then, the function \(\phi\circ f\circ \phi^{-1}\) is a conformal automorphism of \(\mathbb{D}\) or \(\mathbb{C}\) that fixes the origin with multiplier \(\lambda\ .\) A conformal automorphism of \(\mathbb{D}\) or \(\mathbb{C}\) fixing the origin is necessarily a linear map map. Hence \[\phi\circ f \circ\phi^{-1}(z) = \lambda z.\]

In other words: the linearizing coordinate of the Siegel disk is given by the Riemann mapping.

Figure 2: The Siegel disk of some rational map, set in its Julia set

Siegel disks are Fatou components. Suppose that \(f\) is a polynomial, a rational map or a transcendental entire or meromorphic function. If \(a\) is any irrationally indifferent fixed point of \(f\ ,\) then \(a\) belongs to the Fatou set if and only if it is linearizable. Then the Siegel disk around \(a\) is exactly the connected component of the Fatou set of \(f\) (with one exception*). These assertions date back to Fatou and Julia. (The Fatou set is defined as the largest open set on which the iterates of \(f\) are defined and form a normal family in the sense of Montel. One of the many equivalent definitions of the Julia set is as the complement of the Fatou set.)

(*)in the case of a degree one rational map with an irrationally indifferent fixed point, there is a Möbius map conjugating it to a rotation on the whole riemann sphere; therefore it has exactly two fixed points, both are linearizable, the Fatou set is the whole sphere and the Siegel disk of each fixed point is the sphere minus the other fixed point.

 

Siegel disk boundaries

The behavior of irrational rotations, and hence of the dynamics inside a Siegel disk, is completely understood - although it is more interesting than one might at first think. However, things are different with Siegel disk boundaries.

What does such a boundary look like? Where is it? To quote Zakeri's survey (Zakeri, 2002): "What prevents [the Siegel disk] from going further?" Even today, only partial answers are known: many questions remain open.

Siegel disk boundaries are not analytic curves. The boundary of a Siegel disk \(\Delta\) cannot be an analytic closed curve contained in the domain of definition of \(f\ ,\) otherwise the linearizing coordinate could be extended beyond \(\Delta\) by Schwarz's reflection principle. In other words, there are two cases:

• Either the Siegel disk goes to infinity or touches the boundary of the domain of definition of \(f\ .\) Its boundary may or may not be an analytic curve.
• Or its boundary is contained in the the domain of definition of \(f\ .\) Then it can be shown it will be nowhere analytic.

For polynomials and rational maps of degree at least two, and for transcendental entire and meromorphic maps, Siegel disk boundaries are always nowhere analytic.

Topology of Siegel disk boundaries

The word topology here refers to a classification of objects up to homeomorphism.

Bounded Siegel disks.

Definition. A Siegel disk \(\Delta\) of \(f\) is called bounded if it is contained in a compact subset of the domain of definition of \(f\ .\) Otherwise, the Siegel disk is called unbounded.

Note that the domain of definition of a polynomial or a rational function is the entire Riemann sphere. Hence every Siegel disk of such a map is automatically bounded as per the definition above. In the case of a polynomial of degree at least 2, the Siegel disk is bounded in the classical sense as a subset of the plane, since it is disjoint from the basin of infinity and the latter contains a neighborhood of infinity. For general maps, the following theorem gives a dichotomy for the structure of the boundary of a bounded Siegel disk. Recall that an indecomposable continuum is a compact connected metric space that cannot be written as the union of two closed connected proper subspaces.

Theorem. (J.T. Rogers, 1992) Let \(B\) denote the boundary of a bounded Siegel disk \(\Delta\ .\) Then either

• B is tame: the conformal map from \(\Delta\) to the unit disk has a continuous extension to B.
• B is wild: it is an indecomposable continuum.

(In this statement, the words "wild" and "tame" are to be taken as adjectives, not as terminology. They are not used in the same context in Rogers's article. Note also that the definition of tame above is not to be mistaken with the following, which turns out to be equivalent to local connectivity of the boundary of \(\Delta\) by a theorem of Caratheodory: the conformal map from the unit disk to \(\Delta\) has a continuous extension to the boundary of the unit disk.)

A typical example of a boundary in the first case of Rogers's theorem would be any Jordan curve. (A Jordan curve is a subset of the plane homeomorphic to a circle.) A typical example of the second case would be a pseudocircle (caution: several unrelated mathematical objects share this name). Rogers's theorem applies in fact in a more general setting; refer to his article for details.

Lemma. If the boundary \(B\) of a bounded Siegel disk is locally connected, then it is a Jordan curve. (All Jordan curves are locally connected.) Moreover, the conjugacy to a rotation extends to the boundary to a homeomorphism that still conjugates the map to the rotation.

A bounded Siegel disk whose boundary is wild was constructed in (Chéritat, 2010). On the other hand, there is no polynomial, transcendental or rational map for which the Siegel disk is known not to be a Jordan curve, and it may very well be the case that they always are Jordan curves for these maps. Also, if the rotation number is of bounded type (the entries in the continued fraction expansion form a bounded sequence), then it is known in many cases that \(B\) is a Jordan curve (in fact, even a quasicircle). For Siegel disks of polynomials this was proved by Zakeri in degree 3, by Shishikura in any degree; for rational by Zhang, (Zhang, 2008). These results have also been generalized to certain families of entire transcendental functions; compare Geyer, Zakeri and Zhang. However, as explained below, there are also entire functions with bounded-type Siegel disks that are unbounded and whose boundary is emphatically not a Jordan curve.

Figure 3: A computer drawing of the exponential golden mean Siegel disk. Some parts are too thin to be visible

Unbounded Siegel disks. As already mentioned, polynomials and rational maps do not have unbounded Siegel disks. An unbounded Siegel disk may be created from any function with a linearizable fixed point by arbitrarily restricting the domain of definition, but more interesting examples arise from the study of transcendental entire or meromorphic functions. The possible structures of the boundaries of such unbounded Siegel disks appear to be much more difficult to understand than in the bounded case.

The exponential map well illustrates the topological complications of the unbounded case: let \[f(z) = e^{i2\pi\theta}(e^z-1)\ ,\] where \(\theta=(\sqrt{5}+1)/2\) is the golden mean. This function fixes the origin with multiplier \(e^{i2\pi\theta}\ .\) Its Siegel disk \(\Delta\) is shown in Figure 3.

Figure 4: An enhanced computer drawing of \(\Delta\ ,\) revealing many "strands" to infinity

It was conjectured by Baker that infinity is accessible from \(\Delta\ ;\) that is, there exists a curve \(\gamma\) in the Siegel disk that tends to infinity in one direction. Computer experiments certainly seems to suggest that this is the case, but the question is still open. In what follows, let us suppose that Baker's conjecture is indeed true. Then any iterated preimage of the curve \(\gamma\) will also be a curve to infinity, so the Siegel disk is in fact unbounded "in infinitely many directions". (See Figure 4, and compare Baker and Dominguez, 1999.)

It follows easily that (assuming Baker's conjecture) the boundary \(B\) of \(\Delta\) satisfies neither of the two alternatives from Rogers's theorem for bounded Siegel disks.

Figure 5: An approximation of the Julia set, with the curve \(\beta\) indicated in black.

It also seems plausible that \(-\infty\) is accessible from the the Julia set. More precisely, it is believed that there is an injective curve \(\beta:(0,1]\to\mathbb{C}\) such that \(\lim_{t\to 0}\operatorname{Re}(\beta(t))=-\infty\) and such that every point of \(\beta\) converges to infinity under iteration of \(f\ .\) Compare Figure 5.

If this is true in addition to Baker's conjecture, then the following statement about the boundary can be made: \(B\) is not itself indecomposable, but contains uncountably many indecomposable continua, all pairwise disjoint except for the point at infinity. (See Rempe, 2007.) This illustrates the expected topological complexity of unbounded Siegel disks of entire functions.

Obstructions

We now turn to the question of what happens at the boundary of a Siegel disk \(\Delta\) that prevents it from extending further. This section gives a quick overview, a more thorough discussion being done in the subpage Obstructions.

• A Siegel disk which is not compactly contained in the domain of definition cannot be extended.
• A Siegel disk cannot contain critical points, periodic points besides its center.

Iterating \(f\ ,\) it follows that the Siegel disk cannot contain any preimage of a critical point, of a periodic point (except its center), or any point that eventually gets mapped out of \(\operatorname{Def}(f)\ .\) For a global map (a rational map or a transcendental entire or meromorphic map) the boundary of the Siegel disk is contained in the Julia set and thus every point on the boundary is accumulated by preimages of any point (with a few exceptions, like infinity for a polynomial; see Milnor, 1999, exceptional points).

Knowing this, many questions arise:

• Given a Siegel disk, does its boundary contain a critical point? A periodic point? Is it unbounded?
• Are there periodic points whose orbit stay close to the boundary?
• Can one identify other kinds of obstructions?

The situation is quite complicated and there are lots of results.

Critical points on the boundary. For instance if \(P(z)=e^{2i\pi\theta}z+z^2\) and \(\Delta\) denotes its Siegel disk then it has been proved that there exist \(\theta\) such that \(\Delta\) has a critical point in its boundary, and that there exist \(\theta\) for which there is none. For a general map \(f\ ,\) Graczyk and Świątek (2003) have shown that any bounded Siegel disk whose rotation number is of bounded type must contain a critical point on its boundary. Bounded type numbers form a set of Lebesgue measure 0. But for the particular map \(P(z)=e^{2i\pi\theta}z+z^2\ ,\) Petersen and Zakeri proved that for almost all \(\theta\ ,\) the boundary of \(\Delta\) is a Jordan curve, equal to the closure of the orbit of the critical point. (An extension of this to all polynomials of degree at least 2 has been announced by Zhang Gaofei.) On the other hand, Ghys and Herman proved that there are values of $\theta$ such that the Siegel disk of $P$ is a Jordan domain but its boundary does not contain the critical point.

Unbounded Siegel disks. Consider the exponential map\[E_{\theta}(z) = e^{2i\pi\theta}(\exp(z)-1)\] defined on \(\mathbb{C}\ .\) This function fixes the origin with multiplier \(e^{2i\pi\theta}\ .\) It has been proved by Herman that for almost all rotation numbers (this includes all bounded type numbers), the Siegel disk of \(E_{\theta}\) is unbounded. Thus infinity belongs to its boundary. It is an essential singularity of \(E_{\theta}\ ,\) that sort of plays the same role as a critical point. There are pathes going to infinity whose image by \(E_{\theta}\) tends to the omitted value \(s=-e^{2i\pi\theta}\ .\) The complex number \(s\) is called an asymptotic value, and Geyer, Buff and Fagella proved that if the Siegel disk of \(E_{\theta}\) is unbounded, then \(s\) belongs to its boundary, even though it is not known yet if there exists or not a path going to infinity within the Siegel disk.

Periodic points. Periodic points on the boundary seem less likely to exist:

Theorem: (Rogers) If the boundary \(B\) of a Siegel disk of a polynomial of degree \(d\geq 2\) contains a periodic point, then \(B\) is an indecomposable continuum.

There is no polynomial for which it is known to occur, as mentioned in the section on bounded Siegel disks. Despite this, periodic points still seem to be closely related to dynamical obstruction. Indeed, if \(f\) is a rational (or transcendental entire/meromorphic) function, \(B\) is contained in the Julia set of \(f\ ,\) which equals the closure of the set of repelling periodic points of \(f\ .\)

Small cycles. Yoccoz has shown that for quadratic polynomials, in the non-linearizable case, there are small cycles: cycles of arbitrarily high period completely contained in arbitrarily small neighborhoods of the fixed point. Does this adapt to other families? to Siegel disks? These questions are far from being settled. In the non-polynomial case, Perez-Marco has constructed examples of non-linearizable fixed points, and of Siegel disks, without small cycles.

Boundaries of Siegel disks and the postcritical/postsingular set.

If \(f\) is a entire or meromorphic function (this includes polynomials and rational maps), then the set \(S(f)\) of singular values is a generalization of the set of critical values of a polynomial, and plays a similar role in dynamics. For such functions the boundary of the Siegel disk is always closely related to the behavior of singular values. Let the postsingular set of \(f\) be the closure of the union of the orbits of all singular values. For a polynomial or a rational map, the singular set is the set of critical values and the postsingular set is also called the postcritical set. The following theorem is due to Fatou (although he did not state it for transcendental functions).

Theorem: Let \(f\) be a rational map or a transcendental entire or meromorphic function, and suppose that \(f\) has a Siegel disk. Then its boundary is contained in the postsingular set of \(f\ .\)

For rational maps there is a stronger theorem by Mañé: there exists a recurrent critical point (recurrent means that there is a sequence of iterates of \(c\) tending to \(c\)) such that the boundary of the Siegel disk is contained in the closure of its orbit.

Hausdorff dimension

Regarding the geometry of Siegel disk boundaries, there is for instance the following result:

Theorem. (Graczyk and Jones, 2002) If \(\Delta\) is a bounded Siegel disk of a holomorphic function \(f\) whose boundary is a quasicircle and contains at least one critical point, then the rotation number of \(\Delta\) is of bounded type, and the Hausdorff dimension of its boundary is strictly greater than one (and strictly less than two).

On the other hand, Perez-Marco (unpublished) has constructed a bounded Siegel disk whose boundary has Hausdorff dimension equal to two. He also provided the first examples where the Hausdorff dimension is one (see the next section).

Smooth Siegel disks

If the boundary \(B\) of a bounded Siegel disk is a Jordan curve containing a critical point, then this curve cannot be differentiable. However, when there is no critical point, then there is a priori no reason why \(B\) could not be smooth - and indeed such examples exist! (Here smooth means that it is parametrized by a \(C^{\infty}\) function whose derivative vanishes nowhere.)

The first examples of this kind were constructed by Perez-Marco (unpublished). Later, Buff and Chéritat proved that there are quadratic polynomials having Siegel disks with smooth boundaries. This, coincidentally, gives a proof of the existence of Siegel disks whose boundary does not contain the critical point, independent of that of Ghys and Herman. In fact, smooth Siegel disks appear in most families of analytic maps (Avila and Buff, unpublished).

Quadratic Siegel disks

Siegel disks of quadratic polynomials are covered in a separate subpage: Quadratic Siegel disks. This family of maps, for which much more is known, may seem particular but in fact it has universal properties and its Siegel disks are thought to be typical.

Open problems

Here is a selection of open problems, as of 2010.

• What is the exact arithmetic condition for the presence of a critical point in the boundary of the quadratic Siegel disks? Is it Herman numbers? Same question for higher degree polynomials, rational maps, entire maps, etc...
• Is the boundary of a quadratic Siegel disk approximable by cycles of the polynomial? Same question for higher degree polynomials, rational maps, entire maps, etc...
• Is Brjuno's condition a necessary condition for linearizability of higher degree polynomial? Of rational maps? Entire maps, etc...

Beyond

We will briefly mention a few related topics.

Herman rings

Figure 6: Example of Herman ring of a rational map, and the Julia set thereof.

A Herman ring is a maximal annulus shaped region on which a map is conjugated to a rotation. For a rational map, a Herman ring is a Fatou component. There can be no Herman ring for polynomials, because their Fatou component are simply connected (apart the component containing \(\infty\)). The theory of Herman rings has ties with that of circle maps. Shishikura has shown that Herman rings correspond very closely to Siegel disks: one can transform a map with a Siegel disk into a map with a Herman ring and vice-versa (via the technique of Holomorphic surgery).

Hedgehogs

Hedgehogs are a generalization of Siegel disks, invented by Ricardo Perez-Marco. For a given holomorphic map \(f\) with an irrationaly indifferent fixed point \(a\ ,\) he proved in particular that there exists, close to \(a\ ,\) connected subsets containing \(a\ ,\) that are forward and backward invariant by the local branch of \(f\ .\) These are the hedgehogs. They form a nested family of compact sets containing \(a\ .\) In some sense, f behaves like a rotation on each hedgehog.

(In fact there is a hedgehog for any open subset U on which \(f\) and \(f^{-1}\) are injective and extend analytically to the boundary. If moreover the boundary of U is a smooth curve then this set is unique and equal to the following set: the connected component containing the fixed point, of the closure of the set of points that never escape from the closure of U under forward and backward iteration of \(f\ .\))

In the non-linearizable case, a hedgehog has a complicated shape, quite hairy. It is never locally connected. In the linearizable case a hedgehog can be either a Siegel disk, or the union of a Siegel disk and some hair. These objects are still mysterious. There is yet no computer picture of hedgehogs, but some of them probably are homeomorphic to Cantor bouquets (the latter are objects that naturally appear in the dynamics of exponential maps, see Devaney, 1999).

Links with circle maps

After the works of Perez-Marco, the hedgehogs allow to transfer results on fixed points to results on analytic circle diffeomorphisms and conversely. However, subtle differences arise. For instance, the set of rotation numbers so that any analytic circle diffeomorphism with that rotation number is analytically linearizable (conjugate to a rotation by an analytic map in a neighborhood of the circle) is different from the set of rotation numbers such that any fixed point is linearizable. The first is the set of Herman numbers, a strict subset of the second which is the set of Brjuno numbers.

Several dimensions

In more than one complex variable, the differential at a fixed point is a linear endomorphism \(L\) of a dimension \(d\) vector space over \(\mathbb{C}\ .\) One analog in dimension $n$ to the one dimensional Siegel disk situation is the following: when \(L\) is diagonalizable with eigenvalues $\lambda_1, \ldots, \lambda_n$ and there is a subset of the form \(\lambda_1=e^{2i\pi\theta_1}, \ldots, \lambda_s=e^{2i\pi\theta_s}\) for some real numbers \(\theta_1, \ldots, \theta_s\ ;\) moreover these eigenvalues are multiplicatively independent (i.e. the collection \((1,\theta_1,\ldots,\theta_s)\) is linearly independent over \(\Z\)). Then \(L\) restricted to the subspace $E$ generated by the first $s$ eigenvectors is equivalent to \((z_1,\ldots,z_s)\mapsto(\lambda_1 z_1,\ldots \lambda_s z_s)\ ,\) which are just \(s\) independent rotations. Moreover, the orbit of most point of $E$ is a dense subset of an \(s\)-dimensional torus.

The question is then: does there exist an invariant manifold $M$ with tangent space $E$ at the fixed point, and on which the dynamics is conjugated to the action of $L$ on $E$? Such is called a linearization manifold, and also a linearization domain if $s=n$.

Almost nothing is known on linearization domains/manifolds in higher dimension. Their boundaries are complete mysteries. Even the correct optimal arithmetical condition on \((\lambda_1, \ldots, \lambda_n)\) is not known, for which all fixed points with these eigenvalues have linearization manifolds. There is an analog of the Brjuno condition which is sufficient (see Brjuno, 1965 for the case $s=n$, Pöschel, 1986 for the general case), but there are also examples of \((\lambda_1, \ldots, \lambda_n)\) not satisfying this analog, yet for which all fixed points are linearizable.

References

References cited here

• Brjuno, A.D. [1965]: On convergence of transforms of differential equations to the normal form, (Russian) Dokl. Akad. Nauk SSSR 165 987–989.

• Chéritat, A. [2011]: Compactly contained Siegel disks with non-locally connected boundaries, Mathematische Annalen, Volume 349, Issue 3, 529-542.

• Devaney, R. L. [1999]: Cantor bouquets, explosions, and Knaster continua: dynamics of complex exponentials, Publ. Mat. 43, no. 1, 27–54.

• Geyer, L. [2004]: Linearization of saturated polynomials, Preprint.

• Graczyk, J. and Jones, P. [2002]: Dimension of the boundary of quasiconformal Siegel disks, Inventiones mathematicae Volume 148, no. 3, pp. 465-493.

• Graczyk, J. and Świątek, G. [2003]: Siegel disks with critical points in their boundaries, Duke Mathematical Journal Volume 119, no. 1, pp. 189-196.

• McMullen, C. T. [1998]: Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Mathematica Volume 180, no. 2, pp. 247-292.

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• Zakeri, S. [2002]: Old and New on Quadratic Siegel Disks In "Shahyad", a volume dedicated to Siavash Shahshahani on the occasion of his 60th birthday.

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Recommended reading

• Milnor, J. [1999]: Dynamics in one complex variable: Introductory lectures

• Rogers, J.T. [1998] Recent results on the boundaries of Siegel disks in "Progress in Holomorphic Dynamics", Editor: Hartje Kriete, Pitman Research Notes in Mathematics Series (387), pp. 41-49.