Siegel disks

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Author: Dr. Xavier Buff, Laboratoire E. Picard, Université Paul Sabatier
Author: Dr. Arnaud Chéritat, Laboratoire E. Picard, Université Paul Sabatier
Author: Dr. Lasse Rempe, Department of Mathematical Sciences, University of Liverpool

Dr. Xavier Buff accepted the invitation on 24 November 2007 (self-imposed deadline: 24 May 2008).

Dr. Arnaud Chéritat accepted the invitation on 24 November 2007 (self-imposed deadline: 9 July 2008).

Work In Progess

Figure 1: The quadratic golden mean Siegel disk

Figure 1: The quadratic golden mean Siegel disk

Siegel disks are maximal domains of conjugacy to a rotation, of a discrete holomorphic dynamical system, in complex dimension one.

Example

Take 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 2πω and $\omega=\frac{\sqrt{5}+1}{2}$ denotes the golden mean. The origin z=0 is a linearizable fixed point, and the maximal linearization domain looks like the picture on the right. This particular example is known to be bounded by a fractal Jordan curve, whose Hausdorff dimension lies strictly between 1 and 2.

The linearization problem

Siegel disks are closely related to the linearization problem and small divisors, so it is probably useful to mention a few results concerning these topics. However, more detail can be found on the following subpage of the present article: /Linearization.

Statement, first properties

Given a fixed point of a dynamical system, the linearization problem consists in finding a change of coordinates for which the dynamical system becomes linear. The change of coordinate, also called a conjugacy, only has to be defined and work in a neighborhood of the fixed point. Moreover, for a dynamical system f that is regular, in the sense that it is differentiable, one tries to conjugate it to its differential D_af at the fixed point a. In this article, the dynamical system is a map from (part of) a complex dimension 1 manifold to itself. For instance, $z\mapsto z^2+c$ from $\mathbb{C}$ to $\mathbb{C}$ , where $c\in\mathbb{C}$ is any given parameter.

Topologically linearizable $\iff$ holomorphically linearizable
In the particular setting of a holomorphic map of a complex dimension 1 manifold (i.e. a Riemann surface) linearizability by a continuous conjugacy turns out to be equivalent to linearizability by a holomorphic* conjugacy. Any regularity in between is thus also equivalent.

*recall that a holomorphic map (complex-differentiable) is the same thing as an analytic map (power series expandable near every point)

The multiplier and its influence on linearizability
If f is a holomorphic map and a is a fixed point, i.e. f(a)=a, then the multiplier is the complex number λ=f'(a). The multiplier is invariant under conjugacy.

If |λ| = 0 (superattracting fixed point), then f is not linearizable, unless it is constant in a neighborhood of a.
If 0 < |λ| < 1 (attracting not superattracting), or 1 < |λ| (repelling), then a is a linearizable fixed point. This is referred to as Koenig's theorem.
If |λ| = 1 (indifferent), then it depends. Write λ = exp(i2πθ) for some .
- If $\theta\in\mathbb{Q}$ (parabolic fixed point), then f is not linearizable most of times. More precisely, it will be linearizable if and only if f has an iterate equal to the identity.
- If $\theta\notin\mathbb{Q}$ (irrationally indifferent), then we get into a much more difficult question.

The latter case is where Siegel disks arise.

Brjuno's arithmetic condition

Here, $\theta$ refers to an irrational real number: $\theta\in\mathbb{R}\setminus\mathbb{Q}$

Definition: Let $p_n/q_n$ be the sequence of continued fraction convergents of $\theta$ . The number $\theta$ is said to satisfy Brjuno's condition (also called the Brjuno-Rüssmann condition) whenever $\sum_{n=0}^{\infty} \frac{\log q_{n+1}}{q_n} < +\infty$ .

For instance the Diophantine numbers satisfy Brjuno's condition.

Theorem: 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 $e^{2i\pi\theta}$ .

The first part of the previous theorem is due to Brjuno and Rüssmann. Yoccoz gave later an independent proof of it, proved the second part, and moreover specified it as follows:

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

Siegel disks

Consider an irrationally indifferent fixed point $a$ of a holomorphic map $f$ . Assume this fixed point is linearizable. Then there is a maximal open set containing $a$ and on which the map $f$ is conjugated to a rotation. This domain is simply connected, thus homeomorphic to a disk. It is called the Siegel disk of $f$ at $a$ .

Riemann mapping equals linearizer
Let $\lambda=f'(a)$ . By Riemann's mapping or uniformization theorem, there exists a conformal map* $\phi$ from the Siegel disk to either $\mathbb{D}$ or $\mathbb{C}$ , sending $a$ to 0. Then, the conjugate $\phi\circ f\circ \phi^{-1}$ is necessarily the rotation $z\mapsto \lambda z$ , because the only holomorphic bijections of $\mathbb{D}$ that fix the origin are the rotations, and the only holomorphic bijections of $\mathbb{C}$ that fix the origin are linear maps, and the multiplier of the conjugate is the same as that of $f$ .

*conformal map: a bijection that is holomorphic (and whose inverse is therefore holomorphic too)

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

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

Fatou component and Siegel disks
In the case of a rational map* R with a linearizable irrationally indifferent fixed point a, the Siegel disk is equal to the connected component of the Fatou set** of R that contains a. This also holds for entire maps.

*A rational map is a quotient of two polynomials; a polynomial is a special case of rational map.
**The Fatou set is defined as the complement of the Julia set, the latter being the locus of chaos for R.

The dynamics inside a Siegel disk is rather simple (even though a rotation is not as simple as it looks at first). But, what about the boundary?

Boundaries

What does the boundary look like? Where is it? As Written in Zakeri's survey (Zakeri, 2002): "What prevents it from going further"? It is important to realize that even today, the answers are only partly known: many questions remain open.

Not analytic
The boundary cannot be an analytic closed curve contained in the domain of definition of $f$ , otherwise the Siegel disk could be extended beyond by Schwarz's reflection principle. In other words, either the Siegel disk is not compactly contained in the domain of definition of $f$ (it goes to infinity or touches the boundary of the domain of definition) or its boundary is non-analytic at at least one point. In the latter case, it can be shown it will be analytic nowhere. For polynomials, rational maps and entire maps of degree >1, the boundary is nowhere analytic.

Bounded case
A bounded Siegel disk refers to a Siegel disk which is compactly contained in (a.k.a. contained in a compact subset of) the domain of definition of $f$ .

Theorem: (J.T. Rogers) Consider the boundary B of a bounded Siegel disk G. Then either

B is tame*: the conformal map from G to the unit disk has a continuous extension to B.
B is wild*: it is an indecomposable continuum.

An indecomposable continuum is a closed connected set that is not the union of two closed connected proper subsets.
A typical example of the first category of boundaries is any Jordan curve. A typical example of the second category is an object called the pseudocircle (caution: several unrelated mathematical objects share this name). Rogers' theorem applies in fact to a more general class of maps (Rogers, 1995).

*The words "wild" and "tame" are to be taken as adjectives, not as terminology. They are not used in the same context in Rogers' article.

It shall be noted that no examples are known of bounded Siegel disks in the second category (wild). In fact, all examples of bounded Siegel disks for which the category has been determined have in fact a Jordan curve boundary. So it may well be the case for all bounded Siegel disks.

Obstructions

A Siegel disk cannot contain any periodic point other than its center. Indeed, on the Siegel disk, the dynamics is conjugated to an aperiodic rotation.
A Siegel disk cannot contain any critical point or essential singularity of a map. Same reason.

A natural question is then: "Can there be critical points on the boundary of Siegel disks? Singularities? Periodic points?".

Critical points on the boundary

There exist examples of Siegel disks with the critical point on the boundary:

Theorem: Let $\theta$ be a Brjuno number, $P(z)=e^{2i\pi\theta}z+z^2$ and $\Delta$ be its Siegel disk.
(Ghys, Herman) There exist $\theta$ such that $\Delta$ has no critical point in its boundary.
(Douady, Ghys, Herman, Shishikura) There exist $\theta$ such that $\Delta$ has a critical point in its boundary.
In both examples, the Siegel disk is a Jordan curve (even better: it is a quasicircle).

Complement: There is a set of numbers, introduced by Herman and characterized by Yoccoz. These are called Herman numbers an the set thereof is denoted by $\mathcal{H}$ . It is defined in terms of analytic circle diffeomorphisms. Yoccoz gave an arithmetical characterization in terms of the continued fraction. It is a rather complicated one. It contains all bounded type irrationals and even all diophantine irrationals. Herman proved that when $\theta\in\mathcal{H}$ , and $\Delta$ is a Siegel disk with multiplier $e^{2i\pi\theta}$ , and with the supplementary assumption that $\Delta$ is bounded and that $f$ is injective on its boundary $\partial\Delta$ , then there must be a critical point on $\partial\Delta$ . For polynomials, the supplementary assumption is not needed: indeed $\Delta$ is bounded, and Rogers proved that if $f$ is injective on $\partial\Delta$ , then $\partial\Delta$ contains a critical point. Also, when the rotation number has bounded type, the injectivity assumption is not needed (Graczyk and Świątek, 2003).

It is not known whether there exists a polynomial Siegel disk with a critical point on its boundary but with $\theta$ not in $\mathcal{H}$ .

Periodic points on the boundary?

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

However, no examples of such polynomials are known. It may even be the case that it never happens (see the discussion on bounded Siegel disks).

Singularities on the boundary

To be completed Exponentials, and theorems

Accumulation

Recall that a Siegel disk of a rational map R is necessarily a connected component of the Fatou set F(R). Its boundary is thus a subset of the Julia set J(R). The latter is the closure of the set of repelling periodic points (Fatou and Julia). Therefore, every point in the boundary of the Siegel disk is accumulated by repelling periodic points.

Theorem (Fatou) The boundary of the Siegel disk is contained in the closure of the orbit of the critical points.

Mañe has made this statement more precise: there exists at least one critical point which is recurrent and such that the closure of its orbit contains the boundary of the Siegel disk.

To be completed Idem for some transcendental maps.

It has been proved by Yoccoz that a degree 2 polynomial with a non-linearizable indifferent fixed point has small cycles: there are cycles of arbitrarily high period completely contained in arbitrarily small neighborhoods of the fixed point. However, it is not known if an analog holds in the linearizable case near the boundary of the Siegel disk. On the other hand, Perez-Marco has constructed non-polynomial examples of non-linearizable fixed points, and of Siegel disks, without small cycles.

Hausdorff dimension

Theorem (Graczyk and Jones, 2002): If S is a bounded Siegel disk whose boundary is a quasicircle and contains at least one critical point of f then the rotation number of S is of bounded type, and the Hausdorff dimension of its boundary is >1.

To be completedMore?

Smooth Siegel disks

Perez-Marco has constructed the first examples of bounded Siegel disks with smooth boundaries. Here, smooth means that the boundary is a loop which has an injective indefinitely derivable parameterization whose derivative vanishes nowhere. Later, Buff and Chéritat proved that there exist also examples of smooth Siegel disk boundaries in the family of degree 2 polynomials. 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. Smooth Siegel disks in fact appear in most families of analytic maps (Avila and Buff, unpublished).

Inner and Conformal radius

Consider a Siegel disk Δ around the fixed point $a$ of $f$ . The inner radius of D is the radius of the biggest disk centered on $a$ and contained in D. Let ψ be a conjugacy to a rotation. Then ψ is a conformal map from Δ to a round disk D. There exists exactly one conjugacy such that ψ'(0)=1. The conformal radius of the Siegel disk (understood at its fixed point) is the radius of the round disk D, image of Δ by this unique ψ. These 2 notions are somehow close: indeed inner-radius < conformal-radius < 4×inner-radius (this is a consequence of Koebe's 1/4 theorem)

There exists an interpretation of the conformal radius in terms of electrostatic capacity, and also in terms of the hyperbolic metric on Riemann surfaces.

In good cases, the conformal radius coincides with the radius of convergence of the power series expansion $\phi(z)=a+z+a_2 z^2 + a_3 z^3+\ldots$ , where φ is the inverse function of ψ. The good cases include all polynomials. The coefficients of this expansion can be inductively explicitely computed. However, the radius of convergence is not easily understood from these formulaes.

Yoccoz proved that if $f$ is injective on a disk of radius $r$ centered on $a$ and if the rotation number satisfy Brjuno's condition, which is that its Brjuno sum B is finite (see /Linearization), then the inner radius of D is greater than $r/C\exp(B)$ for some universal constant C. He also proved that this bound is optimal in that there exist for all Brjuno rotation number a function $f$ fixing a point $a$ , injective in a disk of radius $r$ centered on $a$ , and whose Siegel disk has an inner radius $<rC'/\exp(B)$ for some universal constant C'.

Quadratic Siegel disks

The terminology quadratic Siegel disks refers to the Siegel disks of period one of degree 2 polynomials. It amounts to look at $P(z)=\rho z + z^2$ with $\rho=\exp(i2\pi\theta)$ for some irrational $\theta\in\R$ .

Universality: Yoccoz proved that given $\theta$ if there exists a holomorphic map fixing a point with multiplier $\rho=\exp(i2\pi\theta)$ and non linearizable, then the polynomial $\rho z + z^2$ is not linearizable either.

Figure 3: Illustration of the quasiconformal model

Figure 3: Illustration of the quasiconformal model

Quasiconformal model: When $\theta$ has bounded type, it is possible to find some $\tau\in\R$ and a quasiconformal map $\phi$ from the complement of the unit disk to the complement of the Siegel disk of $P(z)=\rho z + z^2$ such that $\phi$ conjugates the map $B_\tau$ to P, where $B_\tau(z)=e^{i2\pi\tau} z^2 \frac{z-3}{1-3z}$ . (This is an example of the technique called quasiconformal surjery).

This implies in particular that the Siegel disk is bounded by a quasicircle, and that this boundary contains the critical point.

This model has been used by C.L. Petersen to prove that the Julia set of P is locally connected.

It is also at the heart of the proof by McMullen (McMullen, 1998), of the following results (together with renormalization and other elaborated techniques).

Figure 4: Asymptotic self-similarity at the critical point (on this example, the rotation number is equal to the golden mean)

Figure 4: Asymptotic self-similarity at the critical point (on this example, the rotation number is equal to the golden mean)

Theorem (McMullen): Consider a degree 2 polynomial with an indifferent fixed point whose rotation number is a bounded type irrational. Then
- The Hausdorff dimension of the Julia set is <2 (in fact it is porous).
- In particular the boundary of such a Siegel disk has Hausdorff dimension <2.
- In the disk of radius ε, the proportion of the area taken by the basin of infinity tends to 0 when ε tends to 0. Moreover, if the rotation number has an enventually periodic continued fraction expansion, then:
- The Siegel disk is asymptotically self-similar at the critical point.
- The scaling ratio is a universal constant for many Siegel disks with the same rotation number.

The set of bounded type irrationnals has a Lebesgue measure equal to 0 (i.e. if you choose all the infinitely many digits randomly, the resulting number will be an unbounded type irrational with probability 1). Petersen and Zakeri were able to generalize the surgery to a set of rotation numbers with full measure (Petersen and Zakeri, 2004). The conjugacy is not quasiconformal anymore, yet some properties remain:

Theorem (Petersen and Zakeri): Consider a degree 2 polynomial with an indifferent fixed point whose rotation number $\theta$ has its continued fraction $\theta=a_0+1/(a_1+1/\cdots)$ whose entries $a_n$ satisfy the following inequality: $\log a_n\leq C \sqrt{n}$ for some constant C. Then:
- The Siegel disk is bounded by a Jordan curve that contains the critical point
- The Lebesgue measure of the Julia set is equal to 0.

Figure 5: The graph of the function $\Upsilon$

Concerning the conformal radius of Siegel disks, we have the following theorem (Buff and Cheritat, 2006) . Let $Y(\theta)$ denote Yoccoz's variant of the Brjuno sum: $Y(\theta)=\sum_{n=0}^{\infty} \beta_{n-1} \log \frac{1}{\alpha_n}< +\infty$ where $\alpha_0$ is the fractional part of $\alpha$ , $\alpha_{n+1}$ is the fractional part of $\alpha_n$ , $\beta_{-1}=1$ and $\beta_n=\alpha_0 \cdots \alpha_n$ . Let $r(\theta)$ be the conformal radius of the Siegel disk of $P(z)=e^{i2\pi\theta}z+z^2$ when the fixed point at the origin is linearizable. Let $r(\theta)=0$ when this fixed point is not linearizable.

Theorem (Buff and Cheritat): The function $\Upsilon(\theta) = \log r(\theta) + Y(\theta)$ is the restriction to the set of Brjuno number of a continuous function over $\R$ . In particular it is bounded.

The boundedness was conjectured and partially proved by Yoccoz. The continuity was conjectured by Marmi after computer experiments. There is still an open conjecture formulated by Marmi Moussa and Yoccoz: that $\Upsilon$ is 1/2-Hölder continuous.

Note The functions $r(\theta)$ and $Y(\theta)$ are highly discontinuous. For instance, $r(\theta)$ tends to 0 at every rationnal, whereas it is positive at every Brjuno number. The function r(\theta) is upper semi-continuous, the function $Y(\theta)$ is lower semi-continuous.

Note When $r(\theta)$ is small, it does not mean necessarily that the diameter D of the Siegel disk is small. In fact, so r(\theta) is close to the shortest distance between the fixed point and the boundary points, that we'll denote $r'(\theta)$ in this paragraph. More precisely one has $r'<r(\theta)<4 r'$ . It does happen for some values of $\theta$ that $r'$ is small but not D. Note that even if r' is small, it may still be possible to fit not-small round disk inside the Siegel disk, and in fact this case does happen too. On the other hand, it is known that there are values of theta such that D is arbitrarily small.

Concerning the boundary, there are in the quadratic family Siegel disks whose boundary can have any prescribed regularity (Buff and Cheritat, 2007). As a corollary:

Theorem (Buff and Cheritat): There exist Siegel disks of quadratic polynomials whose boundary
- is a Jordan curve,
- does --not-- contain the critical point,
- and is a C^n curve but not a C^(n+1) curve.
The same statement also holds with the last property replaced by
- and is not a quasicircle.

Digitation

Figure 6: A Siegel disk getting infolded

Figure 6: A Siegel disk getting infolded

This phenomenon will be only informally described here: take a rotation number $\theta=a_0+1/(a_1+1/\ddots)$ and consider the Siegel disk Δ of $P(z)=e^{i2\pi\theta} z+z ^2$ . Replace one of the entries $a_n$ of the continued fraction of θ by a much bigger integer N. This gives a new irrational θ. Then, the Siegel disk Δ' of $P(z)=e^{i2\pi\theta'} z+z ^2$ looks like Δ, which has been infolded: there are $q_n$ digitations going inward (where $q_n$ is the denominator of the approximant $p_n/q_n=a_0+1/(a_1+1/(\ddots+1/a_n))$ ). As N grows, these digitations go deeper, and tend to the fixed point if N tends to $\infty$ .

It is probably more general than just for quadratic polynomials, but up to now this phenomenon is well controlled only for them.

Exponential Siegel disks

Figure 7: Exponential golden mean Siegel disk, strands invisible...

Figure 7: Exponential golden mean Siegel disk, strands invisible...

Figure 8: ... with the strands revealed

Figure 8: ... with the strands revealed

Open problems

Here is a selection of open problems, as of 2008.

Does there exist a map with a bounded* Siegel disk whose boundary is not a Jordan* curve?
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 poylnomial? 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...
Is the function $\Upsilon$ mentioned above a 1/2-Hölder continuous function?

(*) bounded: contained in a compact subset of the domain of definition of the map, (*) Jordan curve: image of a circle by an injective continuous map (*) quadratic Siegel disk: period one Siegel disk of a quadratic polynomial

Beyond

Well briefly mention a few related topics.

Herman rings

Figure 9: Example of Herman ring of a rational map, and the Julia set thereof.

Figure 9: 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 comonent 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 the Siegel disk, invented by Ricardo Perez-Marco. For a given holomorphic map $f$ with irrationnaly 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 $f$ , bigger than the set consiting in $a$ alone. 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. Each hedgehog is also the set of points not escaping some neighborhood of $a$ .

In fact there is a hedgehog U for any open subset on which $f$ and $f^{-1}$ are injective and extend analytically to the boundary, and it is unique if the boundary of U is a smooth curve, in which case the hedgehog is 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 is a complicated shape, quite hairy. It is never locally connected. In the linearizable case a hedgehog can be either a Siegel disk, a subset of 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.

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 all any analytic circle diffeomorphisms 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

To be completed

References

References cited here

Rogers, J.T. [1992]: Singularities in the boundaries of local Siegel disks, Ergodic Theory Dynam. Systems Volume 12, no. 4, pp. 803-821.

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.

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

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

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

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.

Work In Progess

Invited by:

Prof. James Meiss, Applied Mathematics University of Colorado

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