Siegel disks/Quadratic Siegel disks

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    The terminology quadratic polynomials refers to polynomials of degree two, and quadratic Siegel disks to the Siegel disks thereof.

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    Inner and Conformal radius of Siegel disks

    Let \(a\) be a linearizable irrationally indifferent fixed point of \(f\ ,\) and let \(\Delta\) be the Siegel disk around \(a\ .\) Then the inner radius of \(\Delta\) is the radius of the largest disk centered at \(a\) that is contained in \(\Delta\ .\)

    Furthermore, let \(\psi:\Delta\to D\) be a conjugacy to a rotation on a round disk \(D\ .\) This conjugacy (and the disk \(D\)) is unique if we require that \(\psi'(a)=1\ .\) The conformal radius of the Siegel disk is defined to be the radius of \(D\ .\) (There exists an interpretation of the conformal radius in terms of electrostatic capacity, and also in terms of the hyperbolic metric.)

    The notions of inner radius and conformal radius are closely related. Indeed, using Schwarz's inequality on the left and the Koebe quarter theorem on the right, we have \[\text{inner radius} < \text{conformal radius} < 4\cdot \text{inner radius}.\]

    For bounded Siegel disks, and also for unbounded Siegel disks of transcendental entire or meromorphic functions, 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 \(\phi\) is the inverse function of \(\psi\ .\) The coefficients of this expansion can be inductively computed explicitly. However, the radius of convergence is not easily understood from these formulae.

    Yoccoz proved that if \(f\) is injective on a disk of radius \(r\) centered on \(a\) and if the rotation number satisfies Brjuno's condition, then the inner radius of D is greater than \(r/C\exp(B)\) for some universal constant \(C\ ,\) where \(B\) is the Brjuno sum of the rotation number; that is, the sum in the definition of the Brjuno condition; see also Linearization. Yoccoz also proved that this bound is optimal in the following sense: for every Brjuno rotation \(\alpha\ ,\) there is a function \(f\) with a fixed point \(a\) of rotation number \(\alpha\) such that \(f\) is injective in the disk of radius \(r\) around \(a\) and such that the Siegel disk has inner radius less than \(rC'/\exp(B)\) for some universal constant \(C'\ .\) Buff and Chéritat proved later that one can take \(f=e^{2\pi i \alpha} z + z^2\) (see below).

    Quadratic Siegel disks

    Up to an affine change of coordinate \(z\mapsto az+b\ ,\) every quadratic polynomial with an irrationally indifferent fixed point can be written as \[ P_{\alpha}(z) = e^{i2\pi\alpha} z + z^2,\] for some irrational \(\alpha\in\R\ .\)

    Universality

    Yoccoz proved that the quadratic family is universal in the following sense. If \(\alpha\) is a rotation number for which there is any holomorphic map with a fixed point of multiplier \(\rho=\exp(i2\pi\alpha)\) that is not linearizable, then \(P_{\alpha}\) is not linearizable either.

    Quasiconformal models

    Figure 1: Illustration of the quasiconformal model

    When \(\alpha\) 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 \(P_{\alpha}(z)\) such that \(\phi\) conjugates the map \(B_\tau\) to \(P_{\alpha}\ ,\) where \(B_\tau(z)=e^{i2\pi\tau} z^2 \frac{z-3}{1-3z}\ .\) (This is an example of the technique called quasiconformal surgery).

    The preceding result implies, in particular, that any bounded-type quadratic Siegel disk is bounded by a quasicircle containing the critical point. As mentioned in the main article, this fact has since been generalized to the class of all rational functions.

    The quasiconformal model was used by Petersen to prove that the Julia set of \(P_{\alpha}\) is locally connected for \(\alpha\) of bounded type. It is also at the heart of the following results of McMullen (1998) (whose proof uses some additional elaborate techniques such as renormalization).

    Figure 2: Asymptotic self-similarity at the critical point (on this example, the rotation number is equal to the golden mean)

    Theorem. Suppose that \(\alpha\) is of bounded type. Then

    • The Julia set of \(P_{\alpha}\) has Hausdorff dimension strictly less than two. Hence so does the boundary of its Siegel.
    • In the disk of radius \(\varepsilon\) around the origin, the proportion of the area taken by the basin of infinity tends to zero as \(\varepsilon\to 0\ .\)

    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 of this self-similarity is a universal constant for many Siegel disks with the same rotation number.

    The set of bounded type irrationals has a Lebesgue measure equal to 0. Petersen and Zakeri (2004) were able to generalize the surgery that led to the quasiconformal model to a set of rotation numbers of full measure. The conjugacy is not quasiconformal anymore, yet some properties remain:

    Theorem. Suppose that \(\alpha\) has continued fraction expansion \(\alpha=a_0+1/(a_1+1/\cdots)\) where the entries \(a_n\) satisfy the following inequality\[\log a_n\leq C \sqrt{n}\] for some constant \(C\ .\) Then

    • The Siegel disk of \(P_{\alpha}\) is bounded by a Jordan curve that contains the critical point
    • The Julia set of \(P_{\alpha}\) has zero area.

    Conformal radius

    Figure 3: The graph of the function \(\Upsilon\)

    If \(P_{\alpha}\) is linearizable, let \(r_{\alpha}\) denote the conformal radius (as defined above) of the Siegel disk; otherwise, set \(r_{\alpha}=0\ .\) Also let \(Y(\alpha)\) denote Yoccoz's variant of the Brjuno sum, defined as follows: \[Y(\alpha)=\sum_{n=0}^{\infty} \alpha_0\cdots \alpha_{n-1} \log \frac{1}{\alpha_n},\] where \(\alpha_0\) is the fractional part of \(\alpha\) and \(\alpha_{n+1}\) is the fractional part of \(\alpha_n\ .\) This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant. The following theorem was proved by Buff and Chéritat (2006) and describes the dependence of \(r_{\alpha}\) on \(\alpha\ .\)

    Theorem (Buff and Cheritat): The function \(\Upsilon(\alpha) = \log r(\alpha) + Y(\alpha)\) (defined on the set of Brjuno numbers) is the restriction of a continuous function on \(\R\ .\) In particular it is bounded.

    This theorem gives an alternative proof of Yoccoz's result that \(P_{\alpha}\) has a Siegel disk if and only if \(\alpha\) is a Brjuno number.

    The boundedness of \(\Upsilon\) was conjectured and partially proved by Yoccoz. Its 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.

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

    Also note that \(r(\alpha)\) being small does not necessarily mean that the diameter \(D\) of the Siegel disk is small. Indeed, it does happen for some values of \(\alpha\) that the conformal radius (and hence the inner radius) is small, but \(D\) is not. We also remark that, even if the inner radius is small, it may still be possible to fit a large round disk (not centered at the fixed point) inside the Siegel disk; in fact this case can also occur. On the other hand, it is known that there are values of \(\alpha\) such that D is arbitrarily small.

    Boundaries of quadratic Siegel disks

    The boundary of a quadratic Siegel disk can have any prescribed regularity (Buff and Cheritat, 2007). In particular:

    Theorem. For every \(n\ ,\) there are quadratic Siegel disks whose boundary

    • is a Jordan curve not containing the critical point and
    • is a \(C^n\) curve but not a \(C^{n+1}\) curve.

    There are also quadratic Siegel disks whose boundary is a Jordan curve not containing the critical point, yet sufficiently irregular not to be a quasicircle.

    Digitation

    Figure 4: A Siegel disk getting infolded

    This phenomenon will be only informally described here: take a rotation number \(\alpha=a_0+1/(a_1+1/\ddots)\) and consider the Siegel disk Δ of \(P(z)=e^{i2\pi\alpha} z+z ^2\ .\) Replace one of the entries \(a_n\) of the continued fraction of \(\alpha\) by a much bigger integer N. This gives a new irrational \(\alpha'\). Then, the Siegel disk Δ' of \(P(z)=e^{i2\pi\alpha'} z+z ^2\) looks much like Δ but 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.

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