Obstructions

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Author: Prof. 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

In this subpage, the focus is on the following question: What happens at the boundary of a Siegel disk $\Delta$ that prevents it from extending further?

Notation: $\partial\Delta$ denotes the boundary of $\Delta$ .

The Siegel disk has to be contained in the domain of definition $\operatorname{Def}(f)$ of $f$ so the most obvious obstruction is that its boundary meets the boundary of $\operatorname{Def}(f)$ . A more general and intrinsically defined obstruction is that $\Delta$ be unbounded. (In the sense defined above: it is not contained in a compact subset of $\operatorname{Def}(f)$ .)
The Siegel disk cannot contain a critical point (nor any preimage thereof). Indeed, the conjugacy to a rotation implies $f$ is injective on $\Delta$ .
The Siegel disk cannot either contain a periodic point either. Indeed, $f$ is conjugated to an irrational rotation on $\Delta$ .

Iterating $f$ , it follows that the Siegel disk cannot contain any preimage of a critical point, of a periodic point, or any point that eventually gets mapped out of $\operatorname{Def}(f)$ .

Given a Siegel disk, does its boundary contains a critical point? A periodic point? Is it unbounded?

Critical points on the boundary. For example

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).

(Petersen, Zakeri) For almost all $\theta$ , the boundary of the Siegel disk of $P(z)=e^{2i\pi\theta}z+z^2$ is a Jordan curve, equal to the closure of the orbit of the critical point.

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.

Function-theoretic obstructions. Having a critical point on the boundary and being unbounded are embraced by a more general condition: we say that there is a function-theoretic obstruction if there is no simply connected domain $U$ such that both $f$ and $f^{-1}$ are defined on $U$ and such that the closure $\overline{\Delta}$ of the Siegel disk is a compact subset of $U$ . (This language does not seem to appear in the literature, but it is convenient for discussing the questions that arise in this section.) Loosely speaking, the other obstructions must be of dynamical nature.

Lemma. If the Siegel disk $\Delta$ of a map $f$ is bounded, if $f$ is injective on $\partial\Delta$ and if $\partial\Delta$ contains no critical point of $f$ then there is no function-theoretic obstruction.

It is a fundamental theorem that for some rotation numbers, there must be a function-theoretic obstruction:

Theorem. (Herman) There exists a subset $\mathcal{H}$ of the Brjuno numbers, containing all diophantine numbers, with the following property. If a holomorphic function has a Siegel disk $\Delta$ whose rotation number belongs to $\mathcal{H}$ , then $\Delta$ has a function-theoretic obstruction.

( $\mathcal{H}$ is called the Herman class and is defined in terms of linearizability of analytic circle diffeomorphism. Yoccoz later gave an arithmetical characterization of $\mathcal{H}$ in terms of continued fractions. It is a complicated one.)

It is unknown whether there are any polynomials with a Siegel disk whose rotation number is not in $\mathcal{H}$ but with a function-theoretic obstruction (for instance with a critical point on the boundary).

Unicritical polynomials. Injectivity on the boundary of Siegel disks appears to be a difficult problem in general. However, Herman observed that in some cases it is easy to verify, namely when there is only one critical value.

Lemma. Suppose that $f$ is a unicritical polynomial, i.e. $f(z)=z^d+c$ for some $c\in\mathbb{C}$ , and that $f$ has a Siegel disk $\Delta$ . If the critical point $0$ does not belong to $\partial\Delta$ , then $f$ is injective on ${\partial\Delta}$ .

Corollary. Suppose that $f$ is a unicritical polynomial with a Siegel disk $\Delta$ whose rotation number belongs to the Herman class. Then the critical point $0$ belongs to $\partial\Delta$ .

It is unknown whether the above lemma is true for all polynomials.

Siegel disks and singular values of entire functions. If $f$ is a transcendental entire functions, 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. Here $S(f)$ is the smallest closed set such that $f$ is a covering map over $\mathbb{C}\setminus S(f)$ . It can be proved that $S(f)$ is the closure of the set of all critical values and asymptotic values of $f$ .

It is sensible to ask when the boundary of a Siegel disk of an entire functions contains a singular value of $f$ . This is subtle because, as opposed to the polynomial or rational cases, the Siegel disk may be unbounded. Below is given a small sample of recent results, for the simplest transcendental map: the exponential. Let

$E_{\theta}(z) = e^{2i\pi\theta}(\exp(z)-1).$

This function fixes the origin with multiplier $e^{2i\pi\theta}$ and has only one singular value, namely the omitted value $s=-e^{2i\pi\theta}$ .

Here it was already observed by Herman that, if $E_{\theta}$ has a bounded Siegel disk $\Delta_{\theta}$ , then $E_{\theta}$ is injective on $\partial \Delta_{\theta}$ (for the same reason as for unicritical polynomial). Since $E_{\theta}$ has no critical points, it follows from Herman's theorem above that $\Delta_{\theta}$ is unbounded when $\theta\in\mathcal{H}$ .

However, this leaves open the following problem: if $\Delta_{\theta}$ is unbounded, is necessarily $s\in\partial \Delta_{\theta}$ ? This question was originally posed by Herman, Baker and Rippon (Brannan and Hayman 1989, problem 2.86). Rippon (1994) showed that this is true for a full-measure set of $\theta$ , and the question was fully answered positively by Rempe (2004) and independently by Buff and Fagella (unpublished).

Theorem. (Rempe, 2004; Buff and Fagella)If the Siegel disk $\Delta_{\theta}$ of an exponential map $E_{\theta}$ is unbounded, then $s\in\partial\Delta_{\theta}$ .

Generalizations of this theorem can be obtained for more general entire functions with several singular values, but one encounters problems similar to the injectivity question for bounded Siegel disks mentioned above. Compare Rempe, 2008.

Periodic points and small cycles.

Theorem. (Perez-Marco, 1997) If $f$ has a Siegel disk $\Delta$ and there is no function-theoretic obstruction, then $\partial\Delta$ contains no periodic points of $f$ .

Theorem: (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.

By Perez Marco's theorem such a Siegel disk has function-theoretic obstruction. 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).

Despite the above theorem, periodic points still seem to be closely related to dynamical obstruction. Indeed, if $f$ is a rational (or transcendental entire/meromorphic) function, $\partial\Delta$ is contained in the Julia set of $f$ , which equals the closure of the set of repelling periodic points of $f$ .

One may ask whether there are periodic cycles whose entire orbit remains close to $\partial\Delta$ , or whether an even stronger property holds: that $\partial\Delta$ is the limit of a sequence of peridodic orbits. 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. On the other hand 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. As discussed above, the boundary of a Siegel disk need not, in general, contain a critical point (or, more generally, a singular value) of the function $f$ . Nonetheless, for globally defined 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 $\Delta$ . Then $\partial\Delta$ is contained in the postsingular set of $f$ .

For rational functions, Mañe has made this statement more precise: $\partial\Delta$ is contained in the closure of the orbit of a recurrent critical point. For entire functions, this is not known in such generality, although certain extensions of Mañe's theorem have been considered by various authors. In particular, if an exponential map $E_{\theta}$ as above has a Siegel disk $\Delta$ , then the singular value $s$ is recurrent (see Rempe and van Strien 2008).

Invited by:

Prof. James Meiss, Applied Mathematics University of Colorado

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