Siegel disks/Linearization
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Statement
Linearizable at a fixed point \(\implies\) tame
Given a fixed point of a differentiable map, seen as a discrete dynamical system, the linearization problem is the question whether or not the map is locally conjugated to its linear approximation at the fixed point. Since the dynamics of linear maps on finite dimensional real and complex vector spaces is completely understood, the dynamics of a map on a finite dimensional phase space near a linearizable fixed point is tractable.
More precisely the problem is the following: there is a set \(S\ ,\) the phase space, which can be for instance a subset of \(\mathbb{R}^n\) or \(\mathbb{C}^n\) or a manifold, and a map \(f\) from part of \(S\) to part of \(S\ ,\) which represents a discrete dynamical system. We are interested in a fixed point of \(f\ ,\) call it \(a\ .\) The differential of \(f\) at \(a\) is a linear map, call it \(T\ .\) In our example, \(T\) acts respectively on \(\mathbb{R}^n\ ,\) \(\mathbb{C}^n\) and the tangent space of \(S\) at \(a\ .\) Does there exist a neightborhood \(V\) of \(a\) and a homeomorphism \(\phi\) from \(V\) to some neighborhood of the origin such that the local conjugacy (see Topological conjugacy) \(T=\phi\circ f\circ \phi^{-1}\) holds in a (possibly smaller) neighborhood of \(0\ ?\)
Topologically linearizable \(\iff\) holomorphically linearizable
It shall be noted that for a given fixed point of a given map, the answer to this question may or may not depend on the regularity allowed for the conjugacy. However, 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.
The multiplier
If \(f\) is a holomorphic map and \(a\) is a fixed point, i.e. \(f(a)=a\ ,\) then the multiplier is the complex number \(\lambda=f'(a)\ .\) The multiplier is invariant under conjugacy.
Depending on \(\lambda\ ,\) the fixed point \(a\) is termed accordingly:
- for \(|\lambda|>1\ ,\) \(a\) is repelling
- for \(|\lambda|=1\ ,\) \(a\) is indifferent
- for \(0\leq|\lambda|<1\ ,\) \(a\) is attracting
- for \(\lambda=0\ ,\) \(a\) is superattracting
The multiplier, on a Riemann surface
- Let S be a complex dimension 1 manifold (a Riemann surface)
- \(f\) be a holomorphic map from a part of S to a part of S
- \(a\) be a fixed point of \(f\)
- \(T_aS\) be the tangent space of S at \(a\)
- \(D_af: T_aS \to T_aS\) the differential \(f\) at \(a\)
Since we are in dimension 1, \(D_af\) is completely characterized by its unique eigenvalue λ, and equal to the multiplication by λ: \(D_af\)(v) = λv. Identifying \(T_aS\) with the complex plane \(\mathbb{C}\ ,\) \(D_af\) is a similarity of ratio λ. The multiplier is the number λ.
Schröder's equation
Expressed in a chart where the fixed point is at the origin of $\mathbb C$, $f$ is a map from a neighborhood of $0$ in $\mathbb C$ to a possibly different neighborhood of $0$.
Linearizability is equivalent to the existence of a solution $\psi$ of the equation
$$\psi\circ f = \lambda \psi,$$
which is sometimes called Schröder's equation, such that $\psi$ is a holomorphic function defined near $0$, $\psi(0)=0$ and $\psi'(0)\neq 0$.
This is equivalent to finding a solution to the following equation for $\phi=\phi^{-1}$: $$f\circ\phi(z) = \phi(\lambda z)$$ near $0$. The latter equation does not seem to bear someone's name.
Linearizability, depending on the multiplier
- 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 \(\theta\in\mathbb{R}\ .\)
- 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, which is impossible for instance in the case of a rational map of degree at least 2 (this includes polynomials) and for entire maps that are not of the form \(z\mapsto az+b\).
- If \(\theta\notin\mathbb{Q}\) (irrationally indifferent), then we get into a much more difficult question.
The latter case is where Siegel disks arise.
Power series expansions and small divisors
Assume \(f\) fixes the origin (take a chart where the fixed point is at the origin) and consider the power series expansions \[f(z)=\lambda z +\sum_{n=2}^{+\infty} a_n z^n\ .\] The linearization equation consists in finding \[\phi(z)=z+\sum_{n=2}^{+\infty} b_n z^n\] such that \(\phi^{-1} \circ f \circ \phi (z) = \lambda z\) holds near the origin (a problem equivalent to finding \(\psi\) such that \(\psi \circ f \circ \psi^{-1} (z) = \lambda z\)). In other words, \(f\circ \phi(z) = \phi(\lambda z)\ .\) By indentifying power series expansions, one finds a unique solution defined by the recurrence relation on the coefficients \(b_n\) of \(\phi\ :\) \[b_1=1\] \[b_{n+1}=\frac{P_n(a_2,\ldots,a_{n+1},b_2,\ldots,b_{n})}{\lambda^{n+1}-\lambda}\] where \(P_n\) is an explicit, yet complicated, mutivariate polynomial.
Thus for \(\lambda=\exp(i 2\pi\theta)\) with irrational \(\theta\ ,\) the conjugating power series \(\phi\) is always defined as a formal power series. Linearizability of \(f\) is equivalent to the convergence of this series, i.e. to its convergence radius to be positive. Even though the numerator in the recurrence relation giving \(b_{n+1}\) is much more complicated than the denominator, it is the latter which is the potential source of divergence. The term \(\lambda^{n+1}-\lambda\) is called a small divisor. Indeed, for some values of n, typically for n=q where \(p/q\) is a continued fraction rational approximant of \(\theta\ ,\) the quantity \(\lambda^{n+1}-\lambda\) is small.
Estimating the growth rate of \(b_n\) is thus a subtle problem. It requires a good understanding of rational approximations of irrationals.
A reminder on continued fractions
An irrational has a unique continued fraction expansion \(\theta=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots}}\) with \(a_0\in\mathbb{Z}\) and \(a_n\in\mathbb{N}\ .\)
The continued fraction approximants of \(\theta\) are the numbers \(\frac{p_n}{q_n}=a_0+\cfrac{1}{\ddots+\cfrac{1}{a_n}}\) (the notations may vary).
The best rational approximations of an irrational are given by its continued fraction approximants:
- if \(|\theta-p/q|<\frac{1}{q^2\sqrt{5}}\) then p/q is an approximant of \(\theta\)
- if \(p/q\) is an approximant then \(|\theta-p/q|<\frac{1}{q^2}\ .\)
The quantity \(q^2|\theta-p/q|\) can be thought of a measure of the quality of the rational approximation of \(\theta\) by \(p/q\ .\)
The second point can be made more precise: if \(p_n/q_n\) is the n-th approximant then we have:Concerning the small divisors, with \(\lambda=\exp(2i\pi\theta)\ ,\) the quantity \(|\lambda^{q+1}-\lambda|\) is comparable to \(q|\theta-p/q|\ ,\) where p is the integer so that p/q is closest to \(\theta\ .\) More precisely, there is the following theorem: the smallest value of \(|\lambda^k-\lambda|\) for k ranging from 2 to \(1+q_n\) is obtained precisely at \(k=1+q_n\ ,\) and we have for \(k=1+q_n\ :\)
History
Linearizability is closely related to stability.
Poincaré, in studying the stability of the solar system, had to face similar questions. He thought he could prove stability in the simplified problem he was looking at (1889). He latter realized he was wrong, and by correcting this famous mistake opened the field of chaotic behaviour in dynamical systems.
Concerning the center problem (linearization of an irrationnaly indifferent fixed point of a discrete dynamical system in complex dimension 1):
At the International Congress in 1912, E. Kasner conjectured that such a linearization is always possible. Five years later, G. A. Pfeiffer disproved this conjecture by giving a rather complicated description of certain holomorphic functions for which no local linearization is possible. In 1919 Julia claimed to settle the question completely for rational functions of degree two or more by showing that such a linearization is never possible; however, his proof was wrong. H. Cremer put the situation in much clearer perspective in 1927 with a result [...]
—John Milnor, Dynamics in one complex variable (second edition, 2000)
Cremer's argument is indeed simple. It uses irrationnal rotation numbers \(\theta\) which are well approximated by rationnals. A non-linearizable irrationaly indifferent fixed point is nowadays called a Cremer point.
Siegel was the first to be able to prove, in the 1940s, that linearizability does occur. In fact he showed that if the rotation number is Diophantine, then the fixed point is always linearizable.
Then there remained to determine the exact set of values of \(\theta\) for which f is always linearizable. Brjuno and Rüssman found the exact arithmetic condition, but could only prove it is sufficient. This condition is now called the Brjuno condition.
Yoccoz proved the necessity of the condition, i.e. that for an irrationnal \(\theta\) not satisfying the Brjuno condition, there exists at least one non linearizable example with this rotation number \(\theta\ .\) He even proved that the degree 2 polynomial \(f(z)=\exp(2i\pi\theta) z + z^2\) is such an example.
Results
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 \(\sum_{n=0}^{\infty} \frac{\log q_{n+1}}{q_n} < +\infty\ .\)
There are several other equivalent definitions of Brjuno's condition:
- \(\sum_{k=0}^{\infty} \frac{1}{2^k}\log\left(\sup_{2^k\leq n< 2^{k+1}} \frac{1}{|\lambda^n-1|}\right) < +\infty\) where \(\lambda=e^{2i\pi\theta}\)
- \(\sum_{n=0}^{\infty} \beta_{n-1} \log \frac{1}{\alpha_n}< +\infty\) where \(\alpha_0\) is the fractional part of \(\theta\ ,\) \(\alpha_{n+1}\) is the fractional part of \(\alpha_n\ ,\) \(\beta_{-1}=1\) and \(\beta_n=\alpha_0 \cdots \alpha_n\)
For instance the Diophantine numbers satisfy Brjuno's condition. (An irrational number \(\theta\) is Diophantine if there exists \(C>0\) and an exponent \(\delta\geq 2\) such that rational \(\forall p/q\ ,\) \(\left|\theta-\frac{p}{q}\right| \geq \frac{C}{q^\delta}\ ,\) i.e. such an irrational cannot be too well approximated by rationals.)
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 following statement specifies the second case:
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.
References
- Milnor, J. [2000]: Dynamics in one complex variable, second edition
