Linearization

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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

The linearization problem in complex dimension one dynamical systems

1 Statement
2 Power series expansions and small divisors
- 2.1 A reminder on continued fractions
3 History
4 Results
5 References
6 COPIED FROM MAIN ARTICLE
- 6.1 The linearization problem
  - 6.1.1 Statement of the linearization problem and first properties
  - 6.1.2 Brjuno's arithmetic condition

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Statement

Linearizable at a fixed point $\implies$ tame
Given a fixed point of a 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 → 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.

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 .
- 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.

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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 $\phi$ :

$b_1=1$ and $b_{n+1}=\frac{P_n(a_2,\ldots,a_{n+1},b_2,\ldots,b_{n})}{\lambda^{n+1}-\lambda}$

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.

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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:

$\frac{1}{2q_n q_{n+1}}<|\theta-p_n/q_n|<\frac{1}{q_n q_{n+1}}$ .

There is the well-known recurrence relation on the denominators (also satisfied by the numerators $p_n$ ):

$q_{n+1}=a_n q_n + q_{n-1}$ .

Therefore if $a_n$ is big, then $q_n^2|\theta-p_n/q_n|\approx \frac{1}{a_n}$ . Good approximations correspond to big values of $a_n$ .

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 $k=1+q_n$ :

$|\lambda^k-\lambda|\approx\frac{2\pi}{q_{n+1}}$ .

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History

This section certainly needs to be checked and completed by an expert on the history of the center problem. It would also better fit in an article about the center problem on its own.

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.

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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 $\alpha$ , $\alpha_{n+1}$ 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.

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References

Milnor, J. [2000]: Dynamics in one complex variable, second edition

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COPIED FROM MAIN ARTICLE

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The linearization problem

The question of the existence of Siegel disks is closely related to the linearization problem and 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.

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Statement of the linearization problem and first properties

The linearization problem can be stated for fixed points of any dynamical system $f$ . (A fixed point is a periodic point of period one, i.e. a point $a$ such that $f(a)=a$ .) Given such a fixed point $a$ , the question is whether there exists a local change of coordinates under which the dynamical system becomes linear. (A "local" change of coordinate needs only to be defined in a neighborhood of the fixed point under consideration.) This change of coordinates is also called a conjugacy or a linearizing coordinate.

The problem is usually asked in a situation where the system is differentiable in $a$ , in which case one tries to conjugate $f$ to its differential $D_a f$ at the fixed point.

This encyclopedia entry focuses on holomorphic dynamical systems in one variable. (For instance $z\mapsto z^2+c$ from $\mathbb{C}$ to $\mathbb{C}$ , where $c\in\mathbb{C}$ is any given parameter.)

Equivalence of topological and holomorphic linearizability. For holomorphic maps it turns out that linearizability by a continuous conjugacy implies linearizability by a holomorphic conjugacy. (Recall that being holomorphic, i.e. complex-differentiable, is the same thing as being analytic, i.e. being expressed by a power series expansion near every point. In particular, a conjugacy, if it exists, can be chosen to be $C^{\infty}$ .)

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

If $|\lambda|$ = 0, then $a$ is a superattracting fixed point. In this case, $f$ is not linearizable, unless it is constant in a neighborhood of $a$ .
If $0 < |\lambda| < 1$ or $1<|\lambda|$ , then $a$ is attracting (but not superattracting) or repelling, respectively. In this case, $a$ is always linearizable; this is known as Kœnig's theorem.
If , then is an indifferent fixed point. Write for some . The number is called the rotation number.
- If $\theta\in\mathbb{Q}$ , then $a$ is a parabolic fixed point, and $f$ 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 $\theta\notin\mathbb{Q}$ , then $a$ is called irrationally indifferent. Here the question is much more difficult, and in many cases depends on the number-theoretic properties of the multiplier $\lambda$ . This case includes examples where $f$ is linearizable and examples where it is not.

Siegel disks arise in the case of linearizable irrationally indifferent fixed points.

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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. 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 $\sum_{n=0}^{\infty} \frac{\log q_{n+1}}{q_n} < +\infty$ .

It can be checked that all Diophantine numbers satisfy Brjuno's condition.

Theorem. let $\theta$ be irrational.

The first part of the previous theorem is due to Brjuno and Rüssmann. Yoccoz later proved the second part, and also gave an independent proof of the first.

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 that completely settles the linearization question for quadratic polynomials: (a quadratic poylnomial is a polynomial of degree two) Theorem. If $\theta$ does not satisfy Bjruno's condition, then the fixed point $z=0$ of the quadratic polynomial $e^{2i\pi\theta}z+z^2$ is not linearizable.

Following Yoccozz'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.) On the other hand, it is still an open question whether there exist cubic polynomials with non-Brjuno Siegel disks. It is expected that this is not the case; in fact, the following has been conjectured by Douady.

Conjecture. Let $f$ be a rational function and suppose that $a$ is an irrationally indifferent fixed point of $f$ whose rotation number does not satisfy Brjuno's condition. (A rational function is a quotient of two polynomials.) Then $f$ is not linearizable near $a$ .

Invited by:

Prof. James Meiss, Applied Mathematics University of Colorado

Retrieved from "http://www.scholarpedia.org/article/Siegel_disks/Linearization"

Linearization

From Scholarpedia

Contents

Statement

Power series expansions and small divisors

A reminder on continued fractions

History

Results

References

COPIED FROM MAIN ARTICLE

The linearization problem

Statement of the linearization problem and first properties

Brjuno's arithmetic condition

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