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Consider a holomorphic [[Vector field|vector field]] with a singularity, i.e. of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101201.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101202.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101203.png" />. The eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101204.png" /> are said to be resonant if among the eigenvalues there exists a relation of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101207.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s1101209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012010.png" /> is the usual inner product. H. Poincaré proved in his dissertation that if the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012011.png" /> are non-resonant, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012012.png" /> can be reduced to the linear equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012013.png" /> by a formal change of variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012014.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012015.png" />th order term of this change of variable is given in terms of lower-order terms divided by a term of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012016.png" />. Since, in general, such terms can be either zero (in the resonant case) or arbitrarily close to zero (in which case one says that one has [[Small denominators|small denominators]]), the power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012017.png" /> in general does not converge. An example that divergence can occur was already given by L. Euler. The eigenvalue vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012018.png" /> belongs to the Poincaré domain if zero is not in the convex hull of the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012019.png" />; its complement is called the Siegel domain. When the vector of eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012020.png" /> is in the Poincaré domain, the vector field can be reduced to a polynomial vector field (called the normal form) by a biholomorphic change of variables in a neighbourhood of the singularity. If, in addition, there are no resonances, then one can choose the polynomial normal form to be linear. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012021.png" /> is in the Siegel domain, then one says it satisfies a Diophantine condition if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012026.png" />. Siegel's theorem states that, in this case, the vector field can be reduced to its linear part by a holomorphic change of coordinates in a neighbourhood of the singularity. Siegel sketched a proof of this result in 1942, but only in the 1970s complete proofs of this theorem were given (by A.N. Kolmogorov, J. Moser, V.I. Arnol'd, M.R. Herman, J.-C. Yoccoz and many others in various settings, e.g. in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012027.png" /> category). Many problems concerning invariant tori are based on similar small denominator estimates.
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In the discrete case, when one considers a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012028.png" />, the analogous results also holds. In that case, the vector of eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012029.png" /> is in the Poincaré domain if the norm of all the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012030.png" /> are all smaller or all greater than one. The complement is again called the Siegel domain. If the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012031.png" />, then this reduces to the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012032.png" />.
+
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In fact, in one-dimension the Siegel case is completely understood. So, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012033.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012034.png" /> is not a root of unity. Write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012035.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012036.png" /> be the [[Continued fraction|continued fraction]] expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012037.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012038.png" />, then the formal power series of the linearizing coordinates need not converge (Cremer's theorem). Around 1965, A.D. Bryuno proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012039.png" />, then this power series converges. In the late 1980s, Yoccoz proved the converse: if the Bryuno condition is not satisfied, then there exists a holomorphic diffeomorphism with linear part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012040.png" /> which is not holomorphically linearizable. In 1994, Yoccoz was awarded the Field Medal for this achievement.
+
Consider a holomorphic [[Vector field|vector field]] with a singularity, i.e. of the form  $  {\dot{x} } = Ax + \dots $
 +
with  $  x \in \mathbf R  ^ {b} $
 +
or  $  x \in \mathbf C $.
 +
The eigenvalues  $  \lambda _ {1} \dots \lambda _ {n} $
 +
are said to be resonant if among the eigenvalues there exists a relation of the form  $  \lambda _ {s} = ( m , \lambda ) $,  
 +
where  $  \lambda = ( \lambda _ {1} \dots \lambda _ {n} ) $,
 +
$  m = ( m _ {1} \dots m _ {n} ) $
 +
with  $  m _ {i} \geq  0 $,
 +
$  \sum m _ {i} = | m | \geq  2 $
 +
and  $  ( m , \lambda ) $
 +
is the usual inner product. H. Poincaré proved in his dissertation that if the eigenvalues of  $  A $
 +
are non-resonant, then  $  {\dot{x} } = Ax + \dots $
 +
can be reduced to the linear equation  $  {\dot{y} } = Ay $
 +
by a formal change of variable  $  x = y + \dots $.  
 +
The  $  | m | $
 +
th order term of this change of variable is given in terms of lower-order terms divided by a term of the form  $  ( m , \lambda ) - \lambda _ {s} $.  
 +
Since, in general, such terms can be either zero (in the resonant case) or arbitrarily close to zero (in which case one says that one has [[Small denominators|small denominators]]), the power series  $  x = y + \dots $
 +
in general does not converge. An example that divergence can occur was already given by L. Euler. The eigenvalue vector  $  \lambda $
 +
belongs to the Poincaré domain if zero is not in the convex hull of the eigenvalues  $  \lambda _ {j} \in \mathbf C $;
 +
its complement is called the Siegel domain. When the vector of eigenvalues  $  \lambda $
 +
is in the Poincaré domain, the vector field can be reduced to a polynomial vector field (called the normal form) by a biholomorphic change of variables in a neighbourhood of the singularity. If, in addition, there are no resonances, then one can choose the polynomial normal form to be linear. If  $  \lambda $
 +
is in the Siegel domain, then one says it satisfies a Diophantine condition if there exist  $  C > 0 $,
 +
$  \nu > 0 $
 +
such that $  | {\lambda _ {s} - ( m , \lambda ) } | \geq  {C / {| m |  ^  \nu  } } $
 +
for all  $  m _ {i} \geq  0 $
 +
with  $  \sum m _ {i} = | m | \geq  2 $.  
 +
Siegel's theorem states that, in this case, the vector field can be reduced to its linear part by a holomorphic change of coordinates in a neighbourhood of the singularity. Siegel sketched a proof of this result in 1942, but only in the 1970s complete proofs of this theorem were given (by A.N. Kolmogorov, J. Moser, V.I. Arnol'd, M.R. Herman, J.-C. Yoccoz and many others in various settings, e.g. in the $  C  ^ {k} $
 +
category). Many problems concerning invariant tori are based on similar small denominator estimates.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012041.png" /> is a rational mapping on the [[Riemann sphere|Riemann sphere]], then Siegel domains appear in the Fatou–Sullivan classification theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012042.png" /> be the closure of the set of repelling periodic points (cf. also [[Repelling set|Repelling set]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012043.png" /> be its complement. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012044.png" /> is also the set of points that have a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012046.png" /> forms a [[Normal family|normal family]]. Many properties of these sets were already obtained by G. Julia and P. Fatou in the 1920s and 1930s. Using the measurable Riemann mapping theorem (cf. also [[Quasi-conformal mapping|Quasi-conformal mapping]]), D. Sullivan proved in the 1980s that each connected component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012047.png" /> is eventually mapped onto a periodic component, and that the periodic components can be of four types:
+
In the discrete case, when one considers a mapping $  F ( x ) = Ax + \dots $,
 +
the analogous results also holds. In that case, the vector of eigenvalues  $  \lambda $
 +
is in the Poincaré domain if the norm of all the eigenvalues  $  \lambda _ {i} $
 +
are all smaller or all greater than one. The complement is again called the Siegel domain. If the dimension  $  n = 1 $,
 +
then this reduces to the unit circle  $  | \lambda | = 1 $.
  
i) the basin of an attracting periodic point (where the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012048.png" /> of the linear part satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012049.png" />);
+
In fact, in one-dimension the Siegel case is completely understood. So, assume that  $  | \lambda | = 1 $
 +
and that  $  \lambda $
 +
is not a root of unity. Write  $  \lambda = { \mathop{\rm exp} } ( 2 \pi i \alpha ) $
 +
and let  $  { {p _ {n} } / {q _ {n} } } $
 +
be the [[Continued fraction|continued fraction]] expansion of $  \alpha $.
 +
If  $  \sup  _ {n \geq  0 }  { {( { \mathop{\rm log} } q _ {n + 1 }  ) } / {q _ {n} } } = \infty $,
 +
then the formal power series of the linearizing coordinates need not converge (Cremer's theorem). Around 1965, A.D. Bryuno proved that if  $  \sum { {( { \mathop{\rm log} } q _ {n + 1 }  ) } / {q _ {n} } } < \infty $,
 +
then this power series converges. In the late 1980s, Yoccoz proved the converse: if the Bryuno condition is not satisfied, then there exists a holomorphic diffeomorphism with linear part $  z \mapsto \lambda z $
 +
which is not holomorphically linearizable. In 1994, Yoccoz was awarded the Field Medal for this achievement.
  
ii) the basin of a parabolic periodic point (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012050.png" /> is a root of unity);
+
If  $  f $
 +
is a rational mapping on the [[Riemann sphere|Riemann sphere]], then Siegel domains appear in the Fatou–Sullivan classification theorem. Let  $  J ( f ) $
 +
be the closure of the set of repelling periodic points (cf. also [[Repelling set|Repelling set]]) and let  $  F ( f ) $
 +
be its complement.  $  F ( f ) $
 +
is also the set of points that have a neighbourhood  $  U $
 +
such that  $  f  ^ {n} \mid  _ {U} $
 +
forms a [[Normal family|normal family]]. Many properties of these sets were already obtained by G. Julia and P. Fatou in the 1920s and 1930s. Using the measurable Riemann mapping theorem (cf. also [[Quasi-conformal mapping|Quasi-conformal mapping]]), D. Sullivan proved in the 1980s that each connected component of  $  F ( f ) $
 +
is eventually mapped onto a periodic component, and that the periodic components can be of four types:
  
iii) a Siegel domain (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110120/s11012051.png" /> has norm one but is not a root of unity, but still the mapping is holomorphically linearizable on some neighbourhood of the fixed point);
+
i) the basin of an attracting periodic point (where the eigenvalue  $  \lambda $
 +
of the linear part satisfies  $  | \lambda | < 1 $);
 +
 
 +
ii) the basin of a parabolic periodic point (when  $  \lambda $
 +
is a root of unity);
 +
 
 +
iii) a Siegel domain (when $  \lambda $
 +
has norm one but is not a root of unity, but still the mapping is holomorphically linearizable on some neighbourhood of the fixed point);
  
 
iv) a Herman ring (when the component is an annulus on which the mapping is conjugate to a rotation).
 
iv) a Herman ring (when the component is an annulus on which the mapping is conjugate to a rotation).

Latest revision as of 08:13, 6 June 2020


Consider a holomorphic vector field with a singularity, i.e. of the form $ {\dot{x} } = Ax + \dots $ with $ x \in \mathbf R ^ {b} $ or $ x \in \mathbf C $. The eigenvalues $ \lambda _ {1} \dots \lambda _ {n} $ are said to be resonant if among the eigenvalues there exists a relation of the form $ \lambda _ {s} = ( m , \lambda ) $, where $ \lambda = ( \lambda _ {1} \dots \lambda _ {n} ) $, $ m = ( m _ {1} \dots m _ {n} ) $ with $ m _ {i} \geq 0 $, $ \sum m _ {i} = | m | \geq 2 $ and $ ( m , \lambda ) $ is the usual inner product. H. Poincaré proved in his dissertation that if the eigenvalues of $ A $ are non-resonant, then $ {\dot{x} } = Ax + \dots $ can be reduced to the linear equation $ {\dot{y} } = Ay $ by a formal change of variable $ x = y + \dots $. The $ | m | $ th order term of this change of variable is given in terms of lower-order terms divided by a term of the form $ ( m , \lambda ) - \lambda _ {s} $. Since, in general, such terms can be either zero (in the resonant case) or arbitrarily close to zero (in which case one says that one has small denominators), the power series $ x = y + \dots $ in general does not converge. An example that divergence can occur was already given by L. Euler. The eigenvalue vector $ \lambda $ belongs to the Poincaré domain if zero is not in the convex hull of the eigenvalues $ \lambda _ {j} \in \mathbf C $; its complement is called the Siegel domain. When the vector of eigenvalues $ \lambda $ is in the Poincaré domain, the vector field can be reduced to a polynomial vector field (called the normal form) by a biholomorphic change of variables in a neighbourhood of the singularity. If, in addition, there are no resonances, then one can choose the polynomial normal form to be linear. If $ \lambda $ is in the Siegel domain, then one says it satisfies a Diophantine condition if there exist $ C > 0 $, $ \nu > 0 $ such that $ | {\lambda _ {s} - ( m , \lambda ) } | \geq {C / {| m | ^ \nu } } $ for all $ m _ {i} \geq 0 $ with $ \sum m _ {i} = | m | \geq 2 $. Siegel's theorem states that, in this case, the vector field can be reduced to its linear part by a holomorphic change of coordinates in a neighbourhood of the singularity. Siegel sketched a proof of this result in 1942, but only in the 1970s complete proofs of this theorem were given (by A.N. Kolmogorov, J. Moser, V.I. Arnol'd, M.R. Herman, J.-C. Yoccoz and many others in various settings, e.g. in the $ C ^ {k} $ category). Many problems concerning invariant tori are based on similar small denominator estimates.

In the discrete case, when one considers a mapping $ F ( x ) = Ax + \dots $, the analogous results also holds. In that case, the vector of eigenvalues $ \lambda $ is in the Poincaré domain if the norm of all the eigenvalues $ \lambda _ {i} $ are all smaller or all greater than one. The complement is again called the Siegel domain. If the dimension $ n = 1 $, then this reduces to the unit circle $ | \lambda | = 1 $.

In fact, in one-dimension the Siegel case is completely understood. So, assume that $ | \lambda | = 1 $ and that $ \lambda $ is not a root of unity. Write $ \lambda = { \mathop{\rm exp} } ( 2 \pi i \alpha ) $ and let $ { {p _ {n} } / {q _ {n} } } $ be the continued fraction expansion of $ \alpha $. If $ \sup _ {n \geq 0 } { {( { \mathop{\rm log} } q _ {n + 1 } ) } / {q _ {n} } } = \infty $, then the formal power series of the linearizing coordinates need not converge (Cremer's theorem). Around 1965, A.D. Bryuno proved that if $ \sum { {( { \mathop{\rm log} } q _ {n + 1 } ) } / {q _ {n} } } < \infty $, then this power series converges. In the late 1980s, Yoccoz proved the converse: if the Bryuno condition is not satisfied, then there exists a holomorphic diffeomorphism with linear part $ z \mapsto \lambda z $ which is not holomorphically linearizable. In 1994, Yoccoz was awarded the Field Medal for this achievement.

If $ f $ is a rational mapping on the Riemann sphere, then Siegel domains appear in the Fatou–Sullivan classification theorem. Let $ J ( f ) $ be the closure of the set of repelling periodic points (cf. also Repelling set) and let $ F ( f ) $ be its complement. $ F ( f ) $ is also the set of points that have a neighbourhood $ U $ such that $ f ^ {n} \mid _ {U} $ forms a normal family. Many properties of these sets were already obtained by G. Julia and P. Fatou in the 1920s and 1930s. Using the measurable Riemann mapping theorem (cf. also Quasi-conformal mapping), D. Sullivan proved in the 1980s that each connected component of $ F ( f ) $ is eventually mapped onto a periodic component, and that the periodic components can be of four types:

i) the basin of an attracting periodic point (where the eigenvalue $ \lambda $ of the linear part satisfies $ | \lambda | < 1 $);

ii) the basin of a parabolic periodic point (when $ \lambda $ is a root of unity);

iii) a Siegel domain (when $ \lambda $ has norm one but is not a root of unity, but still the mapping is holomorphically linearizable on some neighbourhood of the fixed point);

iv) a Herman ring (when the component is an annulus on which the mapping is conjugate to a rotation).

References

[a1] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a2] A.D. Bryuno, "Analytical form of differential equations" Trans. Moscow Math. Soc. , 25 (1971) pp. 131–288 (In Russian) (Also: 26 (1972), 199–239)
[a3] M.R. Herman, "Recent results and some open questions on Siegel's linearisation theorem of germs of complex analytic diffeomorphisms of near a fixed point" , Proc. VIII Int. Conf. Math. Phys. , World Sci. (1987)
[a4] C.L. Siegel, "Iteration of analytic functions" Ann. of Math. , 43 (1942) pp. 807–812
[a5] C.L. Siegel, J. Moser, "Lectures on celestial mechanics" , Springer (1971)
[a6] J.-C. Yoccoz, "Théorème de Siegel, polynômes quadratiques et nombres de Brjuno" Astérisque , 231 (1995)
How to Cite This Entry:
Siegel disc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Siegel_disc&oldid=48692
This article was adapted from an original article by S. van Strien (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article