# Siegel disc

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).

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