Small divisors in dimension one.
(Petits diviseurs en dimension 1.)

*(French)*Zbl 0836.30001
Astérisque. 231. Paris: Société Math. de France, 242 p. (1995).

This volume consists of two articles.

The first paper is concerned with the question when a holomorphic function with an irrationally indifferent fixed point is linearizable. A preprint of this article goes back to 1987 and its major results were announced in [C. R. Acad. Sci. Paris Ser. I, 306, 55-58 (1988; Zbl 0668.58010)].

Let \(f(z) = \lambda z + O(z^2)\) be holomorphic in a neighborhood of the origin. If \(0 < |\lambda |< 1\) or \(|\lambda |> 1\) there exists, by a classical theorem of Koenigs, a function \(H_f\) holomorphic in a neighborhood of the origin such that \(H_f'(0) = 1\) and \(H_f (\lambda z) = f(H_f(z))\). The cases that \(\lambda = 0\) or that \(\lambda\) is a root of unity are different, but also well understood. In the remaining case \(\lambda = \exp 2 \pi i \alpha\) where \(\alpha\) is real and irrational. Here \(H_f\) exists as a formal power series and the question arises whether it converges in some neighborhood of the origin. If this is the case, \(f\) is called linearizable.

Siegel, in 1942, was the first to give a condition for \(\alpha\) which implies that every holomorphic function \(f(z) = \lambda z + O(z^2)\) is linearizable. (Here and in the following we always write \(\lambda = \exp 2 \pi i \alpha.)\) Siegel’s condition may be written in the form \(\log q_{n + 1} = O (\log q_n)\), where the \(q_n\) are the denominators of the convergents of the continued fractions expansion of \(\alpha\). Brjuno improved this condition to \(\sum^\infty_{n = 0} (\log q_{n + 1})/q_n < \infty\) around 1965. In the opposite direction, Cremer had shown in 1938 that if \(\sup_{n \geq 0} (\log q_{n + 1})/q_n = \infty\), then there exists a holomorphic function \(f(z) = \lambda z + O(z^2)\) which is not linearizable.

The first main result of this paper is that Brjuno’s condition is sharp; that is, if \(\sum^\infty_{n = 0} (\log q_{n + 1})/q_n = \infty\), then there exists a holomorphic function \(f(z) = \lambda z + O (z^2)\) which is not linearizable. The author also gives a new proof of Brjuno’s theorem that \(\sum^{\infty}_{n=0}\inf y (\log q_{n + 1})/q_n < \infty\) implies linearizability and in fact gives a quantative version of this result by estimating the radius of convergence of \(H_f\).

In the second part (of the first paper) the author considers the linearization of the quadratic polynomial \(P_\lambda (z) = \lambda z(1 - z)\). The main result of this part is that \(P_\lambda\) is linearizable if and only if every holomorphic function \(f(z) = \lambda z + O (z^2) \) is linearizable; that is, if \(\sum^\infty_{n = 0} (\log q_{n + 1})/q_n < \infty\). It is also shown that if \(P_\lambda\) is not linearizable, then every neighborhood of the origin contains periodic orbits of \(P_\lambda\).

The second paper of this volume is a slightly revised version of the author’s thesis from 1985. It is concerned with diffeomorphisms of the circle \(T = R/Z\) and in particular with their centralizers and questions concerning their conjugation to rotations. We do not attempt to sketch the history and background of these problems (starting with the work of Poincaré and Denjoy, and important contributions being made by many other people, including Yoccoz’s thesis advisor Herman).

After introducing the necessary notation and recalling some basic results in chapter I of the second paper, the author studies the functional equation \(\psi \circ R_\alpha - \psi = \varphi \circ R_\beta - \varphi\) in chapter II. Here \(\alpha \in T\) and \(\varphi \in C^\infty (T)\) are given, \(R_\alpha (x) = x + \alpha\) is the rotation, and \(\beta \in T\) and \(\psi \in C^\infty (T)\) are to be found. This equation is connected with the study of centralizers.

In chapter III the conjugation to rotations is considered. Let \(F_\alpha^\infty\) be the set of all \(C^\infty\)- diffeomorphisms of \(T\) with rotation number \(\alpha\) and let \(O_\alpha^\infty\) be the set of all diffeomorphisms of \(T\) which are \(C^\infty\)-conjugate to \(R_\alpha\). If \(\alpha\) satisfies a diophantine condition, that is, if there exists \(\gamma > 0\) and \(\beta \geq 0\) such that \(|\alpha - p/q |\geq \gamma q^{- 2 - \beta}\) for all integers \(p\) and \(q\), \(q > 0\), then \(O^\infty_\alpha= F^{\infty}_{\alpha}\). This result of the original thesis was already published in [Ann. Sci. Ec. Norm. Supér, IV. Sér. 17, 333-359 (1984; Zbl 0595.57027)] and is not included here. It is shown here in Chapter III that \(O^\infty_\alpha\) is dense in \(F_\alpha^\infty\) with respect to the \(C^\infty\)-topology for each irrational \(\alpha\).

In chapters IV–VI the centralizers of diffeomorphisms of \(T\) with rational rotation numbers are considered. These results are used in chapter VII and VIII to study the centralizers of diffeomorphisms of \(T\) whose rotation number is a Liouville number; that is, an irrational number that does not satisfy a diophantine condition. One of the principal results is that there exists an irrational \(\alpha\) and \(f \in F^\infty_\alpha\) such that the centralizer of \(f\) consists only of the iterates of \(f\). For generic \(f \in F^\infty_\alpha\), however, this is not the case. Moreover, for generic irrational \(\alpha\) there is no \(f \in F_\alpha^\infty\) with this property.

The two articles constitute major contributions to their respective fields and the results obtained were among the reasons to award Yoccoz a Fields medal in 1994.

The first paper is concerned with the question when a holomorphic function with an irrationally indifferent fixed point is linearizable. A preprint of this article goes back to 1987 and its major results were announced in [C. R. Acad. Sci. Paris Ser. I, 306, 55-58 (1988; Zbl 0668.58010)].

Let \(f(z) = \lambda z + O(z^2)\) be holomorphic in a neighborhood of the origin. If \(0 < |\lambda |< 1\) or \(|\lambda |> 1\) there exists, by a classical theorem of Koenigs, a function \(H_f\) holomorphic in a neighborhood of the origin such that \(H_f'(0) = 1\) and \(H_f (\lambda z) = f(H_f(z))\). The cases that \(\lambda = 0\) or that \(\lambda\) is a root of unity are different, but also well understood. In the remaining case \(\lambda = \exp 2 \pi i \alpha\) where \(\alpha\) is real and irrational. Here \(H_f\) exists as a formal power series and the question arises whether it converges in some neighborhood of the origin. If this is the case, \(f\) is called linearizable.

Siegel, in 1942, was the first to give a condition for \(\alpha\) which implies that every holomorphic function \(f(z) = \lambda z + O(z^2)\) is linearizable. (Here and in the following we always write \(\lambda = \exp 2 \pi i \alpha.)\) Siegel’s condition may be written in the form \(\log q_{n + 1} = O (\log q_n)\), where the \(q_n\) are the denominators of the convergents of the continued fractions expansion of \(\alpha\). Brjuno improved this condition to \(\sum^\infty_{n = 0} (\log q_{n + 1})/q_n < \infty\) around 1965. In the opposite direction, Cremer had shown in 1938 that if \(\sup_{n \geq 0} (\log q_{n + 1})/q_n = \infty\), then there exists a holomorphic function \(f(z) = \lambda z + O(z^2)\) which is not linearizable.

The first main result of this paper is that Brjuno’s condition is sharp; that is, if \(\sum^\infty_{n = 0} (\log q_{n + 1})/q_n = \infty\), then there exists a holomorphic function \(f(z) = \lambda z + O (z^2)\) which is not linearizable. The author also gives a new proof of Brjuno’s theorem that \(\sum^{\infty}_{n=0}\inf y (\log q_{n + 1})/q_n < \infty\) implies linearizability and in fact gives a quantative version of this result by estimating the radius of convergence of \(H_f\).

In the second part (of the first paper) the author considers the linearization of the quadratic polynomial \(P_\lambda (z) = \lambda z(1 - z)\). The main result of this part is that \(P_\lambda\) is linearizable if and only if every holomorphic function \(f(z) = \lambda z + O (z^2) \) is linearizable; that is, if \(\sum^\infty_{n = 0} (\log q_{n + 1})/q_n < \infty\). It is also shown that if \(P_\lambda\) is not linearizable, then every neighborhood of the origin contains periodic orbits of \(P_\lambda\).

The second paper of this volume is a slightly revised version of the author’s thesis from 1985. It is concerned with diffeomorphisms of the circle \(T = R/Z\) and in particular with their centralizers and questions concerning their conjugation to rotations. We do not attempt to sketch the history and background of these problems (starting with the work of Poincaré and Denjoy, and important contributions being made by many other people, including Yoccoz’s thesis advisor Herman).

After introducing the necessary notation and recalling some basic results in chapter I of the second paper, the author studies the functional equation \(\psi \circ R_\alpha - \psi = \varphi \circ R_\beta - \varphi\) in chapter II. Here \(\alpha \in T\) and \(\varphi \in C^\infty (T)\) are given, \(R_\alpha (x) = x + \alpha\) is the rotation, and \(\beta \in T\) and \(\psi \in C^\infty (T)\) are to be found. This equation is connected with the study of centralizers.

In chapter III the conjugation to rotations is considered. Let \(F_\alpha^\infty\) be the set of all \(C^\infty\)- diffeomorphisms of \(T\) with rotation number \(\alpha\) and let \(O_\alpha^\infty\) be the set of all diffeomorphisms of \(T\) which are \(C^\infty\)-conjugate to \(R_\alpha\). If \(\alpha\) satisfies a diophantine condition, that is, if there exists \(\gamma > 0\) and \(\beta \geq 0\) such that \(|\alpha - p/q |\geq \gamma q^{- 2 - \beta}\) for all integers \(p\) and \(q\), \(q > 0\), then \(O^\infty_\alpha= F^{\infty}_{\alpha}\). This result of the original thesis was already published in [Ann. Sci. Ec. Norm. Supér, IV. Sér. 17, 333-359 (1984; Zbl 0595.57027)] and is not included here. It is shown here in Chapter III that \(O^\infty_\alpha\) is dense in \(F_\alpha^\infty\) with respect to the \(C^\infty\)-topology for each irrational \(\alpha\).

In chapters IV–VI the centralizers of diffeomorphisms of \(T\) with rational rotation numbers are considered. These results are used in chapter VII and VIII to study the centralizers of diffeomorphisms of \(T\) whose rotation number is a Liouville number; that is, an irrational number that does not satisfy a diophantine condition. One of the principal results is that there exists an irrational \(\alpha\) and \(f \in F^\infty_\alpha\) such that the centralizer of \(f\) consists only of the iterates of \(f\). For generic \(f \in F^\infty_\alpha\), however, this is not the case. Moreover, for generic irrational \(\alpha\) there is no \(f \in F_\alpha^\infty\) with this property.

The two articles constitute major contributions to their respective fields and the results obtained were among the reasons to award Yoccoz a Fields medal in 1994.

Reviewer: W.Bergweiler (Berlin)

##### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30C10 | Polynomials and rational functions of one complex variable |

30C62 | Quasiconformal mappings in the complex plane |

37E99 | Low-dimensional dynamical systems |

37F99 | Dynamical systems over complex numbers |

37C55 | Periodic and quasi-periodic flows and diffeomorphisms |