Prof. Michael R. Herman has studied the problem of quasi-symmetric conjugacy of orientation preserving homeomorphisms/diffeomorphisms of the circle to the rigid rotation. He obtained a series of results around years 1986-1987, however his manuscripts remain unpublished until today. These results on the circle, together with surgery technique, had important consequences in the study of the bounday of Siegel disks of complex rational maps and entire functions. For the survey of the theory, see Douady's lecture in the Bourbaki seminar [Douady] (and also [Petersen]).
In the year 1988-89 at the Institute for Advanced Study, photocopies of the hand written manuscipts were made and distributed to a limited number of people, including myself. In July 2000, I asked Prof. Herman if I can scan the manuscripts and put the images on the world wide web, and he kindly agreed. The below are the pages that show the scaned images of the manuscripts. The images are kept in a larger size so that the details are readable. Some pages of the copies of the manuscripts were already shrunk in order to fit to letter size papers.
The scanning work was supported by Grant-in-Aid # 11440053 of Ministry of Education of Japan.
Manuscript
1:
Conjugaison quasi symétrique des
homéomorphismes du cercle à des rotations.
(1987?)
In this paper, it is proved that if a real-analytic homeomorphism of the circle (i.e., critical points with odd local order are allowed) has irrational rotation number of constant type, then it is conjugated to a rigid rotation by a quasi-symmetric mapping of the circle. This result implies, via surgery technique, that if a quadratic polynomial has a Siegel disk of period 1 and of irrational rotation number of constant type, then the boundary of the Siegel disk is a quasi-circle, in particular a Jordan curve, and contains the critical point. These results were recaptured in [Petersen].
Manuscript
2:
Conjugaison quasi symétrique des
difféomorphismes du cercle à des rotations et
applications aux disques singuliers de Siegel, I. (1986?)
In this earlier paper, it is shown that if f(x) is a diffeomorphism of the circle (which is identified with R/Z) satisfying a certain condition A0, then there exists a real number t such that ft(x)=f(x)+t is quasi-symmetrically conjugate to an irrational rotation, but the conjugacy is not C2. Again by surgery, it leads to a construction of a Siegel disk (of period 1) for a quadratic polynomial such that its boundary is a quasi-circle, but does not contain the critical point.
Manuscript
3:
Uniformité de la distorstion de
Swiatek pour les familles compactes de produits Blaschke.
(1987?)
This paper shows the unifomity of the estimate in Manuscript 1 when the map is from a compact family of Blaschke products. The crucial point is that when there are collision of critical points from complex plane into the real line, the total contribution to the distortion constant can be estimated. This led me to prove that Siegel disks of polynomials with constant type rotation number (any degree, any period) have quasi-circle boundary.
Reference
[Douady] Adrien Douady, Disques de Siegel et
anneaux de Herman, Séminaire Bourbaki, Vol. 1986/87.
Astérisque No. 152-153 (1987), p.151--172.
[Petersen] C. Petersen, The Herman-Swiatek Theorems with applications, in "The Mandelbrot set, Theme and Variations", Ed. Tan Lei, London Math. Soc. Lecture Note Ser. 274 (2000), Cambridge Univ. Press, p.211--225.
Mitsuhiro
Shishikura
Kyoto University, Department of Mathematics