Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Siegel discs, Herman rings and the Arnold family


Author: Lukas Geyer
Journal: Trans. Amer. Math. Soc. 353 (2001), 3661-3683
MSC (2000): Primary 30D05; Secondary 58F03, 58F08
DOI: https://doi.org/10.1090/S0002-9947-01-02662-9
Published electronically: April 24, 2001
MathSciNet review: 1837254
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

We show that the rotation number of an analytically linearizable element of the Arnold family $f_{a,b}(x)=x+a+b\sin(2\pi x)\pmod 1$, $a,b\in{\mathbb R}$, $0<b<1/(2\pi)$, satisfies the Brjuno condition. Conversely, for every Brjuno rotation number there exists an analytically linearizable element of the Arnold family. Along the way we prove the necessity of the Brjuno condition for linearizability of $P_{\lambda,d}(z)=\lambda z(1+z/d)^d$ and $E_\lambda(z)=\lambda z e^z$, $\lambda=e^{2\pi i\alpha}$, at 0. We also investigate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of $P_{\lambda,d}$ and $E_\lambda$ are quasidiscs with a critical point on the boundary if the rotation number is of constant type.


References [Enhancements On Off] (What's this?)

  • [Ah1] Ahlfors, L.V., Lectures on quasiconformal mappings, Van Nostrand, 1966. MR 34:336
  • [Ah2] Ahlfors, L.V., Conformal Invariants, Topics in Geometric Function Theory, McGraw-Hill, 1973. MR 50:10211
  • [Ar] Arnol'd, V.I., Small denominators. I: Mappings of the circumference onto itself., AMS Translations, Ser. 2, 46 (1965), 213-284.
  • [Ba] Baker I.N., Wandering domains for maps of the punctured plane, Ann. Acad. Sci. Fenn. Ser. A I, 12 (1987), 191-198. MR 89g:30046
  • [Be] Beardon, A.F., Iteration of Rational Functions, Springer-Verlag, 1991. MR 92j:30026
  • [BI] Bojarski, B. and Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in ${\mathbb R}^n$, Ann. Acad. Sci. Fenn. Ser. A I, 8 (1983), 257-324. MR 85h:30023
  • [CG] Carleson, L. and Gamelin, T.W., Complex Dynamics, Springer-Verlag, 1993. MR 94h:30033
  • [Do] Douady, A., Disques de Siegel et anneaux de Herman, Sém. Bourbaki, 39 (1986-87), 151-172. MR 89g:30049
  • [EL] Eremenko, A. E. and Lyubich, M. Yu., Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, 42, 4 (1992), 989-1020. MR 93k:30034
  • [Fa1] Fagella, N., The Complex Standard Family, Preprint.
  • [Fa2] Fagella, N., Limiting Dynamics for the Complex Standard Family, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), no. 3, 673-699. MR 96f:58129
  • [Ge1] Geyer, L., Quasikonforme Deformation in der Iterationstheorie, Diplomarbeit, TU Berlin, 1994.
  • [Ge2] Geyer, L., Linearization of structurally stable polynomials, Progress in holomorphic dynamics, Pitman Research Notes, 387, Longman, 1998, 27-30. MR 99m:58154
  • [GF] Grauert, H. and Fritzsche, K., Einführung in die Funktionentheorie mehrerer Veränderlicher, Springer-Verlag, 1974. MR 51:8448
  • [He1] Herman, M., Sur les conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-234. MR 81h:58039
  • [He2] Herman, M., Conjugaison quasi-symmétrique des homéomorphismes analytiques du cercle a des rotations, Manuscript.
  • [Hi] Hinkkanen, A., Uniformly quasiregular semigroups intwo dimensions, Ann. Acad. Sci. Fenn. Ser. A I, 21 (1996), 205-222. MR 96m:30029
  • [Ke] Keen, L., Topology and growth of a special class of holomorphic self-maps of ${\mathbb C}^*$, Erg. Th. Dyn. Sys., 9 (1989), 321-328. MR 91b:30070
  • [Ko] Kotus, J., Iterated holomorphic maps on the punctured plane, Dynamical systems (Sopron, 1985), Lecture Notes in Econom. and Math. Systems, 287, Springer-Verlag, (1987), 10-28. MR 92i:58152
  • [Kr] Kriete, H., Herman's proof of the existence of critical points on the boundary of singular domains, Progress in holomorphic dynamics, Pitman Research Notes, 387, Longman, 1998, 31-40. MR 99i:30042
  • [LV] Lehto, O. and Virtanen, K., Quasikonforme Abbildungen, Springer-Verlag, 1965. MR 32:5872
  • [Ma] Makienko, P., Iterations of analytic functions in ${\mathbb C}^*$(Russian) Dokl. Akad. Nauk SSSR 297 (1987), no. 1, 35-37; translation in Soviet Math. Dokl. 36 (1988), no. 3, 418-420. MR 88m:30066
  • [Mi] Milnor, J., Dynamics in One Complex Variable: Introductory Lectures, SUNY Stony Brook Preprint 1990/5.
  • [MS] de Melo, W. and van Strien, S., One-Dimensional Dynamics, Springer-Verlag, 1993. MR 95a:58035
  • [PM1] Pérez-Marco, R., Solution complète au problème de Siegel de linéarisation d'une application holomorphe au voisinage d'un point fixe (d'aprés J.-C. Yoccoz), Sém. Bourbaki 44ème année, 753, (1991-92). MR 94g:58190
  • [PM2] Pérez-Marco, R., Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnold, Ann. Sci. École Norm. Sup., 26 (1993), 565-644. MR 94h:30035
  • [Po] Pommerenke, Ch., Boundary Behavior of Conformal Maps, Springer-Verlag, 1992. MR 95b:30008
  • [Sh] Shishikura, M., On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29. MR 88i:58099
  • [St] Steinmetz, N., Rational iteration, de Gruyter Studies in Math., 16, 1993. MR 94h:30035
  • [Su] Sullivan, D., Conformal Dynamical Systems, Geometric Dynamics, Proc. Int. Symp., Rio de Janeiro/Brasil 1981, Lect. Notes Math. 1007, 725-752 (1983). MR 85m:58112
  • [Sw] Swiatek, G., Remarks on critical circle homeomorphisms, Bol. Soc. Bras. Mat., 29 (1998), 329-351.
  • [Yo1] Yoccoz, J.-C., Théorème de Siegel, nombres de Brjuno et polynomes quadratiques, Asterisque, 231 (1995), 3-88. MR 96m:58214
  • [Yo2] Yoccoz, J.-C., Conjugaison des difféomorphismes analytiques du cercle, Preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30D05, 58F03, 58F08

Retrieve articles in all journals with MSC (2000): 30D05, 58F03, 58F08


Additional Information

Lukas Geyer
Affiliation: Universität Dortmund, FB Mathematik, LS IX, 44221 Dortmund, Germany
Email: geyer@math.uni-dortmund.de

DOI: https://doi.org/10.1090/S0002-9947-01-02662-9
Keywords: Arnold family, standard family, linearization, Herman rings, circle diffeomorphisms
Received by editor(s): December 18, 1998
Received by editor(s) in revised form: December 12, 1999
Published electronically: April 24, 2001
Additional Notes: The author wishes to thank “Studienstiftung des deutschen Volkes” and DAAD for financial support.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society