References
[Ah1]Ahlfors, L. V.,Lectures on Quasiconformal Mappings. Wadsworth & Brooks/Cole, Monterey, CA, 1987.
[Ah2]—,Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973.
[Ar1]Arnold, V. I., Small denominators I: Mapping the circle onto itself.Izv. Akad. Nauk. SSSR Ser. Mat., 25 (1961), 21–86; English translation inAmer. Math. Soc. Transl. Ser. 2, 46 (1965), 213–284.
[Ar2]—Chapitres supplémentaires de la théorie des équations différentielles ordinaires. “Mir” Moscow, 1980.
[Bie]Bielefeld, B., Conformal dynamics problem list. Preprint, Stony Brook.
[Bir]Birkhoff, G. D., Surface transformations and their dynamical applications.Acta Math., 43 (1922), 1–119; Also in:Collected Mathematical Papers, Vol. II, pp. 111–229.
[Bo]Bost, J.-B., Tores invariants des systèmes dynamiques hamiltoniens, inSéminaire Bourbaki, vol. 1984/85, exp. no 639Astérisque, 133–134 (1986), 113–157.
[Br]Brjuno, A. D., Analytical form of differential equations.Trans. Moscow Math. Soc., 25 (1971), 131–228; 26 (1972), 199–239.
[Cam]Camacho, C., On the local structure of conformal mapping and holomorphic vector fields in C2.Astérisque, 59–60 (1978), 83–94.
[Car]Cartan, H.,Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes. Hermann, Paris, 1961.
[Ch]Cherry, T. M., A singular case of iteration of analytic functions: a contribution to the small divisors problem, inNon-linear Problems of Engineering (W. F. Ames, ed.), pp. 29–50. Academic Press, New York, 1964.
[CLi]Carthwright, M. L. &Littlewood, J. E., Some fixed point theorems.Ann. of Math., 54 (1951), 1–37; Also in:Collected Papers of J. E. Littlewood, Vol. 1, pp. 38–74.
[CLo]Collingwood, E. F. &Lohwater, A. J.,The Theory of Cluster Sets. Cambridge Univ. Press. Cambridge, 1966.
[Do1]Douady, A., Systemes dynamiques holomorphes, inSéminaire Bourbaki, vol. 1982/83, exp. no 599.Astérisque, 105–106 (1983), 39–63.
[Do2]Douady, A. Disques de Siegel et anneaux de Herman, inSéminaire Bourbaki, vol. 1986/87, exp. no 677.Astérisque, 152–153 (1987), 151–172.
[Du]Dulac, H., Recherches sur les points singuliers des équations différentielles.Journal de l'Ecole Polytechnique Sér. 2, 9 (1904), 1–125.
[Ec]Ecalle, J., Les fonctions résurgentes et leurs applications. Publ. Math. Orsay, t. I, no 81-05; t II no 81-06; t. III, no 85-05.
[Fa]Fatou, P., Sur les équations fonctionnelles.Bull. Math. France, 47 (1919), 161–271; 48 (1920), 33–94; 48 (1920), 208–314.
[Ga]Garnett, J. B.,Applications of Harmonic Measure. University of Arkansas Lecture Notes in the Mathematical Sciences, 8, John Wiley & Sons, New York, 1986.
[Gh]Ghys, E., Transformations holomorphes au voisinage d'une courbe de Jordan.C. R. Acad. Sci. Paris Sér. I Math., 298 (1984), 385–388.
[He1]Herman, M. R., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations.Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–234.
[He2]—, Recent results and some open questions on Siegel's linearisation theorem of germs of complex analytic diffeomorphisms of Cn near a fixed point, inVIII th International Congress on Mathematical Physics (Marseille, 1986), pp. 138–184. World Sci. Publishing, Singapore, 1987.
[He3]—, Are there critical points on the boundaries of singular domains?Comm. Math. Phys., 99 (1985), 593–612.
[HY]Hocking, J. G. &Young, G. S.,Topology, 2nd edition, Dover, New York, 1988.
[Ju]Julia, G., Mémoire sur la permutabilité des substitutions rationnelles.Ann. Sci. École Norm. Sup., 39 (1922), 131–215; Also in:Oeuvres, vol. I, pp. 335–419. Gauthier-Villars, Paris, 1968.
[KH]Katok, A. &Hasselblatt, B.,Introduction to the Modern Theory of Dynamical Systems. Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995.
[La]Lang, S.,Introduction to Diophantine Approximations. Addison-Wesley, Reading, MA-London-Don Mills, ON, 1966.
[Le]Leau, L., Étude sur les equations fonctionnelles à une ou à plusieurs variables.Ann. Fac. Sci. Toulouse Math. Sér. I. 11 (1897), E.1-E.110.
[LV]Lehto, O. &Virtanen, K. I.,Quasiconformal Mappings in the Plane, 2nd edition. Grundlehren Math. Wiss., 126. Springer-Verlag, New York-Heidelberg, 1973.
[Ma]Mather, J., Topological proofs of some purely topological consequences of Carathéodory's theory of prime ends inSelected Studies (Th. M. Rassias, ed.), pp. 225–255. North-Holland, Amsterdam-New York, 1982.
[Mi]Milnor, J., Dynamics in one complex variable: Introductory lectures. Preprint, Stony Brook, 1990/5.
[MM]Mattei, J.-F. &Moussu, R., Holonomie et intégrales premières.Ann Sci. École Norm Sup. (4), 13 (1980), 469–523.
[Mo]Moser, J., On commuting circle mappings and simultaneous diophantine approximations.Math. Z., 205 (1990), 105–121.
[MR]Martinet, J. &Ramis, J.-P., Classification analytique des équations différentielles non linéaires résonnantes du premier ordre.Ann. Sci. École Norm. Sup. (4), 4 (1984), 571–621.
[MZ]Montgomery, D. &Zippin, L.,Topological Transformation Groups. Krieger, Hungtinton, NY, 1974.
[Na]Naishul, V. I., Topological invariants of analytic and area preserving mappings and their application to analytic differential equations in C2 and CP2.Trans. Moscow Math. Soc., 42 (1983), 239–250.
[Ne]Newman, M. H. A.,Elements of the Topology of Plane Sets of Points. Dover, New York, 1992.
[Oh]Ohtsuka, M.,Dirichlet Problem, Extremal Length and Prime-ends. Van Nostrand Reinhold, 1970.
[Pe1]Pérez-Marco, R., Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnold.Ann. Sci. École Norm. Sup. (4), 26 (1993), 565–644; See also: Sur la structure des germes holomorphes non linéarisables.C. R. Acad. Sci. Paris Sér. I. Math., 312 (1991), 533–536.
[Pe2]—, Non-linearizable holomorphic dynamics having an uncountable number of symmetries.Invent. Math., 119 (1995), 67–127; See also: Centralisateurs non dénombrables de germes de difféomorphismes holomorphes non linearisables de (C,0).C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 461–464.
[Pe3]Pérez-Marco, R. Solution complète au problème de Siegel de linéarisation d'une application holomorphe au voisinage d'un point fixe (d'après J.-C. Yoccoz), inSéminaire Bourbaki, vol. 1991/92, exp. no 753.Astérisque, 206 (1992), 273–310.
[Pe4]Pérez-Marco, R. Topology of Julia sets and hedgehogs. Preprint, Université de Paris-Sud, 94–48 (1994).
[Pe5]Pérez-Marco, R. Holomorphic germs of quadratic type. To appear.
[Pe6]Pérez-Marco, R. Hedgehogs dynamics. To appear; A partial account in: Sur une question de Dulac et Fatou.C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1045–1048.
[Pe7]Pérez-Marco, R. Classification dynamique des compacts pleins invariants par un difféomorphisme holomorphe. Manuscript, 1996.
[Po]Pommerenke, Ch.,Boundary Behaviour of Conformal Maps. Grundlehren Math. Wiss., 299. Springer-Verlag, Berlin, 1992.
[PY]Pérez-marco, R. & Yoccoz, J.-Ch., Germes de feuilletages holomorphes à holonomie prescrite, inComplex Analytic Methods in Dynamical Systems (Rio de Janeiro, 1992).Astérisque, 222 (1994), 345–371.
[Ro]Rogers, J., Diophantine conditions imply critical points on the boundaries of Siegel disks of polynomials. Preprint, Tulane University.
[Rü]Rüssmann, H., Kleine Nenner II: Bemerkungen zur Newtonschen Methode.Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II, 1972, 1–10.
[Sch]Schmidt, W. M.,Diophantine Approximation. Lecture Notes in Math., 785. Springer-Verlag, Berlin-New York, 1980.
[Si]Siegel, C. L., Iterations of analytic functions.Ann. of Math., 43 (1942), 807–812.
[SM]Siegel, C. L. &Moser, J.,Lectures on Celestial Mechanics. Grundlehren Math. Wiss., 187. Springer-Verlag, New York-Heidelberg, 1971.
[Su]Sullivan, D., Conformal dynamical systems, inGeometric Dynamics (Rio de Janeiro, 1981) pp. 725–752. Lecture Notes in Math., 1007, Springer-Verlag, Berlin-New York, 1983.
[Ts]Tsuji, M.,Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
[Vo]Voronin, S. M., Analytic classification of germs of conformal mappings (C,0)→(C,0) with identity linear part.Functional Anal. Appl., 15:1 (1981), 1–17.
[Yo1]Yoccoz, J.-Ch., Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Thèse d'Etat, Université de Paris-Sud, 1985.
[Yo2]—, Conjugation différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diphhantienne.Ann. Sci. École Norm. Sup., (4), 17 (1984), 333–359.
[Yo3]—, Theorème de Siegel, polynômes quadratiques et nombres de Brjuno.C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 55–58.
[Yo4]Yoccoz, J.-Ch., Conjugaison des difféomorphismes analytiques du cercle. Manuscript.
[Yo5]—, An introduction to small divisors problems, inFrom Number Theory to Physics (Les Houches, 1989), pp. 659–679. Springer-Verlag, Berlin, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pérez-Marco, R. Fixed points and circle maps. Acta Math. 179, 243–294 (1997). https://doi.org/10.1007/BF02392745
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02392745