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Additional info for Advanced Course Of Mathematical Analysis III: Proceedings of the Third International School La Rabida, Spain, 3 - 7 September 2007
Then the following statements are equivalent: (i) the origin is a Siegel point for the quadratic polynomial fλ (z) = λz+z 2 ; (ii) the origin is a Siegel point for all f ∈ End (C, 0) with multiplier λ; (iii) the number λ satisfies Bryuno’s condition +∞ k=0 1 1 log < +∞. 2 (see also  and references therein), has proved that condition (iii) implies condition (ii). Yoccoz, using a more geometric approach based on conformal and quasi–conformal geometry, has proved that (i) is equivalent to (ii) and that (ii) implies (iii), that is, that if λ does not satisfy (26) then the origin is a Cremer point for some f ∈ End (C, 0) with multiplier λ — and hence, it is a Cremer point for the quadratic polynomial fλ (z).
Orsay 81-05, Universit´e de Paris–Sud, Orsay (1981). ´ 11. J. Ecalle, Les fonctions r´esurgentes. Tome II: Les fonctions r´esurgentes appliqu´ees a ` l’it´eration, Publ. Math. Orsay 81-06, Universit´e de Paris–Sud, Orsay (1981). 12. P. Fatou, Sur les ´equations fonctionnelles, I, Bull. Soc. Math. France 47 (1919), 161–271. 13. P. Fatou, Sur les ´equations fonctionnelles, II, Bull. Soc. Math. France 48 (1920), 33–94. 14. P. Fatou, Sur les ´equations fonctionnelles, III, Bull. Soc. Math. France 48 (1920), 208–314.
Camacho, On the local structure of conformal mappings and holomorphic vector fields, Ast´erisque 59-60 (1978) 83–94. 7. S. Carleson and F. Gamelin, Complex dynamics, Springer, Berlin (1994). 8. H. Cremer, Zum Zentrumproblem, Math. An. 98 (1927), 151–163. ¨ 9. H. Cremer, Uber die H¨ aufigkeit der Nichtzentren, Math. Ann. 115 (1938), 573–580. ´ 10. J. Ecalle, Les fonctions r´esurgentes. Tome I: Les alg`ebres de fonctions r´esurgentes, Publ. Math. Orsay 81-05, Universit´e de Paris–Sud, Orsay (1981).