Small Cycles Property of Some Cremer Rational Maps and Polynomials

Abstract

This paper concerns the linearization problem on rational maps of degree d ≥ 2 and polynomials of degree d > 2 from the perspective of non-linearizability. The authors introduce a set ${{\cal C}_\infty }$ of irrational numbers and show that if $\alpha \in {{\cal C}_\infty }$, then any rational map is not linearizable and has infinitely many cycles in every neighborhood of the fixed point with multiplier $\lambda = {{\rm{e}}^{2\pi {\rm{i}}\alpha }}$. Adding more constraints to cubic polynomials, they discuss the above problems by polynomial-like maps. For the family of polynomials, with the help of Yoccoz’s method, they obtain its maximum dimension of the set in which the polynomials are non-linearizable.

Keywords

Irrationally indifferent fixed point / Linearization problem / Small cycles property

References

[1] Buff X, Chéritat A. A new proof of a conjecture of Yoccoz. Ann. Inst. Fourier (Grenoble), 2011, 61(1): 319-350

[2] Böttcher L E. The principal laws of convergence of iterates and their application to analysis (Russian). Izv, Kazan. fiz.-Mat. Obshch., 1904, 14: 155-234

[3] Brjuno A D. Convergence of transformations of differential equations to normal forms. Dokl. Akad. Nauk USSR, 1965, 165: 987-989 (Soviet Math. Dokl., 1536–1538)

[4] Cremer H. Zum Zentrumproblem. Math. Ann., 1927, 98: 151-163

[5] Douady A. Disques de Siegel et anneaux de Herman. Astérisque, 1987, 152–153: 151-172

[6] Douady A, Hubbard J. On the dynamics of polynomial-like mappings. Ann. Sci. École Norm. Sup., 1985, 18(2): 287-344

[7] Fatou P. Sur les équations fonctionelles. Bull. Soc. Math. France., 1919, 47: 161-271 48, 1920, 33–94, 208–314

[8] Geyer L. Linearization of structurally stable polynomials. Progress in Holomorphic Dynamics, Pitman Res. Notes Math. Ser., 1998, 387: 27-30 Longman Harlow

[9] Geyer L. Siegel discs, Herman rings and the Arnold family. Trans. Amer. Math. Soc., 2001, 353(9): 3661-3683

[10] Geyer L. Linearizability of saturated polynomials. Indiana Univ. Math. J., 2019, 68(5): 1551-1578

[11] Goldberg L, Milnor J. Fixed points of polynomial maps, part II, Fixed point portraits. Ann. Sci. Éc. Norm. Supér., 1993, 26(1): 51-98

[12] Hardy G H, Wright E M. An Introduction to the Theory of Numbers, 1979 5th ed. Oxford: Oxford Univ. Press

[13] Julia G. Sur quelques problèmes relatifs à l’itération des fractions rationnelles. C. R. Acad. Sci., 1919, 168: 147-149

[14] Kiwi J. Non-accessible critical points of Cremer polynomials. Ergodic Theory Dynam. Systems, 2000, 20(5): 1391-1403

[15] Kœnigs, G., Recherches sur les intégrales de certaines équations fonctionelles, Ann. Sci. Éc. Norm. Supér., 1884, 1–41.

[16] Leau L. Étude sur les equations fonctionelles à une ou plusièrs variables. Ann. Fac. Sci. Toulouse, 1897, 11: E1-E110

[17] Manlove, J. M., Allowable rotation numbers for Siegel disks of rational maps, Ph.D. thesis, Montana State University, 2015.

[18] McMullen C T. Complex Dynamics and Renormalization, 1994, New Jersey: Princeton University Press

[19] Milnor J. Local connectivity of Julia sets: Expository Lectures, The Mandelbrot Set Theme and Variations, 2000, Cambridge: Cambridge University Press 67-116

[20] Milnor J. Dynamics in One Complex Variable, 2006 3rd ed. New Jersey: Princeton University Press

[21] Naishul V I. Topological invariants of analytic and area preserving mappings and their application to analytic differential equations in C ² and CP ². Trans. Moscow Math. Soc., 1983, 42: 239-250

[22] Okuyama Y. Non-linearizability of n-sub hyperbolic polynomials at irrationally indifferent fixed points. J. Math. Soc. Japan, 2001, 53(4): 847-874

[23] Pérez-Marco, R., Sur la dynamique des germes de difîéomorphismes holomorphes de ($\mathbb{C}$, 0) et des difféomor phismes analytiques du cercle, Thèse Université de Paris-Sud, décembre, 1990.

[24] 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). Séminaire Bourbaki, n° 753, Astérisque, 1992, 206: 273-310

[25] Pérez-Marco R. Sur les dynamiques holomorphes non-linéarisables et une conjecture de V.I. Arnold. Ann. Sci. Éc. Norm. Supér., 1993, 26(5): 565-644

[26] Pérez-Marco R. Fixed points and circle maps. Acta Math., 1997, 179(2): 243-294

[27] Pérez-Marco R. Total convergence or general divergence in small divisors. Comm. Math. Phys., 2001, 223(3): 451-464

[28] Petracovici L. Cremer fixed points and small cycles. Trans. Amer. Math. Soc., 2005, 357(9): 3481-3491

[29] Pommerenke C. Boundary Behaviour of Conformal Maps, 1992, New York: Springer-Verlage

[30] Rüssmann H. Über die iteration analytischer funktionen. J. Math. Mech., 1967, 17: 523-532

[31] Siegel C L. Iteration of analytic functions. Ann. of Math., 1942, 43(4): 607-612

[32] Steinmetz N. Rational Iteration: Complex Analytic Dynamical Systems, De Gruyter Studies in Mathematics, 1993, Berlin: Walter de Gruyter & Co.

[33] Yoccoz J C. Linérisation des germes de difîéomorphismes holomorphes de ($\mathbb{C}$, 0). C. R Acad. Sci. Paris, 1988, 306: 55-58

[34] Yoccoz J C. Theorème de Siegel, nombres de Brjuno et polynômes quadratiques. Astérisque, 1995, 231: 3-88

[35] Zakeri S. Dynamics of cubic Siegel polynomials. Comm. Math. Phys., 1999, 206: 185-233