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)

 

 

Cremer fixed points and small cycles


Author: Lia Petracovici
Journal: Trans. Amer. Math. Soc. 357 (2005), 3481-3491
MSC (2000): Primary 37F50
Published electronically: August 11, 2004
MathSciNet review: 2146634
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\lambda= e^{2\pi i \alpha}$, $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, and let $(p_n/q_n)$ denote the sequence of convergents to the regular continued fraction of $\alpha$. Let $f$ be a function holomorphic at the origin, with a power series of the form $f(z)= \lambda z+\sum _{n=2}^{\infty}a_nz^n$. We assume that for infinitely many $n$ we simultaneously have (i) $\log \log q_{n+1} \geq 3\log q_n$, (ii) the coefficients $a_{1+q_n}$ stay outside two small disks, and (iii) the series $f(z)$ is lacunary, with $a_j=0$ for $2+q_n\leq j \leq q_n^{1+q_n}-1$. We then prove that $f(z)$ has infinitely many periodic orbits in every neighborhood of the origin.


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

  • 1. Lukas Geyer, Linearization of structurally stable polynomials, Progress in holomorphic dynamics, Pitman Res. Notes Math. Ser., vol. 387, Longman, Harlow, 1998, pp. 27–30. MR 1643012
  • 2. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
  • 3. John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
  • 4. Ricardo Pérez Marco, Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnol′d, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 5, 565–644 (French, with English summary). MR 1241470
  • 5. John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096594
  • 6. Jean-Christophe Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231 (1995), 3–88 (French). Petits diviseurs en dimension 1. MR 1367353

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37F50

Retrieve articles in all journals with MSC (2000): 37F50


Additional Information

Lia Petracovici
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, Western Illinois University, 1 University Circle, Macomb, Illinois 61455
Email: petracvc@math.uiuc.edu, L-Petracovici@wiu.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03539-1
Keywords: Cremer fixed point, periodic orbit
Received by editor(s): May 28, 2002
Received by editor(s) in revised form: October 14, 2003
Published electronically: August 11, 2004
Additional Notes: The author was supported by NSF Grants # DMS-9970281 and # DMS-9983160
Article copyright: © Copyright 2004 American Mathematical Society