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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
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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.

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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

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