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Cremer fixed points and small cycles

Authors:	Lia Petracovici

Affiliation:	Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Abstract:	Let $lambda= e^{2pi i alpha}$ , , and let denote the sequence of convergents to the regular continued fraction of . Let 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 we simultaneously have (i) $log log q_{n+1} geq 3log q_n$ , (ii) the coefficients $a_{1+q_n}$ stay outside two small disks, and (iii) the series is lacunary, with for $2+q_nleq j leq q_n^{1+q_n}-1$ . We then prove that has infinitely many periodic orbits in every neighborhood of the origin.

Keywords:	Cremer fixed point periodic orbit

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