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Singular values and bounded Siegel disks

Published online by Cambridge University Press:  25 May 2017

ANNA MIRIAM BENINI  and
NÚRIA FAGELLA
ANNA MIRIAM BENINI
Affiliation:
Dip. di Matematica, Universita' di Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy. e-mail: ambenini@gmail.com
NÚRIA FAGELLA
Affiliation:
Dept. de Matemàtiques i Informàtica, Barcelona Graduate School of Mathematics (BGSMath), Gran Via 585, 08007 Barcelona, Spain. e-mail: fagella@maia.ub.es
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Abstract

Let f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class $\mathcal{H}$. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 249 - 265
Copyright
Copyright © Cambridge Philosophical Society 2017 

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