arXiv Analytics

Sign in

arXiv:math/0408041 [math.DS]AbstractReferencesReviewsResources

Siegel Disks and Periodic Rays of Entire Functions

Lasse Rempe

Published 2004-08-03, updated 2008-05-20Version 5

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk such that f restricts to a homeomorphism of the boundary. We show that the Siegel disk is bounded. Using a result of Herman, we deduce that if additionally the rotation number of the Siegel disk is Diophantine, then its boundary contains a critical point of f. Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk $U$ whose boundary contains no singular values, then the condition that f is a homeomorphism of the boundary of U is automatically satisfied. We also investigate landing properties of periodic dynamic rays by similar methods.

Comments: 22 pages, 4 figures. A problem with the image quality of some of the figures was fixed. Some minor corrections were also made. Final version
Journal: J. Reine Angew. Math. 624, 81-102 (2008).
Categories: math.DS
Subjects: 37F10, 30D05
Related articles: Most relevant | Search more
arXiv:math/0505652 [math.DS] (Published 2005-05-30, updated 2006-05-17)
On Newton's Method for Entire Functions
arXiv:1505.06291 [math.DS] (Published 2015-05-23)
Baker Omitted Value
arXiv:math/0610453 [math.DS] (Published 2006-10-15, updated 2007-03-16)
On a question of Eremenko concerning escaping components of entire functions

Links

Toolbox