A separation theorem for entire transcendental maps

Details A separation theorem for entire transcendental maps A separation theorem for entire transcendental maps

2011-12-02

arxiv.org/abs/1112.0531v1

Mathematics

Dynamical Systems

30 pages

Scientific paper

We study the distribution of periodic points for a certain class of holomorphic functions. An interior periodic point of period $p$ is a periodic point which is not the landing point of any periodic ray of period less or equal than $p$, like for example an attracting cycle or a Cremer cycle. In polynomial dynamics, it is known by a theorem of Goldberg and Milnor that periodic rays, together with their landing points, separate the plane into regions each containing exactly one interior periodic point or a parabolic immedite basin. We show an analogue of this theorem for a wide class of transcendental entire functions (those with a bounded set of singular values and finite order, or composition thereof), under the assumption that periodic rays land. This result has many corollaries. Among them, it follows that there cannot be Cremer points on the boundary of Siegel disks; that in the absence of wandering domains "hidden components" of a Siegel disk have to be preperiodic to the Siegel disk itself; or that there cannot be infinitely many attracting cycles of period less than any finite constant.

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