A separation theorem for entire transcendental maps

TL;DR

This paper extends the Goldberg-Milnor separation theorem to entire transcendental functions of finite order with bounded singular values, revealing new structural insights about their periodic points and Fatou components.

Contribution

It generalizes the Goldberg-Milnor theorem to a broader class of transcendental maps, establishing a finite partition of the plane by invariant rays and their landing points.

Findings

Invariant rays and landing points separate the plane into regions with unique interior periodic points.

Two periodic Fatou components can be separated by a pair of landing rays.

There are finitely many non-repelling cycles of any given period.

Abstract

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that "hidden components" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.