A separation theorem for entire transcendental maps

Abstract

Agraïments: Anna Miriam Benini was partially supported by the ERC grant HEVO - Holomorphic Evolution Equations n. 277691. Both authors were supported by the European network MRTN-CT-2006-035651-2-CODY MRTN-CT-2006-035651-2-CODY

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 ∈ 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 [GM], 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 disks; that "hidden components" of a bounded Siegel disk have to be either wandering domains or preperiodic to the Siegel disk itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.