A separation theorem for entire transcendental maps
Benini, Anna Miriam (CRM Ennio de Giorgi (Italy))
Fagella Rabionet, Núria (Universitat de Barcelona. Departament de Matemàtica Aplicada i Anàlisi)

Data:	2015
Resum:	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.
Ajuts:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-792 Ministerio de Economía y Competitividad MTM-2008-01486 Ministerio de Economía y Competitividad MTM2006-05849 Ministerio de Economía y Competitividad MTM2011-26995-C02-02
Nota:	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
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Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Publicat a:	Proceedings of the London Mathematical Society. Third Series, Vol. 110 (2015) , p. 291-324, ISSN 0024-6115

DOI: 10.1112/plms/pdu047

Postprint 39 p, 918.6 KB

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