Singular values and non-repelling cycles for entire transcendental maps

2017/12/01 by Benini, Anna Miriam, Fagella, Núria
#30D05 #30D30 #37F10 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.00273

Abstract

Let f be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When f has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.

Related

• A bound on the number of rationally invisible repelling orbits
• A Landing Theorem for Periodic Dynamic Rays for Transcendental Entire Maps with Bounded Post-Singular Set
• Sur les points fixes et les cycles répulsifs au voisinage d'une singularité essentielle isolée à l'instar de la méthode de Zalcman
• Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality
• Infinite-dimensional Thurston theory and transcendental dynamics IV:\n dependence on parameters and escape on (pre-)periodic rays