2019/07/29 by Anna Miriam Benini, Benini, Anna Miriam, Núria Fagella +1
Mathematics · #30D05 #37F10 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:30D05 #msc:37F10
paper · pdf · doi:10.48550/arxiv.1907.12310
arxiv created 2019/07/29 · arxiv updated 2019/07/30
We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are q<∞ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by q. In particular, there are at most q rationally invisible repelling periodic orbits. The techniques presented here also apply to the more general setting in which the function is allowed to have infinitely many singular values.