2008/01/23 by P. J. Rippon, Rippon, P. J., G. M. Stallard +1 · 1 citation
Mathematics · #30D05 #37F10 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS #msc:30D05 #msc:37F10
paper · pdf · doi:10.48550/arxiv.0801.3605
arxiv created 2008/01/23 · arxiv updated 2009/12/01
Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.