Expansivity properties and rigidity for non-recurrent exponential maps

2012/10/04 by Anna Miriam Benini, Benini, Anna Miriam
Mathematics · #37F10 #37F15 #37F20 #37F25 #37F45 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37F10 #msc:37F15 #msc:37F20 #msc:37F25 #msc:37F45

paper · pdf · doi:10.48550/arxiv.1210.1353

arxiv created 2014/08/07 · arxiv updated 2014/08/08

Abstract

We show that an exponential map f_c(z)=e^z+c whose singular value c is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if c is in the Julia set.

Related

• Poincare Series and instability of exponential maps
• Non-recurrent rational maps with disconnected Julia set
• Ergodic exponential maps with escaping singular behaviours
• Dynamics of generalised exponential maps
• Recurrence of entire transcendental functions with simple post singular sets