Geometric Limits of Julia Sets with Parameters on the Circle

2014/11/10 by Kaschner, Scott R., Romero, Reaper, Simmons, David
#37F10 #37F40 #37F45 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.2848

Abstract

We show that the geometric limit as n → ∞ of the filled Julia sets K(P_n,c) for the maps P_n,c(z) = zⁿ + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle, and this is used to show that for certain parameters, the geometric limit of the Julia sets J(P_n,c) is the unit circle.

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