Geometric Limits of Julia Sets of Maps zⁿ + exp(2πiθ) as n → ∞

Abstract

We show that the geometric limit as n → ∞ of the Julia sets J(P n,c ) for the maps P n,c (z) = z n + 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.

On the dynamics of n-circle inversion

“…We are interested in the geometric asymptotical approach that allows us to understand deformations of the Julia set when n goes to infinity, for the family under study. In [14] the authors study the geometric limit of the Julia set and of the connected locus under degree growth for two families of rational maps (see also [18]). Inspired by this result, we prove that…”

On Dynamics of Asymptotically Minimal Polynomials

“…On the other hand, by [2, Theorem 1.2] if |c| < 1 then the filled Julia sets K pn converges to the closed unit disc D; however if |c| > 1 then the filled Julia sets converges to S 1 with respect to Hausdorff topology. Finally, for almost every c ∈ S 1 the filled Julia sets do not converge to any compact set [11]. Moreover, again by [2, Theorem 1.2] for any limit set K ∞ of the filled Julia sets K pn in the Hausdorff topology we have P c(K ∞ ) = D Recall that in section §3, we showed that if the counting measures of zeros of {p n } n are weak*-convergent, then the filled Julia sets K n 's are uniformly bounded.…”