Geometric limits of Julia sets for sums of power maps and polynomials
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RESEARCH
Geometric limits of Julia sets for sums of power maps and polynomials Micah Brame1 · Scott Kaschner1 Received: 17 April 2019 / Accepted: 28 July 2020 © Springer Nature Switzerland AG 2020
Abstract For maps of one complex variable, f, given as the sum of a degree n power map and a degree d polynomial, we provide necessary and sufficient conditions that the geometric limit as n approaches infinity of the set of points that remain bounded under iteration by f is the closed unit disk or the unit circle. We also provide a general description, for many cases, of the limiting set. Keywords Complex dynamics · Geometric limits · Polynomial dynamics Mathematics Subject Classification 37F10 · 37F40
1 Introduction
for any |c| < 1, lim K(fn ) n→∞
Let q be a degree d polynomial; define fn ∶ ℂ → ℂ by n
fn (z) = z + q(z), and note that fn is the sum of a power map (whose power we increase in the limit) and a fixed degree d polynomial, q. For a map f ∶ ℂ → ℂ , the filled Julia set for f, K(f), is the set of points that remain bounded under iteration by f. We use the notation S0 = {z ∈ ℂ ∶ |z| = 1} for the unit circle and 𝔻 = {z ∈ ℂ ∶ |z| ≤ 1} for the closed unit disk. The purpose of this study is to describe the limit of K(fn ) in the Hausdorff topology as n → ∞. This work was inspired the 2012 study by Boyd and Schulz [4] that included a result for the family fn with deg q = 0 ; that is, q(z) = c ∈ ℂ . Among many other things, they proved Theorem 1 (Boyd-Shulz [4]) If q(z) = c, then under the Hausdorff metric,
* Scott Kaschner [email protected] Micah Brame [email protected] 1
Butler University, 4600 Sunset Ave., Indianapolis, IN 46208, USA
= 𝔻;
for any |c| > 1, lim K(fn ) = S0 . n→∞
It comes as little surprise that this phenomena is easily disrupted. It was shown in [13] that when q(z) = c with |c| = 1 , the limiting behavior of K(fn ) depends on numbertheoretic properties of c and the limit almost always fails to exist. Another study by Alves [1], has shown that for maps of the form fn,c (z) = zn + czk for a fixed positive integer k, if |c| < 1 , then the limit of K(fn,c ) as n → ∞ is S0. Returning to the more general case in which q is any polynomial, the limiting behavior of K(fn ) is substantially more interesting. See Fig. 1 for examples of filled Julia sets for fn , where q(z) = z2 + c and |c| < 1 , that very clearly fail to limit to either the closed unit disk or the unit circle. The color gradation in the pictures indicates the number of iterates required to exceed a fixed bound for modulus. Some results from the deg q = 0 cases still hold. If |z| > 1 , we can still expect the image of z under fn to have large modulus for large enough n. Guided by this intuition, we find the following generalization of a lemma from [4]. We omit the proof, as it is similar to [4], and adopt the notation
𝔻r = {z ∈ ℂ ∶ |z| < r} and 𝔻r = {z ∈ ℂ ∶ |z| ≤ r}.
Lemma 1 For any polynomial q and any 𝜖 > 0, there is an N ≥ 2 such that for all n ≥ N,
K(fn ) ⊂ 𝔻1+𝜖 .
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