Geometric limits of Julia sets for sums of power maps and polynomials

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.

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Acknowledgements

The authors are grateful to Roland Roeder at Indiana University Purdue University Indianapolis for his very helpful advice and the Butler University Mathematics Research Camp, where this project began. We are also very grateful to the referee for their insightful and helpful suggestions. All images we created with the Dynamics Explorer [3] program.

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Appendix

Appendix

Proof of Lemma  4

Note that

$$\begin{aligned} \mu =dd^c\log _{+}|z|,\qquad \text{ where }\qquad \log _{+}|z|=\left\{ \begin{array}{rl} \log |z|,&{} \text{ if } |z|\ge 1\\ 0,&{} \text{ if } |z|<1.\end{array}\right. \end{aligned}$$

Let \(Z_q\) be the zero set of q and A be the maximum of |q(z)| on \(\overline{\mathbb D}\). Let K be a compact subset of \({\mathbb C}\backslash (S_0\cup Z_q)\); then there is an \(\epsilon >0\) such that for any \(z\in K\), we have \(|q(z)|>\epsilon\) and either \(|z|\ge 1+\epsilon\) or \(|z|\le 1-\epsilon\).

If \(|z|\ge 1+\epsilon\), then

$$\begin{aligned} \frac{1}{n}\log |f_{n}(z)|=\, & {} \frac{1}{n}\log \left| z^n\left( 1+\frac{q(z)}{z^n}\right) \right| \\=\, & {} \frac{1}{n}\log |z^n|+\frac{1}{n}\log \left| 1+\frac{q(z)}{z^n}\right| \nonumber \ \le \ \log _{+}|z|+\frac{C}{n}, \end{aligned}$$
(1)

where \(C=\log (1+A/(1+\epsilon ))\).

If \(|z|<1-\epsilon\), then there is an N such that for all \(n\ge N\), we have \(z^n\le \max \{\epsilon /2,A\}\). Then

$$\begin{aligned} \frac{\epsilon }{2}\le |q(z)|-|z^n|\ \le \ |f_{n}(z)|\le & {} |z^n|+|q(z)|\le 2A. \end{aligned}$$

Noting that \(\log _{+}|z|=0\) when \(|z|\le 1-\epsilon\), we have for all \(|z|\le 1+\epsilon\) that

$$\begin{aligned} \frac{1}{n}\log |f_{n}(z)|\le & {} \frac{1}{n}\log (2A)\ =\ \log _{+}|z|+\frac{\log (2A)}{n}. \end{aligned}$$
(2)

Using Eqs. (1) and (2), we have \(\frac{1}{n}\log |f_{n}(z)|\rightarrow \log _{+}|z|\) uniformly on K as \(n\rightarrow \infty\); by the compactness theorem for families of subharmonic functions [10][Theorem 4.1.9], it follows that \(\frac{1}{n}\log |f_{n}(z)|\rightarrow \log _{+}|z|\) in \(L^1_{loc}({\mathbb C})\). Note that \(dd^c\log _{+}|z|=\mu\), and we have from the Poincarè-Lelong formula [8] that \(\frac{1}{n}dd^c\log |f_{n}(z)|=\mu _n\). Thus, we have

$$\begin{aligned} \mu _n=\frac{1}{n}dd^c\log |f_{n}(z)|\rightarrow dd^c\log _{+}|z|=\mu \end{aligned}$$

weakly as \(n\rightarrow \infty\). \(\square\)

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Brame, M., Kaschner, S. Geometric limits of Julia sets for sums of power maps and polynomials. Complex Anal Synerg 6, 22 (2020). https://doi.org/10.1007/s40627-020-00059-4

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Keywords

  • Complex dynamics
  • Geometric limits
  • Polynomial dynamics

Mathematics Subject Classification

  • 37F10
  • 37F40