Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn

Abstract

We present a simple proof of Tan’s theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman’s lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan’s theorem.

Appendix: Existence of the Poincaré function

Here we give a proof of the existence of the Poincaré functions associated with repelling periodic points. This is originally shown by using a local linearization theorem by Koenigs. See [14, Cor.8.12]. Our proof is based on the normal family argument and the univalent function theory (see [4] for example), which follows the idea of [11, Lemma 4.7].

Theorem 12

Let $g:\mathbb {C}\rightarrow \mathbb {C}$ be an entire function with $g(0) = 0,~g'(0) = {\lambda }$, and $|{\lambda }| >1$. Then the sequence $\phi _n(w) = g^n(w/{\lambda }^n)$ converges uniformly on compact sets in $\mathbb {C}$. Moreover, the limit function $\phi :\mathbb {C}\rightarrow \mathbb {C}$ satisfies $g \circ \phi (w) = \phi ({\lambda }w)$ and $\phi '(0) = 1$.

Proof

Since $g(z) = {\lambda }z +O(z^2)$ near $z = 0$, there exists a disk $\Delta = \mathbb {D}(\delta ) = {\left\{ z \in \mathbb {C}\,:\,|z| < \delta \right\} }$ such that $g|\Delta$ is univalent and $\Delta \Subset g(\Delta )$. Hence we have a univalent branch $g_0^{-1}$ of g that maps $\Delta$ into itself.

First we show that $\phi _n$ is univalent on $\mathbb {D}(\delta /4)$: Since the map $\phi _n^{-1}:w \mapsto {\lambda }^n g_0^{-n}(w)$ is well-defined on $\Delta = \mathbb {D}(\delta )$ and univalent, its image contains $\mathbb {D}(\delta /4)$ by the Koebe 1/4 theorem. Hence $\phi _n$ is univalent on $\mathbb {D}(\delta /4)$, and by the Koebe distortion theorem, the family ${\left\{ \phi _n \right\} }_{n \ge 0}$ is locally uniformly bounded on $\mathbb {D}(\delta /4)$ and thus equicontinuous.

Next we show that $\phi _n$ has a limit on $\mathbb {D}(\delta /4)$: Fix an arbitrarily large $r > 0$ and an integer N such that $r < \delta |{\lambda }|^N /4$. By using the Koebe 1/4 theorem as above, the function $G_{N,k}(w):= {\lambda }^N g^{k}(w/{\lambda }^{N + k})~(k \in \mathbb {N})$ satisfying $\phi _{N + k} =\phi _N \circ G_{N,k}$ is univalent on the disk $\mathbb {D}(\delta |{\lambda }|^N/4)$. By the Koebe distortion theorem, there exists a constant $C>0$ independent of N and k such that for any $w \in \mathbb {D}(r)$ and sufficiently large N we have $|G_{N,k}'(w)-1| \le C |w|/|{\lambda }|^N$. By integration we have $|G_{N,k}(w)-w| \le C r^2/(2 |{\lambda }|^N)$ on $\mathbb {D}(r)$. In particular, $G_{N,k} \rightarrow \mathrm {id}$ uniformly on $\mathbb {D}(\delta /4)$ as $N \rightarrow \infty$. Since the family ${\left\{ \phi _n \right\} }$ is equicontinuous on $\mathbb {D}(\delta /4)$, the relation $\phi _{N + k} =\phi _N \circ G_{N,k}$ implies that ${\left\{ \phi _n \right\} }_{n \ge 0}$ is Cauchy and has a unique limit $\phi$ on any compact sets in $\mathbb {D}(\delta /4)$.

Let us check that the convergence extends to $\mathbb {C}$: (We will not use the functional equation $g^n \circ \phi (w) = \phi ({\lambda }^n w)$. Compare [14, Cor.8.12].) Since $|\phi _{N + k}(w)-\phi _N(w)| = |\phi _N(G_{N,k}(w))-\phi _N(w)|$ and $|G_{N,k}(w)-w| = C r^2/(2 |{\lambda }|^N)$ on $\mathbb {D}(r)$, it follows that the family ${\left\{ \phi _{N + k} \right\} }_{k \ge 0}$ (with fixed N) is uniformly bounded on $\mathbb {D}(r)$. Hence ${\left\{ \phi _n \right\} }_{n \ge 0}$ is normal on any compact set in $\mathbb {C}$ and any sequential limit coincides with the local limit $\phi$ on $\mathbb {D}(\delta /4)$.

The equation $g \circ \phi (w) = \phi ({\lambda }w)$ and $\phi '(0) = 1$ are immediate from $g \circ \phi _{n}(w) = \phi _{n + 1}({\lambda }w)$ and $\phi _n'(0) = 1$. $\square$

Remark

One can easily extend this proof to the case of meromorphic g by using the spherical metric.