Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn
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Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn Tomoki Kawahira1,2 Dedicated to Lawrence Zalcman on the occasion of his 75th birthday. Received: 31 July 2019 / Revised: 31 July 2019 / Accepted: 13 February 2020 © Springer Nature Switzerland AG 2020
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.
1 Similarity between M and J The aim of this paper is to give a new perspective on a well-known similarity between the Mandelbrot set and Julia sets (Tan’s theorem) in terms of Zalcman’s rescaling principle in non-normal families of meromorphic functions. We start with a simplified proof of Tan’s theorem [22] following [8], which motivates the whole idea of this paper. The Mandelbrot set and the Julia sets Let us consider the quadratic family
f c (z) = z 2 + c : c ∈ C .
Partially supported by JSPS.
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Tomoki Kawahira [email protected]
1
Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
2
Mathematical Science Team, RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan 0123456789().: V,-vol
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T. Kawahira
The Mandelbrot set M is the set of c ∈ C such that the sequence f cn (c) n∈N is bounded. For each c ∈ C, the filled Julia set K c is the set of z ∈ C such that the sequence f cn (z) n∈N is bounded. One can easily check that • c∈ / M if and only if | f cn (c)| > 2 for some n ∈ N; and • for each c ∈ M, z ∈ / K c if and only if | f cn (z)| > 2 for some n ∈ N. The Julia set Jc is the boundary of K c . Note that all M, K c , and Jc are compact, and also non-empty because we can always solve the equations f cn (c) = c and f cn (z) = z. Tan showed in [22] (originally in a chapter of [2]) that when c0 ∈ M is a Misiurewicz parameter (to be defined below), the “shapes” of M and the Julia set Jc0 are asymptotically similar at the same point c0 . For example, (JM1) of Fig. 1 shows M and Jc0 in squares centered at c0 = i, whose widths range from 6.0 to 0.01. We will prove this by finding an entire function that bridges the dynamical plane and the parameter plane (Lemma 1). Misiurewicz parameters and a key lemma Following the terminology of [2] and [22], we say c0 ∈ M is a Misiurewicz parameter if the forward orbit of c0 by f c0 eventually lands on a repelling periodic point. More precisely, there exist minimal l ≥ 1 and p ≥ 1 p p such that a0 := f cl0 (c0 ) satisfies a0 = f c0 (a0 ) and |( f c0 ) (a0 )| > 1. By the implicit function theorem, we can show that the repelling periodic point a0 is stable: that is, there exists a neighborhood V of c0 and a holomorphic
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