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

Lemma 1 Suppose that c0 ∈M is a Misiurewicz parameter as above. For k ∈N, set ρk := 1/(f l+kp c0 )′(c0). Then we have the following. (1) The function φk(w) = f l+kp c0 (c0 + ρkw) converges to a non-constant entire function φ : C →C as k →∞uniformly on any compact sets. (2) There exists a constant Q ̸= 0 such that the function Φk(w) := f l+kp c0+Qρkw(c0 + Qρkw) converges to the same function φ(w) as k →∞uniformly on compact sets of C.

Theorem 12 in Appendix. Such a φ is called a Poincar´e function. Indeed, φ satisfies the functional equation φ(λ0w) = f p c0 ◦φ(w), but we will not use it.) Note that this function satisfies φ(0) = a0 and φ′(0) = 1. Now let us show (1): set A0 := (f l c0)′(c0), where A0 ̸= 0 since otherwise c0 is strictly periodic. We also have (f l+kp c0 )′(c0) = A0λk 0 = 1/ρk. For sufficiently small t ∈C, we have the expansion f l c0(c0 + t) = a0 + A0 · t + o(t). Fix an arbitrarily large compact set E ⊂C and take

Lemma 1 implies that c0 ∈Jc0 = ∂Kc0 and c0 ∈∂M. Indeed, we can find a w ∈C such that |φ(w)| > 2 and hence |φk(w)| > 2 for sufficiently large k. Equivalently, we have c0 + ρkw /∈Kc0 for sufficiently large k, where c0 + ρkw tends to c0 as k →∞. Since c0 ∈Kc0 by definition, we have c0 ∈Jc0. The proof for c0 ∈∂M is analogous. The Hausdorfftopology. Let us briefly recall the Hausdorfftopology of the set of non-empty compact sets Comp∗(C) of C. For a sequence {Kk}k∈N ⊂Comp∗(C), we say Kk converges to K ∈Comp∗(

Theorem 2 (Similarity between M and J) There exist a non-constant entire func- tion φ on C, a sequence ρk →0, and a constant q ̸= 0 such that if we set J := φ−1(Jc0) ⊂C, then for any large constant r > 0, we have (a)  ρ−1 k (Jc0 −c0)  r →[J ]r, and (b)  ρ−1 k q(M −c0)  r →[J ]r

Proposition 3 (J and ∂M as non-normality loci) For the quadratic family fc(z) = z2 + c (c ∈C), the Julia sets and the boundary of the Mandelbrot set are characterized as follows: 6

Proposition 4 (Invariance) For each z0 ∈Jc, the family Zc(z0) satisfies fc ◦Zc(z0) = Zc(z0) = Zc(z0) ◦Aff. More precisely, (1) If φ ∈Zc(z0) then fc ◦φ ∈Zc(z0) and φ = fc ◦φ1 for some φ1 ∈Zc(z0). (2) For any A ∈Affand φ ∈Zc(z0), we have φ ◦A±1 ∈Zc(z0). Note that the universal space U only satisfies fc ◦U ⊂U and U ◦Aff= U.

Proposition 4 implies that we also have fc ◦Zc = Zc = Zc ◦Aff. However, the equality Zc = Zc(z0) holds for any z0 ∈Jc in most cases. To see this, we introduce some terminology: The univalent grand orbit UGO(z0) of z0 ∈C is the set of ζ such that f m c (z0) = f n c (ζ) for some m, n ∈N and there is a univalent branch g of f −n c ◦f m c in a neighborhood of z0 with g(z0) = ζ. The postcritical set Pc of fc is the closure of the orbit {c, fc(c), f 2 c (c), · · ·}. (Note that c is a unique critical va

Theorem 5 If fc satisfies (∗)-condition, then Zc = Zc(z0) for any z0 ∈Jc. Moreover, the set of such c contains C −∂M, which is an open and dense subset of C, and the Misiurewicz parameters in ∂M except c0 = −2.

Proposition 6 (Invariance for P) For each c0 ∈∂M, the family P(c0) satisfies fc0 ◦P(c0) = P(c0) = P(c0) ◦Aff. Hence we only have P = P ◦Afffor the total space P of the parametric Zalcman functions. In a forthcoming paper we will present a general account on dynamical and para- metric Zalcman functions for families of rational functions parametrized by Riemann surfaces. Dynamical-parametric intersection and similarity. By Proposition 4 and

Proposition 6 above, if c0 ∈Jc0 and c0 ∈∂M, then Zc0(c0) and P(c0) exhibit the same invariance in the universal space U. Hence one might expect that there exists some φ ∈Zc0(c0) ∩P(c0) when c0 ∈Jc0 ∩∂M. Indeed, the existence of such an intersection implies asymptotic similarity between Jc0 and M at c0:

Theorem 7 (Intersection implies similarity) Suppose that Zc0(c0)∩P(c0) ̸= ∅for some c0 ∈∂M. More precisely, there exist sequences of affine maps Ak, Bk ∈Affand positive integers mk, nk ∈N such that as k →∞we have (1) Both Ak and Bk converge to the same constant map c0 in U; and (2) Both f nk c0 (Ak(w)) and f nk Bk(w)(Bk(w)) converge to the same entire function φ(w) in U. 8

Theorem 8 (Intersection) For any Misiurewicz parameter c0, Zc0(c0) and P(c0) share at least one element φ ∈U. Note that the set of Misiurewicz parameters is a dense subset of ∂M. (This can be shown by a standard normal family argument. See Levin [Le] or [Ka, Th´eor`eme 1.1 (1)].) In [Ka] the author proved Lemma 1 for a wider class of parameters in ∂M, called semi-hyperbolic parameters. Shishikura proved in [Sh] that the set of semi-hyperbolic parameters is a dense subset of ∂M of Hausdorffdimensi

Theorem 9 (Similarity between T and J∗) There exist an entire function φ on C, a real linear transformation h : C →C, and a sequence ρk →0 such that if we set J ∗:= φ−1(J∗ c0) ⊂C, then for any large constant r > 0, we have (a)  ρ−1 k (J∗ c0 −c0)  r →[J ∗]r, and (b)  ρ−1 k h(T −c0)

Lemma 10 Suppose that c0 ∈T is a Misiurewicz parameter as above. For k ∈N, set ρk := 1/(g2l+2kp c0 )′(c0). Then we have the following. (1) The function φk(w) = g2l+2kp c0 (c0+ρkw) converges to a non-constant entire function φ : C →C as k →∞uniformly on any compact sets. (2) There exists a real linear transformation H(w) = Qw + Q′w with |Q| ̸= |Q′| such that the function Φk(w) := g2l+2kp c0+H(ρkw)(c0 + H(ρkw)) converges to the same function φ(w) as k →∞uniformly on compact sets of C. 10

Theorem 12 in Appendix below.) By the expansion g2l c0(c0 +t) = a0 +A0t+o(t) (t →0) we obtain φ(w) = lim n→∞g2l+2kp c0  c0 + w A0λk 0  . Hence we set ρk := (A0λk 0)−1 = 1/(g2l+2kp

Lemma 11 (Stability and transversality) (1) Stability: There exists a real analytic function c 7→a(c) defined near c0 with a(c0) = a0 such that a(c) is a repelling fixed point of g2p c , for which the multiplier λ(c) := (g2p c )′(a(c)) is also a real analytic function near c0. 12

Theorem 9 can be easily generalized to the unicritical antiholomorphic fam- ily  z 7→zd + c : c ∈C

Lemma 4.7].

Theorem 12 Let g : C →C be an entire function with g(0) = 0, g′(0) = λ, and |λ| > 1. Then the sequence φn(w) = gn(w/λn) converges uniformly on compact sets in C. Moreover, the limit function φ : C →C satisfies g ◦φ(w) = φ(λw) and φ′(0) = 1.