Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:28 https://doi.org/10.1007/s12044-019-0469-9

Cross-sections of the multicorns

XIUMING ZHANG

Department of Mathematics, Nanjing University, No. 22, Hankou Road, Gulou District, Nanjing 210093, Jiangsu, China E-mail: [email protected]

MS received 13 November 2017; accepted 18 June 2018; published online 11 March 2019

Abstract. For each integer d ≥ 2, we identify the intersections of the connectedness + + locus of zd + c with the rays ωR ,whereωd 1 =±1.

Keywords. Multicorns; cross-sections.

2010 Mathematics Subject Classification. Primary: 37F45; Secondary: 37F10, 37F30.

1. Introduction The study of parameter spaces of holomorphic functions is one of the important topics in complex dynamics. Much work has been done on the parameter spaces of the unicritical polynomials z → zd + c and their generalizations, where d ≥ 2 is an integer. In this article, we are interested in the unicritical anti-holomorphic polynomials

d Pc(z) := z + c, where d ≥ 2 and c, z ∈ C.

For each given c ∈ C,thesetofallz ∈ C such that z has a bounded forward orbit under the iterates of Pc is called the filled-in Julia set of Pc, which is denoted by Kc. For each d ≥ 2, the multicorn is defined as

M ∗ := { ∈ C : }. d c Kc is connected

M ∗ An elementary argument (similar to the holomorphic case) shows that d has the following equivalent definition (see [8]):

M ∗ ={ ∈ C : { ◦n( )} }, d c the orbit Pc 0 n∈N is bounded (1)

◦n := ◦···◦ M ∗ where Pc Pc Pc is the n-th iterate of Pc. See fig. 1 for the multicorns d with d = 2, 3, 4 and 5. M ∗ It was Milnor who found the importance of the tricorn 2 in the slice of the parameter M ∗ plane of real cubic polynomials [6]. Later, Nakane proved in [8] that the tricorn 2 is connected. Actually, his proof can be adopted to all d ≥ 2 completely similarly since his

© Indian Academy of Sciences 1 28 Page 2 of 6 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:28

M ∗ = Figure 1. The multicorns d with d 2, 3, 4 and 5 respectively. It can be seen that M ∗ each d has rotation symmetries around the origin and the axes of reflections. method is analogous to Douady and Hubbard’s classical proof of the connectivity of the Mandelbrot set. The multicorns have received much attention in recent years, see [9], [2], [4], [7] and references therein. M ∗ M M There are many differences between the multicorns d and the multibrot sets d ( d d is the connected locus of z → z + c). For example, it is conjectured that Md is locally M ∗ connected but Hubbard and Schleicher proved that d is not path connected [2]. The M M ∗ boundary of the Mandelbrot set 2 has self-similarity everywhere but the tricorn 2 does not have at some boundary points [3]. In this article, we focus our attention on the most basic topological property of the M ∗ multicorns: the symmetry. From fig. 1, one can observe that each d has two kinds of symmetries: the rotation symmetries around the origin and the axes of reflections. By M ∗ + + Lemma 2.1, we know that each d has exactly d 1-fold rotation symmetries and d 1 axes of reflections. The main purpose of this article is to calculate the exact cross-sections M ∗ of d with the symmetric rays. For each d ≥ 2, we define

d 1 − − − αd := (d − 1)d d 1 ,βd := 2 d 1 (2) and

d − − γd := d d 1 (sinh(dξd ) + d sinh(ξd )), (3) Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:28 Page 3 of 6 28 where ξd is the unique positive root of the equation cosh(dξd ) = d cosh(ξd ).

Main Theorem. Let d ≥ 2 be an integer. The following statements hold: ωd+1 = M ∗ ∩ ωR+ ={ω : ∈[ ,α ]} (a) If 1, then d t t 0 d ; ωd+1 =− M ∗ ∩ ωR+ ={ω : ∈[ ,β ]} (b) If d is even and 1, then d t t 0 d ; and ωd+1 =− M ∗ ∩ ωR+ ={ω : ∈[ ,γ ]} (c) If d is odd and 1, then d t t 0 d .

Here we use the notation R+ := [0, +∞). We would like to mention that some similar results on the symmetries of the Multibort sets Md with d ≥ 2 have been established in [1,10,11]. In fact, we will see later that all the cases in the Main Theorem can be reduced to those of the Multibort sets Md , and our main work in this article is to establish these connections. 3 When d = 3, one√ can use the relation cosh(3x) = 4 cosh(x) − 3 cosh(x) to derive the exact formula γ3 = 32/27, which yields the following Corollary.

COROLLARY 1.1  ∗ πi πi M ∩ 4 R ={ 4 :||≤ / }. 3 e e t t 32 27

We collect here some notations which will be used throughout this article. We use N, R and C to denote the natural, real and complex numbers respectively. The positive and negative real axes (with end {0}) are denoted by R+ := [0, +∞) and R− := (−∞, 0] respectively. For a ∈ C and X ⊂ C, we denote aX := {ax : x ∈ X}.

2. Proof of the main theorem M ∗ In this section, we first give all the possible symmetric rays of d , including rotation M ∗ symmetries and axes of reflections. Then we show that the cross-sections of d with these symmetric rays can be reduced to those of Md . Finally, we complete the proof of the Main Theorem by using some known results on Md .

M ∗ + Lemma 2.1. The multicorn d has exactly d 1-fold rotation symmetries around the origin and d + 1 axes of reflections. Specifically, M ∗ = ωM ∗ ωd+1 = (a) d d if and only if 1; and M ∗ = ω2M ∗ ωd+1 =− (b) d d if and only if 1.

Proof. Suppose that ωd+1 = 1 and denote φ(z) := ωz. Then we have

(ω ) (ω )d −1 Pωc z z d φ ◦ Pω ◦ φ(z) = = + c = z + c = P (z). c ω ω c

This means that Pc is conjugate to Pωc by the rotation z → ωz and we have {| ◦n( )|} < ∞ {| ◦n( )|} < ∞ supn Pc 0 if and only if supn Pωc 0 . According to the definition M ∗ ∈ M ∗ {| ◦n( )|} < ∞ ∈ M ∗ of d in (1), we have c d if and only if supn Pc 0 . Hence c d if and ω ∈ M ∗ only if c d . This proves the sufficiency of part (a). 28 Page 4 of 6 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:28

→ M ∗ Since Pc is conjugate to Pc by z z, it means that d is symmetric about the real M ∗ = M ∗ ∈ C axis, i.e. d d . A direct calculation shows that two points z, z are symmetric about a line eiθ R if and only if z = e2iθ z¯, where θ ∈ R. Therefore, if ωd+1 =−1, then (ω2)d+1 = M ∗ = ω2M ∗ = ω2M ∗ M ∗ 1 and by part (a), we have d d d . It follows that d is symmetric about the line ωR if ωd+1 =−1 and the sufficiency of part (b) is proved. The necessity of parts (a) and (b) has been proved in [12] and a simplified argument was given in [5, p. 47]. For completeness, we give a sketch of the proof here. For each d ≥ 2, we define

d := {c ∈ C : Pc has an attracting fixed point}.

Then one can check that d is a Jordan domain and ∂ d contains exactly d +1 cusps which − d are uniformly distributed on the circle centered at the origin with radius (d + 1)d d−1 . M ∗ {ω : This means that d and hence d cannot have other rotation symmetries except ωd+1 = 1}. Then the necessity is proved since additional axes of reflections cannot exist without additional rotation symmetries. 

Recall that Md is the Multibrot set defined as

M := { ∈ C : { ˜ ◦n( )} }, d c the orbit Pc 0 n∈N is bounded

˜ d where Pc(z) = z +c and d ≥ 2. Recall that αd , βd and γd are constants depending only on d defined in (2) and (3). The following results on the cross-sections of Md with symmetric rays can be found in [11, Theorem 3, p. 7], [10, Theorem 2, p. 40] and [1, Theorem 1.1].

Lemma 2.2 [1,10,11]. Let d ≥ 2 be an integer. Then

(a) If d is odd, Md ∩ R =[−αd ,αd ]; (b) If d is even, Md ∩ R =[−βd ,αd ]; and d−1 + (c) If d is odd and ω =−1, Md ∩ ωR ={ωt : t ∈[0,γd ]}.

M ∗ We will use Lemma 2.2 to obtain some corresponding results on d .

≥ M ∗ ∩ R = M ∩ R Lemma 2.3. For each d 2, we have d d .

M ∗ M Proof. By the definitions of d and d ,wehave

M ∗ ∩ R ={ ∈ R : {| ◦n( )|} < ∞} d c sup Pc 0 n ={ ∈ R : {| ˜ ◦n( )|} < ∞} = M ∩ R.  c sup Pc 0 d n Proof of the Main Theorem. ≥ M ∗ ∩ R+ = M ∩ (a) By Lemma 2.2(a), (b) and Lemma 2.3, for all d 2, we have d d R+ =[,α ] M ∗ + 0 d . By Lemma 2.1, d has d 1-fold rotation symmetry. This means that M ∗ ∩ ωR+ ={ω : ∈[ ,α ]} ωd+1 = d t t 0 d , where 1. Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:28 Page 5 of 6 28

≥ M ∗ ∩R− = M ∩R− = (b) By Lemma 2.2(b) and Lemma 2.3, for all even d 2, we have d d [−β , ] M ∗ + M ∗ ∩ ωR+ ={ω : d 0 . Since d has d 1-fold rotation symmetry, it follows that d t d+1 t ∈[0,βd ]}, where d is even and ω =−1. (c) Suppose that d ≥ 3 is an odd integer. If ωd+1 =−1, then ω = 1/ω and ωd =−ω. Writing φ(z) = ωz,wehave

(ω )d + −1 z c φ ◦ P ◦ φ(z) = = Q /ω(z), c ω c where

d Qc(z) := −¯z + c.

Now we define

N ∗ := { ∈ C : { ◦n( )} }. d c the orbit Qc 0 n∈N is bounded

Then we have M ∗ ={ ∈ C : ◦n( ) ∞ →∞} d c Pc 0 as n ={ ∈ C : ◦n ( ) ∞ →∞}=ωN ∗. c Qc/ω 0 as n d

M ∗ ∩ ωR+ = ω(N ∗ ∩ R+) N ∗ ∩ R+ Since d d , it is sufficient to identify d . ˜ d Let Qc(z) := −z + c and we denote

N := { ∈ C : { ˜ ◦n( )} }. d c the orbit Qc 0 n∈N is bounded

∗ + + Similar to the proof of Lemma 2.3, we have N ∩ R = Nd ∩ R . According to [1, + d Lemmas 2.1 and 2.2], we have Nd ∩ R =[0,γd ], where γd is defined in (3). This means M ∗ ∩ ωR+ ={ω : ∈[ ,γ ]} ωd+1 =−  that d t t 0 d if d is odd and 1. Remark 1.

(1) Coincidentally, comparing Lemma 2.2(c) and the Main Theorem (c), we have

(M ∩ ωR+) = (M ∗ ∩ ω∗R+) = γ , Length d Length d d where d ≥ 3 is odd and ωd−1 = (ω∗)d+1 =−1. (2) It was shown in [1, Proposition 3.1] that the value γd has the following asymptotic formula:   1 + (log d)2 d O 2 γd = 2 d as d →∞.

Acknowledgements The author would like to thank Fei Yang for his comments, remarks and careful reading of the text. Part of this work was supported by the National Natural Science Foundation of China (Grant No. 11801305). He would also like to thank the referee for helpful comments. 28 Page 6 of 6 Proc. Indian Acad. Sci. (Math. Sci.) (2019) 129:28

References [1] Baribeau L and Ransford T, Cross-sections of multibrot sets, J. Anal. 24(1) (2016) 95Ð101 [2] Hubbard J H and Schleicher D, Multicorns are not path connected, Frontiers in complex dynamics, pp. 73Ð102, Princeton Math. Ser., 51 (2014) (Princeton: Princeton University Press) [3] Inou H, Self-similarity for the tricorn, Exp. Math. (2018) https://doi.org/10.1080/10586458. 2017.1420502 [4] Inou H and Mukherjee S, Non-landing parameter rays of the multicorns, Invent. Math. 204(3) (2016) 869Ð893 [5] Lau E and Schleicher D, Symmetries of fractals revisited, Math. Intell., 18(1) (1996) 45Ð51 [6] Milnor J, Remarks on iterated cubic maps, Exp. Math. 1(1) (1992) 5Ð24 [7] Mukherjee S, Nakane S and Schleicher D, On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials, Ergod. Theory Dyn. Syst. 37(3) (2017) 859Ð899 [8] Nakane S, Connectedness of the tricorn, Ergod. Theory Dyn. Syst. 13 (1993) 349Ð356 [9] Nakane S and Schleicher D, On multicorns and unicorns I: antiholomorphic dynamics, hyper- bolic components and real cubic polynomials, Int. J. Bifurc. Chaos 13 (2003) 2825Ð2844 [10] Parisé P, Ransford T and Rochon D, Tricomplex dynamical systems generated by polynomials of even degree, Proceedings of Chaos 2016, Chaotic Modeling and Simulation (2017) no. 1, pp. 37Ð48 [11] Parisé P and Rochon D, Tricomplex dynamical systems generated by polynomials of odd degree, Fractals, 25(3) (2017) 1750026 (11 pages) [12] Sheng X and Spurr M J, Symmetries of fractals, Math. Intell. 18(1) (1996) 35Ð42

Communicating Editor: Kaushal Verma