Cross-sections of multibrot sets Line Baribeau · Thomas Ransford

arXiv:1701.05535v1 [math.DS] 19 Jan 2017

Dedicated to David Minda on the occasion of his retirement

Received: date / Accepted: date

Abstract We identify the intersection of the multibrot set of zd + c with the rays R+ ω, where ω d−1 = ±1. Keywords Mandelbrot set · Multibrot set Mathematics Subject Classification (2010) 37F45

1 Introduction Let d be an integer with d ≥ 2. Given c ∈ C, we define pc (z) := zd + c

and

[n]

pc := pc ◦ · · · ◦ pc

(n times).

The corresponding generalized Mandelbrot set, or multibrot set, is defined by n o [n] Md := c ∈ C : sup |pc (0)| < ∞ . n≥0

Of course M2 is just the classical Mandelbrot set. Computer-generated images of M3 and M4 are pictured in Figure 1. Multibrot sets have been extensively studied in the literature. Schleicher’s article [5] contains a wealth of background material on them. TR supported by grants from NSERC and the Canada research chairs program Line Baribeau D´epartement de math´ematiques et de statistique, Universit´e Laval, 1045 avenue de la M´edecine, Qu´ebec (QC), Canada G1V 0A6 E-mail: [email protected] Thomas Ransford D´epartement de math´ematiques et de statistique, Universit´e Laval, 1045 avenue de la M´edecine, Qu´ebec (QC), Canada G1V 0A6 Tel.: +14186562131 ext 2738 Fax: +14186565902 E-mail: [email protected]

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Line Baribeau, Thomas Ransford

Fig. 1 The multibrot sets M3 and M4

We mention here some elementary properties of multibrot sets. First of all, they exhibit (d − 1)-fold rotational invariance, namely Md = ωMd

(ω ∈ C, ω d−1 = 1).

(1) [n]

Indeed, for these ω, writing φ (z) := ωz, we have φ −1 ◦ pc ◦ φ = pc/ω , so pc (0) [n]

remains bounded if and only if pc/ω (0) does. (In fact, the rotations in (1) are the only rotational symmetries of Md . The paper of Lau and Schleicher [1] contains an elementary proof of this fact.) Also, writing D(0, r) for the closed disk with center 0 and radius r, we have the inclusions D(0, α(d)) ⊂ Md ⊂ D(0, β (d)), where α(d) := (d − 1)d −d/(d−1)

and β (d) := 21/(d−1) .

The first inclusion follows from the fact that, if |c| ≤ α(d), then the closed disk [n] D(0, d −1/(d−1) ) is mapped into itself by pc , and consequently the sequence pc (0) is bounded. For the second inclusion, we observe that, if |c| > β (d), then by induction [n+2] |pc (0)| ≥ (2d)n (|c|d − 2|c|) for all n ≥ 0, and the right-hand side of this inequality tends to infinity with n. When d is odd, we have Md ∩ R = [−α(d), α(d)].

(2)

This equality was conjectured by Paris´e and Rochon in [3], and proved by them in [4]. Also, when d is even, we have Md ∩ R = [−β (d), α(d)].

(3)

This equality was also conjectured in [3], and subsequently proved in [2]. When d = 2, it reduces to the well-known equality M2 ∩ R = [−2, 14 ].

Cross-sections of multibrot sets

3

By virtue of the rotation-invariance property (1), the equalities (2) and (3) yield information about the intersection of Md with certain rays emanating from zero. Indeed, if ω d−1 = 1, then Md ∩ R+ ω = {tω : 0 ≤ t ≤ α(d)}, and if ω d−1 = −1 and d is even, then Md ∩ R+ ω = {tω : 0 ≤ t ≤ β (d)}. This leaves open the case when ω d−1 = −1 and d is odd. The purpose of this note is to fill the gap. The following theorem is our main result. Theorem 1.1. If ω d−1 = −1 and d is odd, then Md ∩ R+ ω = {tω : 0 ≤ t ≤ γ(d)}, where γ(d) := d −d/(d−1) sinh(dξd ) + d sinh(ξd ) ,

(4)

and ξd is the unique positive root of the equation cosh(dξd ) = d cosh(ξd ). 3 When d = 3, one can p use the relation cosh(3x) = 4 cosh x − 3 cosh x to derive the exact formula γ(3) = 32/27, which yields p Corollary 1.2. M3 ∩ iR = {iy : |y| ≤ 32/27}. √ In comparison, note that (2) gives M3 ∩ R = {x : |x| ≤ 2/ 27}. See Figure 1. The first few values of α(d), β (d), γ(d) are tabulated in Table 1 for comparison.

Table 1 Values of α(d), β (d), γ(d) for 2 ≤ d ≤ 12 d 2 3 4 5 6 7 8 9 10 11 12

α(d) 0.250000000 0.384900179 0.472470394 0.534992244 0.582355932 0.619731451 0.650122502 0.675409498 0.696837314 0.715266766 0.731314279

β (d) 2.000000000 1.414213562 1.259921050 1.189207115 1.148698355 1.122462048 1.104089514 1.090507733 1.080059739 1.071773463 1.065041089

γ(d) 1.100917369 1.088662108 1.078336651 1.069984489 1.063192242 1.057591279 1.052904317 1.048928539 1.045514971 1.042552690 1.039957793

It can be shown that γ(d) > 1 for all d, and that 2 /d 2 ))

γ(d) = 21/d+O((log d) These statements will be justified later.

as d → ∞.

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Line Baribeau, Thomas Ransford

2 Proof of Theorem 1.1 In this section we suppose that d is an odd integer with d ≥ 3. If ω d−1 = −1, then, writing φ (z) := ωz, we have φ −1 ◦ pc ◦ φ = qc/ω , where qc (z) := −zd + c. Thus Md ∩ R+ ω = ω(Nd ∩ R+ ), where n o [n] Nd := c ∈ C : sup |qc (0)| < ∞ . n≥0

We now seek to identify Nd ∩ R+ . We shall do this in two stages. Lemma 2.1. Let d be an odd integer with d ≥ 3. Then Nd ∩ R+ = [0, µ(d)], where µ(d) := max a − bd : a, b ≥ 0, ad + bd = a + b . Proof. Consider first the case c ∈ [0, 1]. In this case we have qc (0) = c and qc (c) = −cd + c ≥ 0. Since qc is a decreasing function, it follows that qc ([0, c]) ⊂ [0, c], and [n] in particular that qc (0) is bounded. Hence c ∈ Nd for all c ∈ [0, 1]. [2] Consider now the case c ∈ [1, ∞). Then qc (0) = c and qc (0) = −cd + c ≤ 0. [2n] As qc is a decreasing function, it follows that qc (0) is a decreasing sequence and [2n+1] [n] qc (0) is an increasing sequence. If, further, c ∈ Nd , then qc (0) is bounded, and [2n+1] [2n] both of these subsequences converge, say qc (0) → a and qc (0) → −b, where a, b ≥ 0. We then have qc (−b) = a and qc (a) = −b, in other words bd + c = a and ad − c = b. Adding these equations gives ad + bd = a + b. Summarizing what we have proved: if c ∈ Nd ∩ [1, ∞), then c = a − bd , where a, b ≥ 0 and ad + bd = a + b. Conversely, if c is of this form, then qc (−b) = a and qc (a) = −b, so [−b, a] is a qc -invariant interval containing 0, which implies that q[n] (0) remains bounded, and hence c ∈ Nd . Combining these remarks, we have shown that Nd ∩ [1, ∞) = {a − bd : a, b ≥ 0, ad + bd = a + b} ∩ [1, ∞).

(5)

The condition that ad + bd = a + b can be re-written as h(a) = −h(b), where h(x) := xd − x. Viewed this way, it is more or less clear that the right-hand side of (5) is a closed interval containing 1, so Nd ∩ [1, ∞) = [1, µ(d)], where µ(d) is as defined in the statement of the lemma. Finally, putting all of this together, we have shown that Nd ∩ R+ = [0, µ(d)]. Next we identify µ(d) more explicitly. Lemma 2.2. µ(d) = γ(d).

Cross-sections of multibrot sets

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Proof. We reformulate the maximization problem defining µ(d). Set S := {(a, b) ∈ R2 : a, b ≥ 0}, f (a, b) := a − bd , g(a, b) := ad + bd − a − b. We are seeking to maximize f over S ∩ {g = 0}. The set S ∩ {g = 0} is compact and f is continuous, so the maximum is certainly attained, say at (a0 , b0 ). Notice also that ∇g 6= 0 at every point of S ∩ {g = 0}. There are two cases to consider. Case 1: (a0 , b0 ) ∈ ∂ S. The condition that g(a0 , b0 ) = 0 then implies that (a0 , b0 ) = (0, 0), (0, 1) or (1, 0). The corresponding values of f (a0 , b0 ) are 0, −1, 1 respectively. Clearly we can eliminate the first two points from consideration. As for the third, we remark that the directional p derivative of f at (1, 0) along {g = 0} in the direction pointing into S is equal to 1/ 1 + (d − 1)2 , which is strictly positive. So (1, 0) cannot be a maximum of f either. Case 2: (a0 , b0 ) ∈ int(S). In this case, by the standard Lagrange multiplier argument, we must have ∇ f (a0 , b0 ) = λ ∇g(a0 , b0 ) for some λ ∈ R. Writing this out explicitly, we get 1 = λ (da0d−1 − 1), − 1). = λ (dbd−1 −dbd−1 0 0 Dividing the second equation by the first and then simplifying, we obtain a0 b0 = d −2/(d−1) . Thus a0 = d −1/(d−1) eξ and b0 = d −1/(d−1) e−ξ for some ξ ∈ R. With this notation, the constraint g(a0 , b0 ) = 0 translates to cosh(dξ ) = d cosh(ξ ), and the value of f at (a0 , b0 ) is f (a0 , b0 ) = a0 − bd0 =

a0 − b0 ad0 − bd0 + = d −d/(d−1) d sinh(ξ ) + sinh(dξ ) . 2 2

There are precisely two roots of cosh(dξ ) = d cosh(ξ ), one positive and one negative. Necessarily the positive root gives rise to the maximum value of f , thereby showing that µ(d) = γ(d). Remark. Clearly f (1, 0) = 1. The treatment of Case 1 above shows that f does not attain its maximum over S ∩ {g = 0} at (1, 0), and so µ(d) > 1. This shows that γ(d) > 1, thereby justifying a statement made in the introduction. Proof of Theorem 1.1. Combining the various results already obtained in this section, we have Md ∩ R+ ω = ω(Nd ∩ R+ ) = ω[0, µ(d)] = ω[0, γ(d)]. This concludes the proof of Theorem 1.1.

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Line Baribeau, Thomas Ransford

3 An asymptotic formula for γ(d). Our aim is to justify the following statement made in the introduction. Proposition 3.1. If γ is defined as in (4), then 2 /d 2 )

γ(d) = 21/d+O((log d)

as d → ∞.

(6)

There is no need to suppose that d is an integer here. Proof. We begin by deriving an asymptotic formula for ξd as d → ∞. On the one hand, since edξd ≥ cosh(dξd ) = d cosh(ξd ) ≥ d, we certainly have ξd ≥ (log d)/d. On the other hand, since the unimodal function (cosh x)/x takes the same values at ξd and dξd , we must have ξd ≤ η ≤ dξd , where η is the point at which (cosh x)/x assumes its minimum. Thus edξd ≤ cosh(dξd ) = d cosh ξd ≤ d cosh η, 2 whence

1 log d . +O d d This is not yet precise enough. Substituting into the equation cosh(dξd ) = d cosh(ξd ), we obtain 1 (log d)2 edξd +O = d +O , 2 d d whence (log d)2 log(2d) ξd = +O . d d3 This is good enough for our needs. We now estimate γ(d) as d → ∞. First of all, we have (log d)3 d sinh(ξd ) = dξd + O(dξd3 ) = log(2d) + O . d2 Also (log d)2 (log d)2 sinh(dξd ) = sinh log(2d) + O = d +O . 2 d d Hence d log γ(d) = log d sinh(dξd ) + sinh(dξd ) − log d d −1 (log d)2 1 1 = log d + log(2d) + O − 1 + + O 2 log d d d d (log d)2 log d log d log(2d) +O − log d + +O = log d + 2 d d d d2 2 log 2 (log d) = +O . d d2 Finally, taking exponentials of both sides, we get (6). ξd =

Cross-sections of multibrot sets

7

Acknowledgements The first author thanks the organizers of the Conference on Modern Aspects of Complex Geometry, held at the University of Cincinnati in honor of Taft Professor David Minda, for their kind hospitality and financial support.

References 1. Lau, E., Schleicher, D.: Symmetries of fractals revisited. Math. Intelligencer 18(1), 45–51 (1996) 2. Paris´e, P.O., Ransford, T., Rochon, D.: Tricomplex dynamical systems generated by polynomials of odd degree. Preprint (2016) 3. Paris´e, P.O., Rochon, D.: A study of dynamics of the tricomplex polynomial η p + c. Nonlinear Dynam. 82(1-2), 157–171 (2015) 4. Paris´e, P.O., Rochon, D.: Tricomplex dynamical systems generated by polynomials of odd degree. Preprint (2015) 5. Schleicher, D.: On fibers and local connectivity of Mandelbrot and Multibrot sets. In: Fractal geometry and applications: a jubilee of Benoˆıt Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, pp. 477– 517. Amer. Math. Soc., Providence, RI (2004)