A REMARK ON ZAGIER S OBSERVATION OF THE MANDELBROT SET

Shimauchi, H. Osaka J. Math. 52 (205), 737 746 A REMARK ON ZAGIER S OBSERVATION OF THE MANDELBROT SET HIROKAZU SHIMAUCHI (Received April 8, 203, revised March 24, 204) Abstract We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given. Bielefeld, Fisher and Haeseler [] presented Zagier s observation for the growth of the denominator of the coefficients for the Mandelbrot set, and Yamashita [4] verified it. Our study takes into consideration both Zagier s observation and Yamashita s estimate.. Introduction Let be the complex plane, Ç the Riemann sphere, the open unit disk, the exterior of the closed unit disk, and the set of non-negative integers. We denote the n-th iteration of polynomial P by P Æn, which is defined inductively by P Æ(n) P Æ P Æn with P Æ0 (z) z. The Julia set J P of P is the set of all z ¾ Ç such that {P Æn } ½ n0 is not normal in any neighborhood of z. We consider the complex dynamical systems of the polynomials given by P d,c (z)ï z d c on, Ç where c¾ is a parameter and d ¾ ÆÒ{} is fixed. The Multibrot set M d of degree d is the set of all parameters c ¾ for which J Pd,c is connected. The original Mandelbrot set is derived in the case d 2. It is known that M d is compact and is contained in the closed disk of radius 2 (d ) with center 0 (see e.g. [, pp. 20 22] and [8]). Constructing a conformal homeomorphism 2 Ï Ç Ò M 2 that satisfies 2 (z)z as z ½, Douady and Hubbard [2] proved the connectedness of the Mandelbrot set. The statement can be generalized to d 2 in the same way. For more details, see [] for d 2 and [4, 2] for the generalized statement. Theorem. ([2, Theorem 2]). There exists a conformal homeomorphism d Ï Ç ÒM d that satisfies lim z½ d (z)z. 200 Mathematics Subject Classification. Primary 37F45; Secondary 30D05.

738 H. SHIMAUCHI We define the map d Ï Ç Ò M d by d (z) d (z), where d is the inverse map of d. Let b d,m be the coefficients of the Laurent expansion of d as d (z) z ½ m0 b d,m z m. If d extends continuously to the unit circle, then, according to Carathéodory s continuity theorem, the Mandelbrot set is locally connected (see e.g. [, p. 54]). There is an important conjecture which states that the Mandelbrot set is locally connected; actually, this conjecture includes the conjecture for the density of hyperbolic dynamics in the quadratic family (see [3, p. 5]). It should be noted that the convergence of the Laurent series on the unit circle implies that M d is locally connected. With this in mind, we investigate the arithmetic properties of the above-stated coefficients b d,m. 2. Zagier s observation of the denominator Jungreis [7] presented a method to compute the coefficients b 2,m of 2. Levin [0] showed an algorithm for computing the coefficients of the function ¼ 2 2. Using Levin s algorithm, b 2,m can be computed. Jungreis s method can be generalized to d ¾ Æ Ò {} similarly (see [3, 4, 2]). We can make a program for this procedure and derive the exact value of b d,m, because the following proposition holds. We call a real number x d-adic rational if there exist k ¾ and n ¾ such that x kd n. Proposition 2. ([7]). The coefficients b d,m are d-adic rational numbers. Moreover, Jungreis [7] showed that some of these coefficients are 0, which we call zero-coefficients. Several detailed investigations on b d,m are presented in [, 4, 5, 9] for the case d 2, and in [0, 8, 3, 4] for general d. In particular, Bielefeld, Fisher and Haeseler presented Zagier s observations in [, pp. 32 33]. One of the observations is the necessary and sufficient condition for the coefficients b 2,m to be zero. Some information about the zero-coefficients can be found in [, 9, 0, 8, 3, 4]. However, the necessary and sufficient condition for the zero-coefficients is still unknown. The other observations are related to the growth of the denominator of b 2,m. In this paper, we focus on the denominator of b d,m. We prepare the notation about the p-adic valuation. For any prime number p and every non-zero rational number x, there exists a unique integer Ú such that x p Ú rq with integers r and q that are not divisible by p. The p-adic valuation p Ï ÉÒ{0} is defined as p (x) Ú. We extend p to É as follows: p (x) Ú for x ¾ ÉÒ{0}, ½ for x 0.

ZAGIER S OBSERVATION OF MANDELBROT SET 739 For any real number x, we set the floor function x Ï max{m ¾ Ï m x}. We would like to note that the floor function and the p-adic valuation have the following properties (see e.g. [6, pp. 69 72, and pp. 02 4]). Lemma 2.2. Let x, y ¾ Ê and p be a prime number. The floor function and the p-adic valuation satisfy the following: (2.) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) xm x m for m ¾, xy x y, x m x for m ¾, m p (mn) p (m) p (n) for m, n ¾, p (mn) p (m) p (n) for m ¾, n ¾ Ò{0}, p (m n) min{ p (m), p (n)} for m, n ¾, p (m!) ½ l m p l for m ¾. Zagier s observation for the growth of the denominator is as follows: Observation 2.3 ([, Observation (ii)]). For 0 m 000, 2 (b 2,m ) 2 ((2m 2)!) holds. Furthermore, equality holds if and only if m 0 or m is odd. We would like to note the following inequality presented by Ewing and Schober in [5]. Theorem 2.4 ([5, Theorem ]). For any m ¾, 2 (b 2,m ) 2m holds. Furthermore Levin [0] showed that equality holds in Zagier s observation if m is odd. Theorem 2.5 ([0, Theorem on p. 3520]). If m ¾ is an odd integer, then 2 (b 2,m ) 2 ((2m 2)!) holds. Yamashita [4] defined the order of integers with respect to d ¾ Æ Ò {} and gave an estimate of the growth of the denominator of b d,m (see [4, Theorem 4.30]). However, the theorem was incorrect. We learnt from Yamashita that there was a mistake in the calculation. However, if we restrict d to be a prime number, his result is valid. Using the p-adic valuation, his statement can be presented in the following way.

740 H. SHIMAUCHI Theorem 2.6 ([4, Theorem 4.30]). (p ) (m ), Let m ¾ and p be a prime number. If p (b p,m ) p ((pm p)!) p holds. Otherwise, p (b p,m ) ½ holds. Furthermore, if (p ) (m ), equality holds precisely when m p 2 or p m. Hence Observation 2.3 holds for general m. In this paper, we give an estimate for the prime factor of the denominator of b d,m. Our estimate includes Observation 2.3 and is equal to Yamashita s estimate if d is a prime number. Actually the general statement of Observation 2.3 and Theorem 2.6 are given as a corollary of our main results. Furthermore we obtain another corollary for the growth of the denominator of b d,m. 3. Main result First, we introduce Ewing and Schober s coefficients formula for general d. The result for d 2 is given in [4]; however, the argument can be generalized to d 2 similarly. For more details, see [4, Lemma 4.8]. Lemma 3. ([4, Theorem ]). sufficiently large. Then, (3.) Using the recursion P Æn d,c For n ¾, let m d n 3 and R 0 be b d,m Pd,z Æn 2mi (z)mdn dz. zr ) (z) (PÆ(n d,c (z)) d c for (3.), we obtain the following formula for coefficients in the same way (see [4, Corollary 4.20]). Let C j (a) denote the general binomial coefficient, i.e., C j (a) for any a ¾ Ê and j ¾ Æ with C 0 (a). Corollary 3.2 ([5, Corollary]). b d,m m a(a )(a 2) (a j ) j( j )( j 2) () Let n ¾ Æ and m d n 3. Then, m m m C j C d n j2 d j d n C j3 d 2 j d n 2 d j 2 m C jn d dn j d n 2 j 2 d j n, where the sum is over all non-negative indices j,, j n (d n ) j 2 (d n 2 ) j 3 (d ) j n m. such that (d n ) j

ZAGIER S OBSERVATION OF MANDELBROT SET 74 Levin [0], Lau and Schleicher [8], and Yamashita [3] proved the following lemma independently. Furthermore, using Corollary 3.2, Yamashita presented an alternative short proof of Lemma 3.3 in [4]. Lemma 3.3 ([0, Theorem on p. 355], [8, p. 47], [3, Theorem 2]). Let d 3 and m ¾. If (d ) (m ), then b d,m 0 holds. Our main result is given as below. Theorem 3.4. Let d ¾ Æ Ò {}, m ¾ with (d ) (m ) and set a (m )(d ). Assume that d is factorized as d p t p t 2 2 p t s s, where s ¾ Æ, t, t 2,, t s ¾ Æ and p, p 2,, p s are distinct prime numbers. Then, pi (b d,m ) pi (a!)t i a holds for any p i ¾ {p, p 2,, p s }. Furthermore, equality holds precisely when m d 2 or p i m. REMARK 3.5. We disregard the case (d ) (m ) because pi (b d,m ) ½ holds for any p i by Lemma 3.3. There are some zero-coefficients which are not included in Lemma 3.3 (see [3]). Proof of Theorem 3.4. Let m ¾ and assume that d is factoraized as d p t p t 2 2 p t s s where s ¾ Æ, t, t 2,, t s ¾ Æ and p, p 2,, p s are distinct prime numbers. From direct calculation, b 2,0 2 (see [7, ]). Furthermore b d,0 0 for d 3 by Lemma 3.3. Hence, the statement holds for m 0. For fixed p i ¾ {p, p 2,, p s }, we take n ¾ Æ such that m d n 3 and pi (m) nt i. We define the set  as:  {( j, j 2,, j n ) ¾ ( ) Ï n (d n ) j (d n ) j 2 (d ) j n m }. Let k ¾ {, 2,, n}, ( j, j 2,, j n ) ¾  and set and «md n k d k j d k 2 j 2 d j k d n k «m d n j d n j 2 d n k2 j k. Then we have (3.2) C jk («) «(«)(«2) («( j k )) j k! ( dn k )( 2d n k ) ( ( j k )d n k ). d j k(n k) j k!

742 H. SHIMAUCHI The denominator of the fractional expression (3.2) satisfies the following equality: (3.3) pi (d j k(n k) j k!) (2.4) pi ( j k!) j k t i (n k ). On the other hand we have pi ( (l )d n k ) pi (m d n j d n j 2 d n k2 j k (l )d n k ) pi (m d n k (d k j d k 2 j 2 d j k (l ))) (2.6) min{ pi (m), t i (n k )} for all l ¾ {, 2,, j k }. Hence the numerator of the fractional expression (3.2) satisfies the following inequality: (3.4) pi ( ( d n k )( 2d n k ) ( ( j k )d n k )) (2.4) j k min{ pi (m), t i (n k )}. Combining (3.3) and (3.4), we obtain (3.5) pi (C jk («)) (2.5) pi ( j k!) j k t i (n k ) j k min{ pi (m), t i (n k )} pi ( j k!) j k max{t i (n k ) pi (m), 0}. By Corollary 3.2 and (3.5), Due to pi (b d,m ) pi (m) max ( j, j 2,, j n )¾Â max ( j, j 2,, j n )¾Â n k n k pi ( j k!) µ j k max{t i (n k ) pi (m), 0} µ. (3.6) (d n ) j (d n ) j 2 (d ) j n m

ZAGIER S OBSERVATION OF MANDELBROT SET 743 for all ( j, j 2,, j n ) ¾  and (2.2), we have n k pi ( j k!) (2.7) n (2.2) (3.6) ½ k l jk p l i j j n ½ p l l i ½ m l pi (a!) (2.7) (d )p l i for all k n. Equality holds if j j 2 j n 0, j n (m )(d ) a. Conversely if equality holds, then j j 2 j n 0, j n a, because ( j, j 2,, j n ) ¾  satisfies and hence (d n d n 2 ) j (d ) j n j n m d max {j {j,, j n j n } m )¾Â d is valid precisely when j j 2 j n 0, j n a. We consider the case of p i m. Since pi (m) 0 for p i m, we have n k n j k max{t i (n k ) pi (m), 0} k n k t i a (3.6) j k t i (n k ) j k t i d n k d for all ( j, j 2,, j n )¾Â. Further, equality holds if and only if j j 2 j n 0, j n a. We note that if j j 2 j n 0, j n a, then C 0 (md n )C 0 (md n ) C a (md) C a (md) 0. Considering the elements of the sum in the formula in Corollary 3.2, we have b d,m 0 and pi (b d,m ) pi (a!)t i a.

744 H. SHIMAUCHI We consider the other case p i m. Since pi (m) nt i, there exists n ¼ n and ( j, j 2,, j n ) ¾  such that On the one hand, max ( j, j 2,, j n )¾Â n ¼ k n¼ k n¼ k n k j k max{t i (n k ) pi (m), 0} j k (t i (n k ) pi (m)) j k t i (n k n¼ ) j k t i (n k ) n k k max ( j, j 2,, j n )¾Â t i a. j k pi (m). j k t i (n k ) n k µ j k t i d n k d Thus, equality holds if and only if j j 2 j n 0, j n a. On the other hand, n ¼ k j k pi (m) pi (m). The necessary and sufficient condition for equality is j j 2 j n. Hence, n ¼ k j k t i (n k n¼ ) k j k pi (m) t i a pi (m). Equality holds if and only if j j 2 j n 0, j n, which means that m d 2. Therefore, if p i m, pi (b d,m ) pi (m) pi (a!)t i a pi (m) pi (a!)t i a, and equality holds if and only if m d 2. Considering Theorem 3.4, we can verify Observation 2.3 for the denominator of nonzero-coefficients and Theorem 2.6 by the same calculation of Yamashita [4]. µ

ZAGIER S OBSERVATION OF MANDELBROT SET 745 Proof of Observation 2.3 and Theorem 2.6. Let p be a prime number. By Lemma 3.3 and that b 2,0 2, we may assume (p ) (m ). We obtain p (b p,m ) m p p m p! from Theorem 3.4. By the direct calculation which is the same as Yamashita s calculation in [4], m m p p p! (2.7) m ½ m p (p )p l (2.3) m p p (2.) p (2.7) l ½ m ± p l l ½ m ± p l l0 p ((pm p)!) p Furthermore, we obtain the following estimate for the growth of the denominator of b d,m. For any real number x, we set the ceiling function xï min{m ¾ Ï m x}. Corollary 3.6. Let m ¾, d ¾ ÆÒ{}, (d )(m) and a (m)(d ). Assume that d is factorized as p t p t 2 2 p t s s, where s ¾ Æ, t,t 2,,t s ¾ Æ and p, p 2,, p s are distinct prime numbers. Then,. where x(m) Ï max is b d,m d x(m) ¾, pi (a!)t i a t i. ACKNOWLEDGEMENT. The author would like to thank Professor Yohei Komori and Osamu Yamashita for their valuable advice and encouragement. He further expresses his sincere gratitude to Professor Toshiyuki Sugawa for his helpful suggestions and constant encouragement. Last but not least, the author is grateful to the referee for his/her careful reading of this manuscript and for his/her helpful comments.

746 H. SHIMAUCHI References [] B. Bielefeld, Y. Fisher and F. Haeseler: Computing the Laurent series of the map Ï C D C M, Adv. in Appl. Math. 4 (993), 25 38. [2] A. Douady and J.H. Hubbard: Itération des polynômes quadratiques complexes, C.R. Acad. Sci. Paris Sér. I Math. 294 (982), 23 26. [3] A. Douady and J.H. Hubbard: Exploring the Mandelbrot Set, The Orsay Notes, 985. [4] J.H. Ewing and G. Schober: On the coefficients of the mapping to the exterior of the Mandelbrot set, Michigan Math. J. 37 (990), 35 320. [5] J.H. Ewing and G. Schober: The area of the Mandelbrot set, Numer. Math. 6 (992), 59 72. [6] R.L. Graham, D.E. Knuth and O. Patashnik: Concrete Mathematics, second edition, Addison- Wesley, Reading, MA, 994. [7] I. Jungreis: The uniformization of the complement of the Mandelbrot set, Duke Math. J. 52 (985), 935 938. [8] E. Lau and D. Schleicher: Symmetries of fractals revisited, Math. Intelligencer 8 (996), 45 5. [9] G.M. Levin: Arithmetic properties of a sequence of polynomials, Russian Math. Surveys 43 (988), 245 246. [0] G.M. Levin: On the theory of iterations of polynomial families in the complex plane, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen 5 (989), 94 06 (Russian), translation in J. Soviet Math. 52 (990), 352 3522. [] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda: Holomorphic Dynamics, Cambridge Univ. Press, Cambridge, 2000. [2] H. Shimauchi: On the coefficients of the Riemann mapping function for the exterior of the multibrot set; in Topics in Finite or Infinite Dimensional Complex Analysis, Tohoku University Press, Sendai, 203, 237 248. [3] O. Yamashita: On the coefficients of the map Ï Ç Ò D Ç Ò M, Sci. Bull. Josai Univ. Special Issue 4 (998), 65 69. [4] O. Yamashita, On the coefficients of the mapping to the exterior of the Mandelbrot set, Master thesis, Graduate School of Information Science, Nagoya University, (998), (Japanese), http://www.math.human.nagoya-u.ac.jp/master.thesis/997.html. Graduate School of Information Sciences Tohoku University Sendai, 980-8579 Japan e-mail: shimauchi@ims.is.tohoku.ac.jp e-mail: hirokazu.shimauchi@gmail.com