Non Locally-Connected Julia Sets constructed by iterated tuning

Transcription

1 Non Locally-Connected Julia Sets constructed by iterated tuning John Milnor Stony Brook University Revised May 26, 2006

2 Notations: Every quadratic map f c (z) = z 2 + c has two fixed points, α and β, where α + β = 1, αβ = c, R(α) R(β). The multiplier µ = f c(α) = 2α will be used as an alternative parameter for the quadratic family. Here c and µ determine each other: c = c(µ) = α(1 α) with α = µ/2, µ = µ(c) = 1 1 4c, with R(µ) 1. The map f c(µ) corresponding to µ will be denoted by fµ (z) = z 2 + c(µ). 2

3 The connectedness locus M, consisting of all µ in the half-plane R(µ) 1 with K ( f µ ) connected, will be called the rounded Mandelbrot set. Its period one hyperbolic component, the set of all µ M for which f µ has an attracting fixed point, is the open unit disk D. 3

4 H(1/3) 1/4 0 H(1/3) H(1/4) 2/7 H(1/4) 0 1/4 2/5 3/8 1/3 Unit Disk Region in M and Disk Approximation There is a satellite hyperbolic component H(n/p) of period p attached to D at each p-th root of unity e 2πin/p. Similarly, there are satellites H(n/p) H(n /p ) of period pp attached to H(n/p) at corresponding boundary points; and so on. Empirically, each iterated satellite H(n 1 /p 1 ) H(n k /p k ) can be approximated by a round disk of radius 1/(p 1 p k ) 2. Question: How can this be made precise? 4

5 Tuning (in parameter space). The Douady-Hubbard tuning construction assigns to each hyperbolic component H M a homeomorphism H : M = (H M) M from M onto a small copy of M. Each H maps hyperbolic components to hyperbolic components, with per(h H ) = per(h) per(h ). The set of all H forms a free non-commutative monoid, with D as identity element. Each H : D H is holomorphic, and yields the canonical Douady-Hubbard parametrization of H: For each µ D, the attracting periodic orbit for the map f H µ has multiplier µ. The image H 1 H is called the root point of H. 5

6 Tuning in the dynamic plane (intuitive picture). = To obtain the filled Julia set for H 1 µ 2, choose µ 1 H 1, then replace every Fatou component of K ( f µ1 ) by a copy of K ( f µ2 ). (In the figure, f µ2 is the Chebyshev map z z 2 2, and K ( f µ2 ) is the line segment [ 2, 2].) 6

7 Constructing a non locally-connected K ( f µ ) H(t ) 1 r 1 r 3 r2 H(t ) 1 H(t ) 2 Unit disk r 0 = 1 (Small disk sizes exaggerated.) Choose any sequence of rational angles t 1, t 2,... 0, and let r k be the root point of H(t 1 ) H(t k ). 7

8 The Theorem. Let ω M be any limit point for the sequence of root points ( ) H(t 1 ) H(t k ) as k. r k Theorem (Douady, Hubbard, Sørensen). If the sequence { t j } converges to zero sufficiently rapidly, then the filled Julia set K ( f ω ) is not locally-connected. The proof will be based on external rays and separating periodic orbits. 8

9 1/5 1/7 4/15 2/7 H(1/3) Two nested wakes W [1/7, 2/7] W [1/5, 4/15]. Every root point H 1 M is the landing point of exactly two external rays, with angles 0 a < b 1. These rays cut the parameter plane into two halves. Definition. The half containing H = H [a,b] is called the wake W [a,b], and [a, b] is called its characteristic interval. 9

10 In the Dynamic Plane: b = 4/15 c a = 1/5 For every hyperbolic component H = H [a,b] of period p > 1 and every µ H, the external rays of angle a and b for K ( f µ ) land at a common repelling periodic point. I will write z 1 = z 1( [a, b], µ ) = l a (µ) = l b (µ) K ( f µ ). 10

11 2 a b (ι = 1) a 16a 8b (2) 2 b (ι = 0 4) 4 b 8 a (3) 4 a Number the periodic Fatou components of f µ as 0 U 0 U 1. Then the orbit of z 1 consists of points z ι = z ι( [a, b], µ ) = l 2 ι 1 a (µ) = l 2 ι 1 b (µ) U ι called dynamic root points, indexed by ι Z/p. 11

12 b/2 z 0 a/2 The a/2 and b/2 rays land on opposite sides of U 0 : l a/2 (µ) = ±z 0 ([a, b], µ) and l b/2 (µ) = z 0 ([a, b], µ). 12

13 z 1 z 2 z 0 = z 4 z 3 More generally, these periodic points z ι( [a, b], µ ) K ( f µ ) are defined, and vary holomorphically with µ, for all µ in the wake W [a,b], even when µ M. 13

14 Given any infinite sequence of hyperbolic components H k period p k > 1, any point in the nested intersection, of ω k (H 1 H k M), is said to be infinitely renormalizable. Let [a k, b k ] be the characteristic interval for the k-fold tuning product H k = H 1 H k. I will consider only the case where the nested intersection [a k, b k ] consists of a single angle θ. (An equivalent condition would be that H k R = for infinitely many k.) Note that the points z k (ω) = z 0( [a k, b k ], ω) and their negatives cut the filled Julia set K ( fω ) into countably many pieces. 14

15 (1+a 1 )/2 (1+a 2 )/2... (1+θ)/2... (1+b 2 )/2 X b 1 /2 b 2 /2... θ/2... a 2 /2 (1+b 1 )/2 In this schematic diagram, externals rays are orange, equipotentials are blue, and the Julia set is black. Let X be the connected component of zero in the set K ( fω ) {±zk (ω)}. Ḷemma 1. X is compact, connected, and cellular. Every limit point of {±z k (ω)}, and every limit point of the θ/2 and (1 + θ)/2 rays, belongs to X. But every other ray is bounded away from X. If K ( f ω ) is locally connected then X = {0}. Conjecture. Conversely, if X = {0}, then K ( f ω ) is locally connected. 15 a 1 /2

16 Choosing the Angles. Now assume that the H k are satellite hyperbolic components H(n k /p k ). Again let H k = H 1 H 2 H k, with characteristic interval [a k, b k ], and let r k be the root point of H k. Then the periodic point z j (r k ) = z 0( [a j, b j ], r k ) K ( f rk ). is defined for all j k. (This point has period parabolic for j = k, and repelling for j < k.) p 1 p j 1. It is Lemma 2. We can choose the angles t j = n j /p j 0 inductively so that these points z j (r k ) are within some specified neighborhood of z 1 (r 1 ) for all j k. Start Proof. Suppose H 1,..., H k have already been chosen. We must show that each z j (r k+1 ) with j k + 1 depends continuously on the choice of r k+1, and hence can be placed arbitrarily close to z j (r k ) by choosing r k+1 close to r k. 16

17 H k H k-1 r k r k+1? Since r 1,..., r k have been chosen, z k (r k ) is a well defined parabolic point of period p 1 p k 1 and multiplier e 2πin k /p k. For µ in a small neighborhood of r k, the orbit of z k (r k ) splits into an orbit of the same period p 1 p k 1 with multiplier e 2πin k /p k, and a nearby orbit of period p 1 p k with multiplier +1. Take µ = r k+1 to be a point at rational angle along the boundary of H k. Then this new orbit will again be parabolic, and the orbit point z k+1 (r k+1 ) will converge to z k (r k ) as r k+1 converges to r k. Since the points z j (r k+1 ) with j k clearly vary continuously with r k+1, this proves Lemma 2. 17

18 Proof of the Theorem. Recall that: The sequence {r k } of root points has ω as limit point. The function µ z j (µ) is continuous for µ W [aj, b j ]. Therefore, for each fixed j, the sequence of points z j (r k ) has z j (ω) as a limit point. By Lemma 2, we can choose the H k uniformly bounded away from 0. so that the z j (r k ) are Hence the points z j (ω) are also bounded away from 0. Therefore, by Lemma 1, K ( f ω ) cannot be locally connected. 18

19 (θ+1)/2 b 1 /2 b 2 / θ/2 a 2 /2 a 1 /2 If t k > 0 for all angles t k, then the θ/2 and (θ + 1)/2 rays spiral around each other without landing, in a paper clip pattern as sketched above, with the Julia set spiraling between them. Z -Z Z -Z Z Here is a schematic picture close to the (θ/2)-ray, which has been straightened out. All the points z j (ω) are assumed to lie in the region Z, while their negatives lie in Z. 19 b 1 /2 θ/2 a 1 /2

20 The sin(1/x) model. (1+a 1 )/2 b 1 /2 (1+a 2 )/2... (1+b 2 )/2 (1+b 1 )/ (1+θ)/2 θ/2... Z Z b 2 / a 2 /2 a 1 /2 On the hand, if the signs of the t k alternate, then the Julia set (indicated here in black) contains a sin(1/x)-like curve. Compare Sørensen. 20

21 How rapidly must t k 0? What rate of convergence is needed to guarantee that the z k (ω) do not converge to zero? Here is a wild guess. Perhaps there are order of magnitude estimates something like the following log z k(ω) z k+1 (ω) log z k (r k ) z k+1 (r k+1 ) t 1/p k k+1 so that {z k (ω)} converges to zero if and only if t 1/p k k+1 =. (??) k For example, if t k = 1/p k with p k+1 = (k + 1) p k, then p 1 = 1, p 2 = 2, p 3 = 9, p 4 = , p ,... tending rapidly to infinity. Yet t 1/p k k+1 = 1/(k + 1) with sum +. Conjecturally, this {p k } does not increase fast enough! 21

22 To conclude: Four Pictures It is probably impossible to make any real picture of one of these non locally-connected Julia sets. However, we may get some intuitive idea by looking at relatively modest iterated satellite tunings. In the first two pictures, the separating periodic points z 1 and z 2 are circled. The rays of angle a 1 /2 = 1/14 and b 1 /2 = 1/7 are shown, but those of angle a 2 /2 b 2 /2 are too close to distinguish from 1/14 respectively 1/7. In the last two pictures, z 1, z 2 and z 3 are defined and circled. (As the angles t 2, t 3 tend to zero, these circled points would converge towards each other.) In these cases, the rays of angle a 1 /2 = 1/14 < a 2 /2 < b 2 /2 < b 1 /2 = 1/7 can be distinguished. (Assuming only that p k 3 for all k, it follows that the differences b k a k 2 p 1 p k tend faster than exponentially to zero as k.) 22

23 H(1/3) H(1/20) 0 23

24 H(1/3) H( 1/20) 0 24

25 H(1/3) H(1/7) H(1/13) 0 25

26 H(1/3) H( 1/7) H(1/13) 0 26

27 References A. Douady, Algorithms for computing angles in the Mandelbrot set, pp of Chaotic Dynamics and Fractals, ed. Barnsley & Demko, Acad. Press, 1986 A. Douady, Chirurgie sur les applications holomorphes, Proc. Int. Cong. Math. Berkeley, A.M.S. 1987, A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Sup. (Paris) 18 (1985) C. McMullen, Complex Dynamics and Renormalization, Princeton U. Press, J.Milnor, Self-similarity and hairiness in the Mandelbrot set, in Computers in Geometry and Topology, (M.C. Tangora, ed.), Marcel Dekker, New York, Basel, 1989, pp

28 J.Milnor, Local connectivity of Julia sets: expository lectures, pp of The Mandelbrot set, Theme and Variations edit: Tan Lei, LMS Lecture Note Series 274, Cambridge U. Press, J.Milnor, Periodic orbits, external rays and the Mandelbrot set: An expository account, Asterisque 261 (2000) Geometrie Complexe et Systemes Dynamiques, pp J. Milnor, Dynamics in One Complex Variable, (3rd ed.) Princeton U. Press, D. E. K. Sørensen, Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set, J. Geom. Anal. 10 (2000),