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1 Topology Proceedings Web: Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume 30, No. 1, 2006 Pages TWISTED SOLENOIDS AND MAPS OF R 2 WHOSE MINIMAL SETS ARE CANTOR SETS WILLIAM BASENER AND CARL V. LUTZER Abstract. We construct homeomorphisms of R 2 that have a minimal Cantor set similar to cross sections of solenoids. It is proven that certain of these homeomorphisms are C and the rest are not continuously differentiable. The differentiability depends on the number of components of the n th stage in the construction of the Cantor set. 1. Introduction Suppose X is a topological space and f : X X is a homeomorphism. A nonempty compact subset Y X is said to be a minimal set for f if, for every y Y, the orbit of y under iterations of f is dense in Y. The set Y is said to be an exceptional set if it is both a minimal set and a Cantor set. Denjoy showed that any diffeomorphism of S 1 that has an exceptional set cannot be C 2. The case for maps of the plane is different. In particular, the return map for a cross section of a solenoid (see [11]) is a C map of the plane with an exceptional set. In this paper, we define a collection of mappings of the plane which are solenoid-like and have an exceptional set. Each map is determined by a sequence of integers {p i }, similar to the standard solenoids (see [1], [11]). In contrast to solenoids, there is a dichotomy for our maps in terms of differentiability. Our main result, Theorem 1, says that the map is C if and only if the series Key words and phrases. minimal set, solenoid. 69

3 70 W. BASENER AND C. V. LUTZER i=1 1 p i converges and, moreover, is not continuously differentiable if the series diverges. The only if portion of this theorem follows directly from the very nice paper of J.-M. Gambaudo, D. Sullivan, and C. Tresser [4]. In their notation, the number n i=1 1 p i is called the n th average linking number, denoted l n. It is proven in Theorem 1 of [4] that the sequence l n must converge if the the map is continuously differentiable. 2. The construction We begin by introducing a family of disks (Figure 1) and an associated family of smooth functions. Suppose p 1, p 2,... is an infinite sequence of positive integers which are pairwise relatively prime, D is the unit disk in R 2, and f 1 C (R 2 ) rotates D by 2π/p 1. D 1 D D2 D 0 Figure 1. The first three levels of nested disks with p 1 = 3, p 2 = 5, p 3 = 7. First, index a set of p 1 disjoint, closed disks D (i) that are properly contained in D such that f 1 maps D (i) bijectively onto D (i+1 mod p1 ). For each i = 0, 1,..., (p 1 1), we define a closed disk E (i) such that D (i) E (i) (strict containment) and E (i) E (i ) = when i i. Lastly, in this step, we choose a smooth function f 2 that rotates each D (i) by 2π/p 2 and that is the identity outside of i E (i).

4 TWISTED SOLENOIDS 71 Second, index a set of p 2 disjoint, closed disks D (i,j) that are properly contained in D (i), for each i, such that f 2 takes D (i,j) to D (i,j+1 mod p2 ). Hence, f 2 f 1 : D (i,j) D (i+1 mod p1,j+1 mod p 2 ). As before, we choose a collection of closed disks E (i,j) that are pairwise disjoint such that D (i,j) E (i,j) (strict containment), and we select a smooth function f 3 that rotates D (i,j) by 2π/p 3. Continuing this pattern, we define closed disjoint disks D (x1,...,x i ), where (x 1,..., x i ) Z p1 Z pi 1 Z pi, and closed disks E (x1,...,x i ) D (x1,...,x i ) (strict containment) such that E (x1,...,x i ) E (y1,...,y i ) = when (x 1,..., x i ) (y 1,..., y i ). And for each i N we define a homeomorphism f i : R 2 R 2 that satisfies the following conditions: (a) The function f i rotates every D (x1,...,x i 1 ) by 2π/p i. (b) The function f i is the identity of off the E (x1,...,x i 1 ). (c) The composition f i f i 1 f 2 f 1 maps D (x1,...,x i 1,x i ) to D (x1 +1 mod (p 1 ),...,x i 1 +1 mod (p i 1 ),x i +1 mod (p i )). Our objects of study are the limit function F, defined by (1) F = lim i f i f i 1 f 2 f 1 and its exceptional set. We can now state our main result. Theorem 1. The function F defined by equation (1) is C if and only if the sum 1 p i=1 i converges. Moreover, if the sum diverges then the map is not continuously differentiable. If the sum i=1 1 p i converges, then we call the suspension of F a twisted solenoid and if this sum diverges, then we call F an over-twisted solenoid. 3. The exceptional set of F Toward describing the exceptional set of F, let C i = D (x1,x 2,...,x i ). {(x 1,x 2,...,x i ) x k {0,1,...,p k 1}}

5 72 W. BASENER AND C. V. LUTZER It is easy to verify that the set C = i=1 is a Cantor set. We will use the group G = i=1 Z p i to index the points in C via the map I : G C, I : (x 1, x 2,...) D (x1 ) D (x1,x 2 ) D (x1,x 2,x 3 ) C i and, henceforth, will identify (x) and I(x). Proposition 1. The set C is a minimal set for the map F. Proof: It follows from our definition that for each (x) C, (2) F (x) = (x mod (p 1 ), x mod (p 2 ),... ) C. Because diamd (x1,...,x i ) 0 as i, for any ε > 0 there is an N N such that for two points (x 1, x 2, x 3,... ), (y 1, y 2, y 3,... ) C, (x 1, x 2, x 3,... ) (y 1, y 2, y 3,... ) < ε if x i = y i for all i < N. Thus, the proof will be complete if we can demonstrate that, for any points (y 1, y 2, y 3,... ) and (x 1, x 2, x 3,... ) C and positive integer N, there exists a positive integer k such that the first N entries of F k (x 1, x 2, x 3,... ) agree with the first N entries of (y 1, y 2, y 3,... ). This follows easily from (2) because the p i are pairwise relatively prime. We note that the bijection I determines a conjugacy between F C and a map on G. To define this conjugacy, we equip G with the cylinder topology, where the basic open sets are of the form B b1,...,b k = {(x 1, x 2,...) G, x i = b i for i = 1,..., k}. Then the map F restricted to C is conjugate to the map α : G G defined by α(x) = (x) + (1). Specifically, I((x) + (1)) = F (I(x)). This follows directly from condition (c) and equation (1) on each f i. By comparison, the return map to a solenoid flow is conjugate (on its exceptional set) to an adding machine, or odometer. The adding machine for a sequence of integers {p 1, p 2,...} (not necessarily pairwise prime) is the map β : G G, defined by β(x i, x 2,...) = (y 1, y 2,...) where { x i + 1 mod p i if x j = p j 1 for all j < i, y i = x i otherwise.

6 TWISTED SOLENOIDS 73 Notice that the map α has more twisting than β. That is, the early coordinates of points in G change more under the action of α than under the action of β. In the setting of maps of R 2, it means that the points are rotated more by F than by the return map for the solenoid. For this reason, we refer to the suspension of F as a twisted solenoid and to the case where i=1 1 p i diverges as an over-twisted solenoid. 4. Analysis of F The argument that F is smooth relies on Theorem 2, which is an extension of Theorem 7.17 in [13]. Before stating the theorem, we ask the reader to recall the following definition. Definition 1. We say an ordered pair α of non-negative integers is a multi-index with length α = α 1 + α 2 and define the differential operator α by α F α F = x α. 1 1 xα 2 2 Theorem 2. For each i N let f i : R 2 R 2 be a smooth function and suppose the sum f i f i+1 C i i=1 converges where, for F (x) = (F 1 (x), F 2 (x)), (3) F C m = sup F j (x) + j {1,2} sup 1 α m sup α F j (x). j {1,2} Then there is a function f C such that f i f 0. For convenience, we will identify R 2 with C and use the notation of the complex plane to state the following technical lemma. Lemma 1. For any positive integers p, k, real numbers 0 < a < b < 1, and any real number ε > 0 there exists a C diffeomorphism φ : C C that satisfies the following: (1) There is a prime number q > p so that φ(z) = ze 2πi/q for all z C with z a. (2) φ(z) = z for all z C such that z b. (3) φ Id C k < ε, where Id(z) = z is the identity function.

7 74 W. BASENER AND C. V. LUTZER Proof: Let ρ : R R be a C function that is monotonically decreasing on (a, b) such that ρ(r) = 1 for r < a 2 and ρ(r) = 0 for r > b 2, and define the matrix valued function A : R 2 R 2 2 by ( ) ( ) (4) A(x) = cos 2π q ρ(r2 ) sin 2π q ρ(r2 ) ( ) ( ), sin 2π q ρ(r2 ) cos 2π q ρ(r2 ) where r 2 = x x2 2. Then the function φ defined by φ(x) = A(x)x satisfies (1) and (2) from the lemma. We will show that φ(x) also satisfies (3) when q is chosen large enough. From Theorem 2(3), we have (5) φ Id C k = sup j {1,2} (φ Id) j + sup sup 1 α k j {1,2} α (φ Id) j We proceed by showing that each of the terms on the right side of this equation can be made less than ε/2 if q is chosen sufficiently large. We begin by considering the first component of (φ Id)(x), which is part of the first summand of (5): (φ Id) 1 (x) ( ) ( ) 2π 2π = x 1 cos q ρ(r2 ) x 2 sin q ρ(r2 ) x 1 ( ) ) ( ) 2π 2π = x 1 (cos q ρ(r2 ) 1 x 2 sin q ρ(r2 ). Note that (φ Id) 1 (x) = 0 when x > 1 since ρ(r) = 0 for r > 1 > b. It follows that (φ Id) 1 sup (φ Id) 1 (x). x 1 That is, we may assume that x x This allows us to choose q 1 sufficiently large that, when q q 1, ( ) 2π cos q ρ(r2 ) 1 < ε 4 and sin ( 2π q ρ(r2 ) ) < ε 4.

8 TWISTED SOLENOIDS 75 Similarly, there is a q 1 associated with the second component of (φ Id), and by choosing q max{q 1, q 1 }, we ensure that the first summand of (5) is bounded above by ε 2. Next, we address the second summand of equation (5). As before, we need only consider x x The case of α = 1 is illuminating, so we begin by calculating (6) (φ Id) 1 = x [ 1 ( ) ] 2π cos q ρ(r2 ) 1 ( ) 2π 4πx 2 sin q ρ(r2 ) 1 ρ (r 2 ) q ( ) 2π 4πx1 x 2 ρ (r 2 ) cos q ρ(r2 ) q We can make the first summand on the right-hand side of equation (6) arbitrarily small by choosing q sufficiently large. Further, because ρ is smooth with compact support, its derivatives are bounded. Along with the fact that x x2 2 1, this allows us to control the magnitude of the second and third summands of (6). Higher order derivatives of (φ Id) result in more factors of ρ and higher order derivatives of ρ, but each is divided by at least one factor of q. Thus, by choosing q sufficiently large, we can guarantee that sup 1 α k α (φ Id) j < ε 2 for j {1, 2}, and this concludes the proof. Denote the smooth diffeomorphism of Lemma 1(1) by φ (a,b);p,k,ε and define f 1 := φ (1,1.1);1,1,0.5. We note that f 1 rotates the unit disk by 2π/q 1 for some prime number q 1 > 1, and that f 1 is the identity outside of the disk of radius 1.1 centered at the origin. Further, f 1 Id C 1 < 0.5. We will use the n th roots of unity to guide our development of f j, for j > 1. Denote these by u n,k := ( cos ( ) ( 2πk n, sin 2πk )) n, 0 k (n 1). Toward defining f 2, choose numbers 0 < a 1 < b 1 < 1 such that u q1,i u q1,j > 4b 1 whenever i j. We define the points c i = 1 2 u q 1,i for i = 0, 1,..., (q 1 1)

9 76 W. BASENER AND C. V. LUTZER and the disks D i = B a1 (c i ) for i = 0, 1,..., (q 1 1) E i = B b1 (c i ) for i = 0, 1,..., (q 1 1) where B r (c) is the closed ball of radius r centered at the point c. Notice that f 1 (D i ) = D i+1 mod q1. Choose some number p 2 > q 1 and, continuing to use the notation of C, define ψ i (z) = φ (a1,b 1 );p 2,2,0.25(z c i ) + c i, 0 i q 1 1. Then set f 2 = ψ q1 1 ψ q1 2 ψ 0. The function f 2 rotates each D i by 2π/q 2, where q 2 is prime and q 2 > p 2 > q 1, and it is the identity function outside the disks E i. Further, f 2 Id C 2 < For each i and j, 0 i q 1 1 and 0 j q 2 1, define c (i,j) = c i + a 1 2 u q 2,j. Notice that f 1 (c (i,j) ) = c (i+1 mod q1, j) and f 2 (c (i,j) ) = c (i,j+1 mod q2 ), so f 2 f 1 (c (i,j) ) = c (i+1 mod q1, j+1 mod q 2 ). Choose 0 < a 2 < b 2 < 1 so that c (i1,j 1 ) c (i2,j 2 ) > 4b 2 whenever (i 1, j 1 ) (i 2, j 2 ) and for each i define the disks D (i,j) = B a2 (c (i,j) ) for j = 0, 2,..., (q 2 1) E (i,j) = B b2 (c (i,j) ) for j = 0, 2,..., (q 2 1). Following the centers c (i,j), we have f 2 f 1 (D (i,j) ) = D (i+1 mod q1,j+1 mod q 2 ). We continue by induction. Suppose the maps f j, 1 j (n 1), and the disks D (x1,x 2,...,x j ) and E (x1,x 2,...,x j ) (centered at the point c (x1,x 2,...,x j )) satisfy the following desiderata: For each j > 1, the function f j rotates each D (x1,x 2,...,x j ) by 2π/q j for some prime number q j > q j 1. For every j, f j Id C j < 2 j. For every j, the function f j is the identity outside of the disks E (x1,x 2,...,x j ). The intersection E (x1,x 2,...,x j ) E (y1,y 2,...,y j ) = when (x 1, x 2,..., x j ) (y 1, y 2,..., y j ).

10 TWISTED SOLENOIDS 77 For every (x 1, x 2,..., x j ) Z q1 Z q2 Z qj f j f j 1 f 1 (D (x1,x 2,...,x j )) = D (x1 +1 mod q 1,x 2 +1 mod q 2,...,x j +1 mod q j ). we have There exist positive numbers a n 1 < b n 1 < 1 such that c (x1,x 2,...,x n 1 ) c (y1,y 2,...,y n 1 ) > 2b n 1 for all (x 1, x 2,..., x j ) (y 1, y 2,..., y j ). We choose some number p n > q n 1 and set G n := Z q1 Z q2 Z qn 1. Then for each γ G n, we define ψ γ (x) = φ (an 1,b n 1 );p n,n,2 n(x c γ) + c γ. The function f n is defined as (7) f n = γ Gn ψ γ. That is, f n is the composition of all ψ (x1,x 2,...,x n 1 ), where (x 1, x 2,..., x n 1 ) Z q1 Z q2 Z qn 1. (The order of composition does not matter because the set of points on which ψ γ1 is not the identity is disjoint from the set of points on which ψ γ2 is not the identity when γ 1, γ 2 are distinct elements of G n.) The function f n rotates each disk D (x1,x 2,...,x n 1 ) by 2π/q n for some prime number q n > p n > q n 1, is the identity outside of γ Gn E γ, and f n Id C n < 2 n. For each (x 1, x 2,..., x n ) Z q1 Z q2 Z qn, define c (x1,x 2,...,x n) = c (x1,x 2,...,x n 1 ) + a n 1 2 u q n,x n. Then f n (c (x1,x 2,...,x n )) = c (x1,x 2,...,x n+1 mod q n ) and f n f n 1 f 1 (c (x1,x 2,...,x n)) = c (x1 +1 mod q 1,x 2 +1 mod q 2,...,x n+1 mod q n). Choose positive numbers a n < b n < 1 so that c (x1,x 2,...,x n) c (y1,y 2,...,y n ) > 4b n whenever (x 1, x 2,..., x n ) (y 1, y 2,..., y n ) and, for each (x 1, x 2,..., x n ) Z q1 Z q2 Z qn, define the disks D (x1,x 2,...,x n ) = B an (c (x1,x 2,...,x n )) E (x1,x 2,...,x n) = B bn (c (x1,x 2,...,x n)).

11 78 W. BASENER AND C. V. LUTZER Notice that f n (D (x1,x 2,...,x n)) = D (x1,x 2,...,x n+1 mod q n). Hence, f n f n 1 f 1 (D (x1,x 2,...,x n )) = D (x1 +1 mod q 1,x 2 +1 mod q 2,...,x n+1 mod q n). We now define F n = f n f n 1 f 1. Note that F n differs from F n 1 on only the disks E (x1,x 2,...,x n ), and then only by the rotation affected by f n, so F n 1 F n C n = f n Id C n < 2 n. Thus, F n 1 F n C n <, n=1 and Theorem 2 guarantees the existence of a smooth function F such that F n F uniformly. We have thus proven Theorem Remarks 5.1. A homeomorphism on R 3 with Antoine s necklace as a minimal set Antoine s necklace can be defined in the following way (see [12]). Let q 1, q 2,... be an infinite sequence of positive integers. Let T = S 1 D 2 be a solid torus coordinatized by (θ, φ, r) where θ [0, 2π] mod 2π is the coordinate on S 1 and (φ, r) are polar coordinates on D 2. Inside T define a chain of solid tori T 0,..., T q1 1 as follows. For each i {0, 1,..., q 1 1}, let p i = (i2π/q 1, 0, 0), and let γ i be the circle of radius 3π/4q 1 that is centered at p i and contained in {(θ, φ, r) φ = i2π/q 1 }. (See Figure 2.) Note that the linking number of γ i with γ j is ±1 if i j mod q 1 = 1 and that γ i is the image of γ i 1 mod q 1 under the rotation of T by (θ, φ, r) (θ + i2π/q 1, φ + i2π/q 1, r). From now on we refer to this homeomorphism simply as rotation by (i2π/q 1, i2π/q 1 ). For each i, define T i to be a torus neighborhood of γ i such that all of the T i are disjoint and a rotation of T by (i2π/q 1, i2π/q 1 ) takes T i to T i+1 mod q 1. Denote the union of these tori by C 1 = q 1 1 i=0 T i. In T 0, define a chain of q 2 pairwise disjoint solid tori, T (0,0),..., T (0,q2 1), such that a rotation of T 0 by (2π/q 2, 2π/q 2 ) takes T (0,i) to T (0,i+1 mod q 2 ). Let T (i,j) be the images of T (0,j) under rotations of T by (2π/q 1, 2π/q 1 ) and let C 2 = q 2 1 q1 1 j=0 i=0 T (i,j). Continuing

12 TWISTED SOLENOIDS 79 Figure 2. The first two steps in creating Antoine s necklace. in this manner results in a nested sequence of compact sets C 2 C 1 and Antoine s necklace is the set A = C i. i=1 The topologically interesting property of Antoine s necklace is that it is a Cantor set and its complement is not simply connected when imbedded in R 3. The natural homeomorphism J : G A is J(x 1, x 2,... ) = T T x1 T (x1,x 2 ). One could define a map that has A as an exceptional set by mimicking the construction outlined in the previous sections The D-function An important topological invariant of a minimal set is the D- function, developed by Xiang Dong Ye in [14]. Suppose that f : X X is a continuous map of a compact Hausdorff space and that Y is a minimal set for f. The D-function for Y is the function

13 80 W. BASENER AND C. V. LUTZER f Y : N N which takes the natural number n to the number of distinct minimal sets of f n which are contained in Y. If p 1, p 2,... is the sequence of pairwise relatively prime positive integers defining F, then the D-function of F on C is F C (n) = p i. {p i :p i n} References [1] Harold Bell and Kenneth R. Meyer, Limit periodic functions, adding machines, and solenoids, J. Dynam. Differential Equations 7 (1995), no. 3, [2] Christian Bonatti, Jean-Marc Gambaudo, Jean-Marie Lion, and Charles Tresser, Wandering domains for infinitely renormalizable diffeomorphisms of the disk, Proc. Amer. Math. Soc. 122 (1994), no. 4, [3] César Camacho and Alcides Lins Neto, Geometric Theory of Foliations. Boston, MA: Birkhäuser Boston, Inc., [4] J.-M. Gambaudo, D. Sullivan, and C. Tresser, Infinite cascades of braids and smooth dynamical systems, Topology 33 (1994), no. 1, [5] J.-M. Gambaudo, S. van Strien, and C. Tresser, Hénon-like maps with strange attractors: there exists C Kupka-Smale diffeomorphisms on S 2 with neither sinks nor sources, Nonlinearity 2 (1989), no. 2, [6] Jean-Marc Gambaudo and Charles Tresser, Diffeomorphisms with infinitely many strange attractors, J. Complexity 6 (1990), no. 4, [7], Self-similar constructions in smooth dynamics: rigidity, smoothness and dimension, Comm. Math. Phys. 150 (1992), no. 1, [8] W. H. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), [9] Glen Richard Hall, A C Denjoy counterexample, Ergodic Theory Dynamical Systems 1 (1981), no. 3, [10] Anatole Katok and Boris Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge, UK: Cambridge University Press, [11] L. Markus and Kenneth R. Meyer, Periodic orbits and solenoids in generic Hamiltonian dynamical systems, Amer. J. Math 102 (1980), no. 1, [12] Dennis Roseman, Elementary Topology. Upper Saddle River, NJ: Prentice Hall, [13] Walter Rudin, Principles of Mathematical Analysis. 3rd ed. International Series in Pure and Applied Mathematics. New York-Auckland-Dsseldorf: McGraw-Hill Book Co., 1976.

14 TWISTED SOLENOIDS 81 [14] Xiang Dong Ye, D-function of a minimal set and an extension of Sharkovskiĭ s theorem to minimal sets, Ergodic Theory Dynamical Systems 12 (1992), no. 2, [15] Zhi Fen Zhang, A topological dynamical system in R 3 with Antoine s necklace as a minimal set, Sci. Sinica Ser. A 25 (1982), no. 9, (Basener) Department of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY address: wfbsma@rit.edu (Lutzer) Department of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY address: cvlsma@rit.edu