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3-manifolds that admit knotted solenoids as attractors

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.


Introduction
The solenoids are first defined in mathematics by Vietoris in 1927 for 2-adic case and by others later in general case, which can be presented either in an abstract way (inverse limit of self-coverings of circles) or in a geometric way (nested intersections of solid tori). The solenoids are introduced into dynamics by Smale as hyperbolic attractors in his celebrated paper [S].
Standard notions in dynamics and in 3-manifold topology will be given in Section 2. The new definitions are the following: Let N = S 1 × D 2 , where S 1 is the unit circle and D 2 is the unit disc. Both S 1 and D 2 admit "linear structures". Let e : N → N be a "linear", D 2 -level-preserving embedding such that (a) e(S 1 × * ) is a w-string braid in N for each * ∈ D 2 , where w > 1 in an integer; (b) for each θ ∈ S 1 , the radius of e(θ × D 2 ) is 1/w 2 .
Definition. Let M be a 3-manifold and f : M → M be a diffeomorphism. If there is a solid torus N ⊂ M such that f |N (resp. f −1 |N ) conjugates e : N → N above, we call S = ∩ ∞ h=1 f h (N ) (resp. S = ∩ ∞ h=1 f −h (N )) a Smale solenoid, which is a hyperbolic attractor (resp. repeller, or negative attractor) of f , and we also say M admits S as a Smale solenoid attractor and N is a defining solid torus of S.
Smale solenoid in the above definition carries more information than a solenoid as a topological space. It also carries the information of braiding of e(N ) in N and the knotting and framing of N in M , in addition to the information that it is a hyperbolic attractor of a diffeomorphism f : M → M .
Say a Smale solenoid S ⊂ M is trivial, if the core of a defining solid torus N bounds a disc in M , otherwise we say S is knotted.
Theorem 1. Suppose M is a closed orientable 3-manifold. There is a diffeomorphism f : M → M such that the non-wandering set Ω(f ) contains a knotted Smale solenoid IF and ONLY IF the manifold M has a lens space L(p, q), with p = 0, ±1, as a prime factor.
Theorem 2. Suppose M is a closed orientable 3-manifold. There is a diffeomorphism f : M → M with the non-wandering set Ω(f ) a union of finitely many Smale solenoids, IF and ONLY IF the manifold M is a lens space L(p, q), p = 0.
Corollary. The diffeomorphism f constructed in the IF part of Theorem 2 is Ω-stable, but is not structurally stable.

From Morse theory
Let f : M → R be a non-degenerate Morse function. Then the gradient vector field gradf is a dynamical system on M with hyperbolic Ω(gradf ). An important aspect of Morse theory is to use the global information of the singularities of f , or equivalently, the information of Ω(gradf ), to provide topological information of the manifold M . The classical examples are: if Ω(gradf ) consists of two points, then M is the sphere by Reeb in 1952 [R], and if Ω(gradf ) consists of three points, then M is a projective plane like manifold of dimension 2, 4, 8 or 16 proved by Eells and Kuiper in 1961 [EK]. The ONLY IF part of Theorems 1 and 2 are results of this style.

From dynamics of Smale's school
In [S], for a diffeomorphism f : M → M , Smale introduced the Axiom A, the strong transversality condition and the no cycle condition for Ω(f ). Important results in the dynamics school of Smale are the equivalences between those conditions and various stabilities. For an Axiom A system f , Smale proved (Spectral Decomposition Theorem) Ω(f ) can be decomposed into the so-called basic sets. He posed several types of basic sets: (a) Zero dimensional ones such as isolated points and Smale Horse Shoe; (b) Anosov maps and maps derived from Anosov; (c) expansive ones such as Smale solenoids.
All those results and notions need examples to testify. Most known examples are local. It is natural to ask a global question where topology and dynamics interact: For which manifold M , is there an f : M → M such that all the basic sets of Ω(f ) belong to a single type above?
There is no restriction when Ω(f ) is zero dimensional. The answer to the question for Anosov map was given by Porteous in 1974 [Po]. The ONLY IF part of Theorem 2 gives an answer about Smale solenoids for 3-manifolds. The Corollary also provides 3-dimensional global examples to testify the notions of stability.
We would also like to point out that there are many nice results on the interplay of topology and dynamics, mostly for flows. See [F], [Su] and [T] for examples.

Searching symmetries of manifolds with stability.
A manifold M admitting a dynamics f such that Ω(f ) consists of two hyperbolic attractors presents a symmetry of the manifold with certain stability. The sphere, the simplest closed manifold, admits a hyperbolic dynamics f such that Ω(f ) consists of exactly two points, one is a source, and the other is a sink. The attractors in this example are the simplest in three senses: (1) The topology of the attractors are trivial, (2) the embedding of attractors into the manifolds are trivial, (3) the restriction of the dynamics f on the attractors are trivial. The IF part of Theorem 2 and the Corollary show more manifolds with such symmetry when we consider more complicated attractors suitably embedded into the manifolds.
Indeed we believe that many more 3-manifolds admit such symmetries if we replace the Smale solenoid by its generalization, the so-called Smale-Williams solenoid [W] (the name is suggested in [Pe]).

The structure of the paper
For the convenience of the readers from both dynamics and 3-manifold topology, we list the needed notions and facts in dynamics and in 3-manifold topology in Section 2. Sections 3, 4 and 5 are devoted respectively to the proofs of the ONLY IF parts of Theorems 1 and 2, the IF parts of Theorems 1 and 2, and the Corollary. Most notions in dynamics mentioned in Section 2 are only used in Section 5. To the authors, the most interesting part of the paper is the discovery of the IF part of Theorem 2 and its explicit constructive proof. Since such an explicit constructive proof is difficult to generalize to the case of Smale-Williams solenoids, we wonder if there is an alternative proof for the IF part of Theorem 2.

Notions and facts in dynamics and in 3-manifold topology From Dynamics
Everything in this part can be found in [Ni], unless otherwise indicated.
Then Ω(f ), the non-wandering set of f , defined as the set of all non-wandering points, is an for some fixed C > 0 and λ > 1.
, Ω(f ) can be decomposed in a unique way into finitely many disjoint sets B 1 , ..., B k , so that each B i is closed, f -invariant and contains a dense f -orbit.
The B i in the decomposition above are usually referred to as basic sets.
x and E u x respectively. Moreover, they are tangent to E s x and E u x at x respectively. W s (x, f ) and W u (x, f ) in the theorem are known as the stable and unstable manifold of f at x.
The Strong Transversality Condition. For all x, y ∈ Ω(f ), the stable and unstable manifolds, W s (x, f ) and W u (y, f ), are transverse.
The No Cycle Condition. An n-cycle of Axiom A system is a sequence of basic sets Ω 0 , Ω 1 , ..., Ω n with Ω 0 = Ω n and Ω i = Ω j otherwise, and such that W u (Ω i−1 ) ∩ W s (Ω i ) = ∅. An Axiom A system satisfies the no-cycle condition if it has no n-cycle for all n ≥ 1.
Stability Theorem. (See the survey paper [Ha]) (a) the Axiom A and the strong transversality condition of Ω(f ) are equivalent to the structural stability of f . (b) the Axiom A and the no cycle condition of Ω(f ) are equivalent to the Ω-stability of f .

From 3-manifold theory
Everything in this part can be found in [He], unless otherwise indicated.
Let M be a 3-manifold and S an embedded 2-sphere separating M . Let M 1 and M 2 be the two 3-manifolds obtained by splitting M along S and capping-off the two resulting 2-sphere boundary components by two 3-cells.
Let F be a connected compact 2-sided surface properly embedded in M . F is said to be compressible if either F bounds a 3-ball, or there is an essential, simple closed curve on F which bounds a disk in M ; otherwise, F is said to be incompressible.
The following three results in 3-manifold topology are fundamental. Kneser-Milnor's Prime Decomposition Theorem. Every closed orientable 3-manifold M = S 3 can be expressed as a connected sum of a finite number of prime factors. Furthermore, the decomposition is unique up to order and homeomorphism.
Haken's Finiteness Theorem. Let M be a compact orientable 3manifold. Then the maximum number of pairwise disjoint, non-parallel, closed connected incompressible surfaces in M , denoted by h(M ), is a finite integer ≥ 0.
Papakyriakopoulos's Loop Theorem. Let M be a compact orientable 3-manifold and S ⊂ M a closed orientable surface. If the homomorphism i * : π 1 (S) → π 1 (M ) induced by the embedding i : S → M is not injective, then there is an embedded disc D ⊂ M such that D ∩ S = ∂D and ∂D is an essential circle in S.
For the definition of the lens space L(p, q), see Section 4.

Proof of the ONLY IF parts of Theorems 1 and 2
We first prove the ONLY IF part of Theorem 1.
Proof. Suppose f : M → M has a knotted Smale solenoid S as an attractor. If L is S 3 , then it is easy to see the core of N bounds a disc, which contradicts the assumption that S is knotted. If L = S 2 × S 1 , then N carries a generator α of π 1 (S 2 × S 1 ) = Z. Since f (N ) is a w-string braid in N , we have f * (α) = wα. Since f is a homeomorphism, f * is an isomorphism. Hence w = 1, and we reach a contradiction.
We have finished the proof of the ONLY IF part of Theorem 1.
We are going to prove the ONLY IF part of Theorem 2.
Suppose Ω(f ) is a union of Smale solenoids S 1 , ..., S n . Then for each i = 1, ..., n, it is known (more or less directly from the definition) that (i) f |S i is hyperbolic and the periodic points of f are dense in S i ; (ii) S i is an f -invariant closed set and there is a dense f -orbit in S i . Then f satisfies the Axiom A by (i). By Spectral Decomposition Theorem, Ω(f ) can be decomposed in a unique way into finitely many disjoint basic sets B 1 , ..., B k , so that each B i is closed, f -invariant and contains a dense f -orbit.
By (ii), each S i ⊂ B l for some l = 1, ..., k. Then from the facts that S i is an attractor of f (or of f −1 ) and that B l contains a dense f -orbit, there is a point x ∈ Int U i so that its f -orbit o(x) is dense in B l , where U i is a closed neighborhood of S i mentioned in the definition of an attractor. Then it is clear that x ∈ Ω(f |U i ). Hence x ∈ S i , thus B l = o(x) ⊂ S i , so we must have S i = B l . Hence each S i is a basic set of Ω(f ) and in particular, Ω(f ) is a disjoint union of finitely many Smale solenoids. Now the ONLY IF part of Theorem 2 follows from the following Lemma 1 and Lemma 2.
Lemma 1. Suppose f : M → M is a diffeomorphism and Ω(f ) is a disjoint union of finitely many Smale solenoids.
Then Ω(f ) is a union of two solenoids, one is an attractor of f and the other is an attractor of f −1 .

Proof.
Suppose is a disjoint union of solenoids, where either Without loss of generality, we assume that the N i 's have been chosen so that , and some S i is an attractor of f (otherwise replace f by f −1 ). Then by re-indexing if necessary we assume that S 1 , ..., S k are attractors of f and the remaining S j are attractors of f −1 ; henceforth, k is the number of attracting solenoids. So we can assume and and by the assumptions (2) and We have proved that f has exactly one attractor. By the same reason f −1 also has exactly one attractor, therefore n = 2 and the lemma is proved.
Lemma 2. Let M be a closed orientable 3-manifolds. If f : M → M is a diffeomorphism with Ω(f ) a union of two disjoint Smale solenoids, then M is a lens space and M is not S 2 × S 1 .

Proof.
Suppose Ω(f ) is a union of two disjoint solenoids S 1 and S 2 . We may further assume that It follows that f n (∂N 2 ) ⊂ Int N 1 , M −N 1 ⊂ f n (N 2 ) for some large integer n > 1. (7) Since H 2 (N 1 , Z) = 0, ∂f n (N 2 ) separates N 1 into two parts Y ′ and Y ′′ with ∂Y ′ = ∂f n (N 2 ) and ∂Y ′′ has two components.
The homomorphism i * : π 1 (∂f n (N 2 )) → π 1 (N 1 ) induced by the embedding i : ∂f n (N 2 ) → N 1 is not injective, since π 1 (N 1 ) = Z and π 1 (f n (∂N 2 )) = Z ⊕ Z. By the Loop Theorem, ∂f n (N 2 ) is compressible in N 1 , that is, there is an embedded disc D ⊂ N 1 such that D ∩ ∂N 2 = ∂D and ∂D is an essential circle in ∂f n (N 2 ). Since the solid torus N 1 is irreducible, a standard argument shows that ∂f n (N 2 ) bounds a solid torus N ′ in N 1 , and therefore we have N ′ = Y ′ . Then by (7), we have (8) Hence M is obtained by identifying two solid tori f n (N 2 ) and N ′ along their common boundary. So M is a lens space.
Since f is a homeomorphism, N ′′ = f −n (N ′ ) is also a solid torus and M is obtained by identifying two solid tori N 2 and N ′′ along their boundary. Now f −1 (N 2 ) is a w-string braid in N 2 , w > 1. That M is not S 2 × S 1 can be proved as before.

Proof of the IF part of Theorem 2
The IF part of Theorem 2 is equivalent to the following Claim. Suppose M is a lens space L(p, q), p > 0. Then there is a diffeomorphism f : M → M with Ω(f ) a union of two (p + 1)-adic solenoids, one is an attractor, the other is a repeller.
We are going to prove this Claim. Denote the oriented cores of N 1 , N 2 by c 1 , c 2 (c i is homologous to λ i in N i ) respectively. We do the following operations to c 1 , as indicated in Figure 1: Lens space L(p, q) as union of solid tori N 1 ∪ ϕ N 2 PSfrag replacements Figure 2: Writhe the core c 1 in N 1 Figure 3: Closed braid β 1 in N 1 and core c 2 of N 2 PSfrag replacements Figure 4: Core c 1 of N 1 and closed braid β 2 in N 2 Figure 2: Writhe c 1 locally, moving a subarc ab toward ∂N 1 and identify it with a subarc of ϕ(µ 2 ). Since µ 2 bounds a meridian disk in N 2 , we can push ab across the disk, the effect seen in N 1 is to replace ab with its complement in ϕ(µ 2 ), see Figure 2. Finally, pushing the obtained curve into Int N 1 , we get a closed braid β 1 in N 1 , as indicated in Figure 3. (In fact, β 1 is the "connected sum" of the "writhed" c −1 with ϕ(µ 2 ) in N 1 .) Do similar operations to c 2 in N 2 , we get a closed braid β 2 in N 2 , as indicated in Figure 4. Now β 1 ⊔ c 2 and c 1 ⊔ β 2 are two links in M .
Proof. Recall that β 1 is obtained by isotoping c 1 , thus if we perform the inverse of the above isotopy, we can transform β 1 into c 1 . We will show that the same isotopy also transforms c 2 into β 2 . From now on, we only use local pictures (represented in the rectangular frame in Figures 1-4) to show changes in both N 1 and N 2 simultaneously. The initial local picture of β 1 ⊔ c 2 is shown in Figure 5-1. In N 1 (on the left), β 1 is a closed braid in N 1 , and a segment of c 2 is shown outside of ∂N 1 . On the right, most of β 1 coincides with µ 2 , along with the part slightly outside ∂N 2 , and c 2 is the core of N 2 . Our isotopy consists of the following three steps: Step 1. µ 2 bounds a meridian disk in N 2 , so we can pull β 1 across the disk. At the same time, a subarc of c 2 is pulled into N 1 , as indicated in Figure 5-2.
Step 2. In the local picture Figure 5-2, β 1 has a self-crossing. A local half twist will eliminate this self-crossing, as indicated in Figure 5-3. Take care so that a subarc cd of c 2 lies on ϕ −1 (µ 1 ). Now compare Figure 5-2 and Figure 5-3, we find an interesting fact: except for the colors and labels, the left/right part of Figure 5-2 is the same as the right/left part of Figure 5-3. This symmetry suggests that the next step is a kind of inverse to Step 1.
Step 3. Push the subarc cd across the meridian disk of N 1 , as indicated in Figure 5-4. We see that β 1 is deformed to c 1 , and c 2 is deformed to β 2 .
Proof of the Claim. Note that β i is a (p + 1)-string braid in N i = S 1 × D 2 i . Isotope β i in N i to meet all fiber discs * × D 2 i transversely. Let N (γ) denote the closed tubular neighborhood of a closed curve γ, and think of N i as N (c i ).
Choose N (β i ) to be a disc bundle over β i embedded into N i so that each disc fiber ⊂ * × D 2 (for * ∈ S 1 ) and has diameter < 1/(p + 1) 2 . Moreover we may assume that N (β i ) misses the core c i .
Pull across the meridian disk of N 2 PSfrag replacements Push across the meridian disk of N 1 PSfrag replacements Figure 5: Local pictures of the 3-step isotopy from β 1 ⊔ c 2 to c 1 ⊔ β 2 The isotopy provided by Lemma 3 that sends β 1 ⊔ c 2 to c 1 ⊔ β 2 can be adjusted to send N (β 1 )⊔N 2 to N 1 ⊔N (β 2 ), and to be "linear" and "disc-fiber preserving" on N (β 1 )⊔ N 2 . Then extend it to a diffeomorphism f : M → M which sends N 1 − N (β 1 ) to N 2 − N (β 2 ). Now the (p + 1)-adic solenoids S 1 = ∩ ∞ h=1 f −h (N 1 ) and S 2 = ∩ ∞ h=1 f h (N 2 ) are the repeller and the attractor of f respectively. Moreover for each x / ∈ S 1 ∪ S 2 , f n (x) approaches to S 2 as n approaches to infinity, hence Ω(f ) = S 1 ∪ S 2 .
We have finished the proof of the Claim, therefore the IF part of Theorem 2.
Remark. By repeating the operations in the proof, we see that in the IF part of Theorem 2, the Ω(f ) can be chosen to be two explicit (mp + 1)-adic solenoids, mp + 1 = 0, ±1.

Proof of the IF part of Theorem 1
Suppose M = N #L(p, q). It is easy to see that the isotopy above that sends β 1 ⊔ c 2 to c 1 ⊔ β 2 can be adjusted to send N 2 to N (β 2 ), to be "linear" and "disc-fiber preserving" on N 2 , and to be the identity on a 3-ball B 3 in N 1 . Therefore there is a diffeomorphism on the L(p, q) − intB 3 which has a knotted solenoid as a hyperbolic attractor and is identity on its 2-sphere boundary. Such a diffeomorphism can be extended to M by the identity on the punctured N .
We have proved the IF part of Theorem 1.

Proof of the Corollary
We start from the end of the proof of the IF part of Theorem 2.
Since Ω(f ) consists of two Smale solenoids S 1 and S 2 , Ω(f ) meets Axiom A.
To prove the Corollary, we need the following explicit description of stable and unstable manifolds of Ω(f ).
First, S 1 is the union of stable manifolds of points in S 1 , and S 2 is the union of unstable manifolds of points in S 2 . Moreover, since f −1 |N 1 (resp. f |N 2 ) preserves the disc fibers of N 1 (resp. N 2 ), F 1 = ∪f n (S 1 × D 1 ) (resp. F 2 = ∪f −n (S 1 × D 2 )) provides an R 2 -foliation of L(p, q) − S 2 (resp. R 2foliation of L(p, q) − S 1 ), which is the union of unstable manifolds of points in S 1 (resp. the union of stable manifolds of points in S 2 ). Hence we have W s (S 1 ) = S 1 , W u (S 1 ) = F 1 , W u (S 2 ) = S 2 , W s (S 2 ) = F 2 .

Therefore
It is clear that f meets the no cycle condition. Hence f is Ω-stable by (b) of the Stability Theorem.
This f is not structurally stable by the Stability Theorem (a) and the following Lemma 4.
Lemma 4. F 1 and F 2 do not meet transversely.
Proof. We need only to prove that F 1 | and F 2 |, the restrictions of F 1 and F 2 on N 1 − N (β 1 ) respectively, do not meet transversely.
It is easy to see that the restrictions of two fibrations F 1 | and F 2 | on N 1 − N (β 1 ) meet transversely on ∂N 1 − N (β 1 ).
Let F 1 be a fiber of F 1 |, which is a (p + 1)-punctured disc. Suppose F 1 | and F 2 | meet transversely on N 1 − N (β 1 ). Then the intersections of F 1 and F 2 | provide a codimension one foliation on F 1 which meets ∂F 1 transversely. Now the genus (p + 1) closed surface D(F 1 ), the double of F 1 , will admit a codimension one foliation, which is impossible since |p + 1| > 1.
We have completed the Proof of the Corollary.