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Genus two Smale-Williams solenoids in 3-manifolds

Using alternating Heegaard diagrams, we construct some 3-manifolds which admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms are composed of Smale-Williams solenoid attractors and repellers, an interesting example is the truncated-cube space. In addition, we prove that if the nonwandering set of the diffeomorphism consists of genus two Smale-Williams solenoids, then the Heegaard genus of the closed manifold is at most two.

Some notions in 3-dimensional manifolds theory and in dynamics will be given in Section 2, for the definition of alternating Heegaard splitting, see Section 4.

Notions and facts in 3-dimensional manifolds theory and in dynamics
For fundamental facts about 3-manifolds see [H] and [J]. Let G be a finite graph in R 3 , then a regular neighborhood H of G in R 3 is called a handlebody, it is a 3-manifold with boundary, the genus of its boundary is called the genus of H, denoted by g (H). Let M be a closed orientable 3-manifold, if there is a closed orientable surface S in M which separates M into two handlebodies H 1 and H 2 , then we say M = H 1 ∪ S H 2 is a Heegaard splitting of M , S is called a Heegaard surface.
Any closed orientable 3-manifold has infinitely many Heegaard splittings, and the minimum of the genus of the Heegaard surfaces is called the Heegaard genus of the 3-manifold M .
A properly embedded 2-sided surface F in a 3-manifold M is called an incompressible surface if it is π 1 -injective, otherwise, it is a compressible surface.
The following theorems will be used in the paper: Haken Finiteness Theorem. Let M be a compact orientable 3-manifold. 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 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 disk D ⊂ M such that D ∩ S = ∂D and ∂D is an essential circle in S.
We recall some facts about Smale-Williams solenoid from the famous paper [W1].
Definition 2.1: A branched 1-manifold L, is just like a smooth 1-manifold, but there are two type of coordinate neighborhoods are allowed. These are the real line R and Y = {(x, y) ∈ R 2 : y = 0 or y = ϕ(x)}. Here ϕ : R −→ R is a fixed C ∞ function such that ϕ(x) = 0 for x ≤ 0 and The first Betti number β 1 (L) is called the genus of the branched 1-manifold, and which is just the genus of the handlebody that L induced(See Example 2.6).
Note that a branched 1-manifold L has a tangent bundle T (L), and a differentiable map f : L → L ′ between branched 1-manifolds induces a map Df : T (L) → T (L ′ ) of their tangent bundles.  Axiom 3. Any point p of L has a neighborhood U (p), such that g(U (p)) is an arc; Axiom 4. There is a finite set A ⊂ L, such that, g(A ∪ B) ⊂ A.
Definition 2.4: Let Σ be the inverse limit of the sequence where L is a branched 1-manifold, g is a Williams expansion map on L. For a point a=(a 0 , a 1 , a 2 , ...) Example 2.5: See [W1], Figure 1 contains all of the branched 1-manifolds with two branched points. Only the first two allow immersions satisfing Axiom 2. Let K be the first one and define the Williams expansion map g : K → K on its oriented 1-cells by: It is easy to check that g satisfies Axiom 1, 3, 4 of Definition 2.3. For Axiom 2, for some semi-conjugacy reason we only need to check that the induced symbolic dynamical system matrix X is irreducible, that is, for all 1 ≤ i, j ≤ dim(X), there exists N (i, j) > 0 such that the ij th entry of X N (i,j) is positive, see [BH]. The induced matrix X of g is Example 2.6: Figure 2(a) contains a genus two handlebody N with disk foliation and a self embedding f . In fact, we take N as a "neighborhood" of K, and there is a natural projection π from N to K, and the embedding f is induced by g in Example 2.5. We define Λ = ∞ n=1 f n (N ), then (Λ, f ) is conjugate to (Σ, h) via T constructed below. We have the following commutative diagram, ∀x ∈ Λ, T (x) is defined to be (π(x), π(f −1 (x)), ..., π(f −n (x)), ...). As we shall see in Proposition 2.7, T is the conjugate map. (1) There is a projection map π : N −→ L which gives a disc foliation structure of N by π −1 (p), ∀p ∈ L. f preserves the foliation structure, furthermore, the area satisfies Proof : This proof is based on the Example 2.6.
Step a. Take a "neighborhood" of L, which is a handlebody N , and we foliated N by disks as in Figure 2(a), so there is a natural projection π : N → L.
Step b. Obviously there is a local immersion map i : in N such that we get an embedding map i 1 : L ֒→ N such that π • i 1 = g. Now we take a neighborhood of i 1 (L), say N 1 , such that N 1 is embedded into N . Just like step a, it is easy to get an embedding map f : N ֒→ N such that, f (N ) = N 1 and satisfies condition (1) of Proposition 2.7.

Conjugation.
Let Σ be the Smale-Williams solenoid defined by (L, g), and h is the shift map. Define T : Since constructed above is an attractor model. Q.E.D.
This type of attractors have important meaning in the study of attractors, see Williams [W2].

Restriction on the Heegaard genus
We term the first two branched 1-manifolds type I and type II respectively, and discuss their Williams expansion map using their geometrical model introduced above. We call this type of attractor type I(type II) Smale-Williams solenoid attractor.
Proof: We prove the lemma only for type I Smale-Williams solenoid attractor, and the proof for the case of type II is similar.
Let K be a type I branched 1-manifold as Figure 1 shows and g be the induced map of f on K, which is a train track map, see [BH].
We choose a base point P for Figure 1). Since Im(g * ) is an abelian group, we have that is a legal path in K and g is a train track map, we have g( is also a legal path, so it can not be a homotopic trivial path in K(See [BH]). This means g * (xyx −1 y −1 ) = 1, it is a contradiction. Q.E.D. Proof : If the nonwandering set Ω(f ) consists of genus two Smale-Williams solenoids, then standard arguments in dynamics theory shows Ω(f ) = Λ 1 ⊔ Λ 2 where Λ 1 is an attractor, Λ 2 is a repeller (See [JNW]). In addition, Λ 1 (Λ 2 ) is realized by genus two handlebody N 1 (N (2-handle addition along c), and denote the resulting manifold by N * 2 : Case 1. If c is non-separating in ∂N 2 , ∂N * 2 = S is a torus. In this case N 1 is divided into two parts W 1 and W 2 by S, ∂W 2 = S ⊔ ∂N 1 and ∂W 1 = S: Subcase 1.1. S is compressible in W 1 , so W 1 is a solid torus, and N 1 − N 2 is a genus two handlebody. Hence g(M ) ≤ 2.
Subcase 1.2. S is incompressible in W 1 , since N 1 is a handlebody, S must be compressible in W 2 . Compressing S in W 2 , we get a 2-sphere P in W 2 , so P is also in N * 2 . P separates W 2 into A and B, where S ⊂ ∂A and ∂N 1 ⊂ ∂B. And we have N * 2 = A * ♯B * , where A * is obtained from A by capping off P , and B * is obtained from B ∪ N 2 − N 1 by capping off P . W 1 ∪ A is in the handlebody N 1 , so W 1 ∪ A is a 3-ball, π 1 (A * ) is nontrivial. And the rank of π 1 (N * 2 ) is at most two, we get that the rank of π 1 (B * ) is at most one. So B * has a genus one Heegaard splitting by the fact that the Heegaard genus of N * 2 is at most two. So M = B * has Heegaard genus at most one.
Case 2. If c is separating in ∂N 2 , ∂N * 2 = S is composed of two tori S 1 and S 2 , so there are two manifolds W 1 and W 2 in N 1 − N 2 with ∂W i = S i : Subcase 2.1. One of W 1 and W 2 is a solid torus. Then there is a nonseparating simple closed curve in ∂N 2 which bound a disk in N 1 − N 2 , and turn to Case 1.
Subcase 2.2. S i is incompressible in W i , so S i must be compressible in N 1 − W i . Let E be a compressible disk of S 1 in N 1 − W 1 , we can assume E ∩ D = ∅, E ∩ S 2 is a set of simple closed curves, and |E ∩ S 2 | is minimal along all such compressible disk of S 1 . If E ∩ S 2 = ∅ , take an innermost disk, say ∆, of E, which is a compressible disk of S 2 in N 1 − W 2 , which is also disjoint from D. Compressing S 2 along ∆, we get a 2-sphere P 2 , which bounds a 3-ball B 2 in N 1 , and W 2 ⊂ B 2 . From this 3-ball, and the compressibility of S 1 in N 1 − W 1 , we can get a 3-ball B 1 , which is disjoint from D and B 2 , and contains W 1 , ∂B 1 = P 1 . Connecting P 1 and P 2 by D, we get a separating 2-sphere P in N 1 ∩ N 2 , which separates M into two components, each component is    Figure 2(a), let N be a handlebody, K be a branched 1-manifold of type I, π : N → K be the natural projection, then there are three disks in N , say E 1 , E 2 and E 3 , which separate N into two 3-balls, and E 3 is separating in N , the image of E i in K i is an interior point in K i . Let c be a simple closed curve in ∂N , we say c is alternating with respect to (E 1 , E 2 , E 3 ), if the intersection points of E i occur in c alternatively about E 3 , that is, along c, we see E 3 , E i 1 , E 3 , E i 2 , E 3 , E i 3 E 3 ...., where i j = 1 or 2 and i j · i j+1 = 2 to all j.
Let M = N 1 ∪ N 2 be a genus two Heegaard splitting, if there are three disks D 1 , D 2 , D 3 separate N 1 into two 3-ball, where D 3 is separating in N 1 , and there are three disks C 1 , C 2 , C 3 separate N 2 into two 3-ball, where C 3 is separating in N 2 , moreover ∂C i is alternating with respect to (D 1 , D 2 , D 3 ), and ∂D i is alternating with respect to (C 1 , C 2 , C 3 ), then we say the Heegaard splitting is alternating of type I. Definition 4.3(Alternating Heegaard splitting of type II): As in Figure 2(b), let N be a handlebody, L be a branched 1-manifold of type II, π : N → L be the natural projection, then there are three disks in N , say E 1 , E 2 and E 3 , which separate N into two 3-balls, none of E i is separating in N , the image of E i in L i is an interior point in L i . Let c be a simple closed curve in ∂N , we say c is alternating with respect to (E 1 , E 2 , E 3 ), if the intersection points occur in c alternatively, that is, along c, we see Let M = N 1 ∪ N 2 be a genus two Heegaard splitting, if there are three disks D 1 , D 2 , D 3 separate N 1 into two 3-balls, none of D i is separating in N 1 , and there are three disks C 1 , C 2 , C 3 separate N 2 into two 3-balls, none of C i is separating in N 2 . Moreover ∂C i is alternating with respect to (D 1 , D 2 , D 3 ), and ∂D i is alternating with respect to (C 1 , C 2 , C 3 ), then we say the Heegaard splitting is alternating of type II. Proof : The proof is an explicit construction, we construct the diffeomorphism in the type I case, the type II case is similar.
We choose three parallel curves c 1 , c 2 , c 3 in ∂N which are parallel to c. The branched 1manifold J is a spine of N , which is composed of oriented 1-cells: J 1 , J 2 , J 3 as Figure 4 shows.
Note that J induces a disk foliation of N , so π : N → J is the projection map. We do the following operations to J: Operation 1. As indicated in Figure(4-1)→ Figure(4-2). We take a subarc J 1,1 of J 1 , half-twist and move it toward ∂N and identify it with a subarc c 1,1 of c 1 , just like Figure (4-2) shows, this process is a band move in Definition 4.1. Since c 1 bounds a disk in M − N , we can push c 1,1 across the disk to the arc of the subarc c 1,2 , where c 1,2 = c 1 − c 1,1 . We do the same surgeries to J 2 , J 3 .
In the end, we get a branched 1-manifold J * which is isotopic to J in M .   Figure(4-3). Slide one end of J * 2 along a subarc of J * 1 , also one end of J * 3 along another subarc of J * 1 , locally as Figure 5 shows. We get a new branched 1-manifold J * * which is isotopic to J in M obviously.
So we can choose an f ∈ Dif f (M ) which is isotopic to the identity such that: (1) f : N ֒→ N ; (2) f (J) = J * * ; (3) f preserve the disk foliation structure of N . For every leaf D of the foliation, the area satisfies S f (D) /S D ≤ ǫ for some ǫ > 0 small enough.
(4) Let g = π • f , we get the following diagram, N N 1 ∪ N 2 . The left figure depicts the diagram seen from out of N 1 , d i bounds a disk D i in N 1 , c i bounds a disk C i in N 2 . The right figure depicts the diagram seen from out of N 2 , it comes from the left diagram by Dehn twists D e 3 D e 2 D e 2 D e 1 in Figure 7 and then follows by a mirror symmetry. And now, by the diagram, we construct a self-diffeomorphism f of RP 3 such that Ω(f ) consists of two type I Smale-Williams solenoids. Let J be the natural spine of N 1 with respect to the three disks D 1 , D 2 and D 3 bounded by d 1 , d 2 and d 3 , and L be the spine of N 2 with respect to the three disks C 1 , C 2 and C 3 bounded by c 1 , c 2 and c 3 .
We isotopy J to J * in M by performing three band moves along v 1 , v 2 and v 3 , similarly we can isotopy L to L * by performing three band moves along w 1 , w 2 and w 3 . By Figure 8, we can see that L * ⊔ J is isotopic to J * ⊔ L: after three local half twist surgeries and pushing moves to J * ⊔ L (see Figure 8-2), we get L * ⊔ J(see Figure 8-3), all the surgery can be regarded as appearing in three mutually disjoint 3-balls N (v i ⊔ w i ) in M (see Figure 9), so this progress is an isotopy move, hence L * ⊔ J is isotopic to J * ⊔ L. This process is similar with [JNW].  Now, we construct f . As Figures 10 and 11 show, we get J * * (L * * ) from J * (L * ) by slide operations. Since J * * ⊔ L is isotopic to L * * ⊔ J and as in the proof of Proposition 4.3, we can construct f ∈ Dif f (M ) which is isotopic to identity, such that f (J) = J * * , f (L * * ) = L and Ω(f ) is composed of type I Smale-Williams solenoids. Note that here the alternating condition is used to show that the induced matrix is irreducible, so Axiom 2 of Smale-Williams solenoid follows.
Theorem 4.5: If a Heegaard splitting M = N 1 ∪ N 2 of the closed orientable 3-manifolds M is a genus two alternating Heegaard splitting, then there is a diffeomorphism f , such that Ω(f ) consists of two Smale-Williams solenoids.
Proof : It is the same as Example 4.4. Q.E.D.
Example 4.6(The Truncated-Cube Space): Figure 12 is an alternating Heegaard diagram of a closed 3-manifold M = N 1 ∪ N 2 see from outside of N 1 .
Its fundamental group, π 1 (M ) =< x 1 , x 2 ; x 1 x 2 x 1 x −1 2 x −1 1 x −1 2 , x 1 x 2 x −1 1 x 2 x 1 x −1 2 >. Let a = x 2 and b = x 2 x 1 , we have π 1 (M ) =< a, b; a 4 = b 3 = (ab) 2 >, it is the extended triangle group, |π 1 (M )| = 48, so M is a genus two Seifert manifold with base surface S 2 and three singular fibers, M is called the truncated-cube space, see [M] and [T].   Acknowledgements: The authors would like to thank Professor Shicheng Wang for his many helpful discussions. Some of this work was completed while the first named author was a postdoctoral fellow at Peking University, and he thanks the department for its hospitality.