ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS MICHAEL C. SULLIVAN Abstract. The Zeeman number is an invariant of flow equivalence of suspensions of shifts of finite type. We prove that the Zeeman number increases by one when splitting a periodic orbit of a Smale flow. This allows us to define the Zeeman number of any non-anomalous Anosov flow. 1. Motivation A flow on a manifold with hyperbolic chain recurrent set may have basic sets that are suspensions of shifts of finite type. These have been modeled by two dimensional branched manifolds call templates. When one cuts a template along a closed orbit a new template is produced. The dynamics of the new template have likely changed; more specifically, the flow equivalence class of the under lying suspended shift of finite type may have changed. The Parry-Sullivan number and Bowen-Franks group, which determine flow equivalence, often change abruptly. Is there a nontrivial invariant that is conserved under orbit surgery or that at least does not change dramatically? Here we show that the Zeeman number increases by one under orbit surgery. This has one obvious application: orbit surgery cannot yield a system flow equivalent to the original. Another application is that one can now define unambiguously the Zeeman number for higher dimensional basic sets, e.g. non-anomalous Anosov flows. A Smale flow can be derived from a non-anomalous Anosov flow via two orbit surgeries. The derived Smale flow will have a single nontrivial saddle type basic set, an attractor and a repeller; the later two must be isolated closed orbits. The Zeeman number of the basic saddle set well be independent of the pair of orbits used. Date: August 3, 2010. 1

2 MICHAEL C. SULLIVAN 2. Introduction For an integer matrix M let M 2 denote the matrix of ones and zeros that is the mod 2 reduction of M. We use Z 2 to denote the two element field. The Zeeman number was first defined in [12]. Definition 2.1. Let A be an n n integer matrix. Then Z(A) will denote the Zeeman number of A which is defined to be the dimension of the kernel of the map (I A) 2 : Z n 2 Zn 2. If A and B are irreducible positive incidence matrices for shifts of finite type that have topologically equivalent suspension flows then a theorem of Bowen and Franks [3] shows Z(A) = Z(B). If B is a suspension of a shift or finite type we define Z(B) to be the Zeeman number of any incidence matrix of any cross section s first return map. A nonsingular Smale flow is a structurally stable flow with onedimensional basic sets. Each basic set is a suspension of a shift of finite type. These maybe attracting, repelling or saddle types isolated closed orbits or chaotic saddle sets containing infinitely many closed orbits. In the latter case any incidence matrix for a Markov partition of a cross section will be an irreducible non-permutation matrix. Orbit surgery, defined below, splits a chaotic saddle basic set along a closed orbit creating a new attractor or repeller and alters the chaotic saddle set. Theorem 4.1 shows that the Zeeman number of the new saddle set is exactly one greater than the Zeeman number of the original saddle set. Consider a hyperbolic flow ψ on a 3-manifold M whose chain recurrent set is M; so ψ is an Anosov flow. One can perform two orbit surgeries on ψ to produce a Smale flow that will have three basic sets [2, 4], an attracting closed orbit, a repelling closed orbit and a nontrivial saddle set B. Theorem 5.1 shows that the Zeeman number of B is independent of the choice of the two orbits the surgery is preformed on. Thus we can unambiguously assign a Zeeman number to ψ. 3. Background Most of the background material is in [6], [5] and [4]. 3.1. Flows. A flow f on a manifold M is a map φ : M R M such that φ(x, 0) = x and φ(x, s + t) = φ(φ(x, s), t) for all x M and s, t R. All flows here are assumed to be smooth nonsingular and on compact 3-manifolds. For flows that have a hyperbolic structure on their chain recurrent set Smale s Special Decomposition Theorem

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS 3 holds and thus the chain recurrent set is the union of finite number of basic sets each of which is a compact invariant set with a dense orbit. A basic set has dimension 0, 1, 2, or 3 and is either an attractor, a repeller or a saddle set. We are assuming there are no zero dimensional basic sets. A one dimensional basic is either an isolated closed orbit or a suspension of a shift of finite type. A nonsingular Smale flow has only one dimensional basic sets; if the basic sets are all isolated closed orbits it is a nonsingular Morse-Smale flow. A two dimensional basic set is either an attractor or repeller although individual orbits have saddle type. A suspension of the Plykin attractor is an example. By an non-anomalous Anosov flow we mean an Anosov flow where the chain recurrent set is all of M; see [8]. 3.2. Orbit Splitting. Orbit splitting is the flow version of the DAmap, see [9]. Let B be a basic set of a flow with hyperbolic chain recurrent set such that each orbit of B is of saddle type. A closed orbit γ in B is interior if there is a neighborhood N of γ such that N B is dense in N. If γ is a closed orbit of B that is not interior it is a boundary orbit. Let γ be any interior closed orbit in B. When we split along γ we produce a new flow such that (i) it has an additional basic set that is either an attracting or repelling closed orbit with the same knot type as γ and (ii) B is replaced by B which is identical to it except that γ is replaced by either has two copies of γ or a double cover of γ as boundary orbits. If B is n > 1 dimensional then B is n 1 dimensional. If B is one dimensional then B is also one dimensional. For details see [4] or [2]. 3.3. Handles. We describe a set of rectangles in a surface that are used to capture the symbolic dynamics of a diffeomorphism. It is a type of Markov partition. Definition 3.1. Let C K be compact surfaces and let f : C K be an embedding. A handle h is an embedded copy of the unit square in C, h = µ([0, 1] 2 ) C. Let H = {h 1,..., h n } be a disjoint set of handles; we sometimes also use H for the union of the h i s. If p = (x, y) h i we define Wi s (p) to be the image of the segment [0, 1] {y} under µ i. Similarly Wi u(p) is the image of the segment {x} [0, 1] under µ i. If p = µ i (1/2, 1/2) then Wi s (p) is the core interval and Wi u (p) is the transverse interval of h i. We say that f : C K is hyperbolic with respect to a set of handles H, or that H is a hyperbolic handle set for f if for i = 1,..., n the following hold.

4 MICHAEL C. SULLIVAN (i) For every p h i if f(p) h j then Wj u u (f(p)) int(f(wi (p))) and f(wi s (p)) int(wj s (f(p)). (ii) There exists an ǫ (0, 1) such that for each p h i with f(p) H and each v T p (Wi s(p)) and w T p(wi u (p)) we have df(v) ǫ v and df(w) 1/ǫ w. It is a standard fact that if H is a hyperbolic handle set for f then k= fk (H) has a hyperbolic structure and that for each p h i H we have W s i (p) W s (p) and W u i (p) W u (p). Definition 3.2. Let C K be compact surfaces and let f : C K be an embedding. Suppose H = {h 1,...h n } is a handle set for f and assume that the images of the transverse intervals are transverse to the core intervals. (This is certainly true if the handle set is hyperbolic.) Then the geometric intersection matrix G is given letting G ij be the the number of points in the f(wj u ) Wi s. Proposition 3.3 (Franks, [6, Remark B.5]). Let C K be compact surfaces and let f : C K be an embedding. Suppose H = {h 1,...h n } is a handle set for f and assume that the images of the transverse intervals are transverse to the core intervals. Then f can be isotoped to an embedding for which H is hyperbolic and has the same geometric intersection matrix as f. Proposition 3.4 (Smale, [10] or [6, Theorem 3.12]). Let C K be compact surfaces and let f : C K be an embedding. Suppose H = {h 1,...h n } is a hyperbolic handle set for f. Then f restricted to k= fk (H) is topologically conjugate to the shift of finite type whose incidence matrix is the geometric intersection matrix of f with respect to H. 3.4. Templates. Templates do for flows what handles do for maps. Suppose we have a diffeomorphism f on a surface F with hyperbolic handle set H and consider the suspension flow on F [0, 1] mod (x, 0) (f(x), 1). Each handle h i H sweeps out a block B i = x h i, t [0, 1] mod (x, 0) (f(x), 1). The union of these blocks determines a closed neighborhood of the invariant set of the flow. This is called a thickened template. If we collapse the components of the stable manifolds we get a branched 2-manifold called a template. While some information is lost in the collapsing, the periodic orbits along with their knotting and linking is preserved. Two examples are show in Figure 8 Templates are a handy visual aid [11]. We can perform orbit surgery on a template. In Figure 1 we show how a cross section of a thickened template changes under the two

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS 5 types of orbit splittings. Because we collapsed the stable direction the sink-saddle type is easier to visualize, but the effect on the template is the same for either type. new sink new source Figure 1. Cross sectional view of two types of orbit splittings 3.5. Flow Equivalence of Matrices. Two non-negative integer square matrices are flow equivalent if the suspension flows of their corresponding shifts of finite type are topologically equivalent. For non-permutation irreducible matrices flow equivalence is determined by two invariants: the Parry-Sullivan number and the Bowen Franks group [7]. The Zeeman number is also an invariant of flow equivalence as mentioned above. A geometric understanding of flow equivalence will be needed in a subtle step in the proof of Theorem 4.1. Flow equivalence is generated by two type of changes and their inverses to a Markov partition along with permutations. The first is that partition elements can be subdivided; inversely sometimes partition elements can be amalgamated. These generate strong shift equivalence. The period of each closed orbit with respect to the old and new Markov partitions is unchanged. The second type of change is the duplication of partition element; inversely sometimes it is possible to identify two partition elements; one can also think of this as deleting a redundant partition element. In this case the period of closed orbits can change. These concepts are defined precisely in the literature, see [6] or [7]. We illustrate them in Figure 2; recall that the intersection of a nontrivial basic set with a handle is a Cantor set. We will need the proposition below.

6 MICHAEL C. SULLIVAN Subdivision Identification Amalgamation Duplcation Figure 2. Generating moves for flow equivalence Proposition 3.5 (Franks, [5, Proposition 1.13]). Let A be an irreducible square matrix of non-negative integers that is not a permutation matrix. For any positive integer n there is a matrix A that is flow equivalent to A which has every entry larger the n and all non-diagonal entries even. Notice that the mod two reduction of A is a diagonal matrix. 4. Main result Theorem 4.1. Let A be an incidence matrix for a Markov partition of the first return map of a cross section of a nontrivial basic set B in a flow on a 3-manifold M. Let γ be a closed orbit of B and let B be result of splitting along γ. Let A be an incidence matrix for some cross section of A. Then Z(A ) = Z(A) + 1. Basic idea: We will be constructing isotopies of cross sections. They change the return map via inducing or undoing various foldings that are not detected in mod 2 calculations and saddle node bifurcations that change the Zeeman number by one. Proof. We claim there exists a set of cross sectional disks, a Markov partition, {d 1,..., d q } in M for B such that d i γ is one point. Start with any Markov partition for =B; denote the disks by {a 1,..., a p }. Call the first return map r. Select a point x 0 γ a i for some i. Let x l = r l (x 0 ) for l = 1,..., per(x 0 ). Subdivide a i into small disks, {a i1,...a ik }, so that for some j we have x 0 = γ a ij. Renumber the partition members with a 1 = a ij. Now if x 1 = r(x 0 ) is in say a v, subdivide a v so that r(a 1 ) is now a partition element. We can now identify a 1 with r(a 1 ). The period if x 0 has dropped by one. Repeat as needed.

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS 7 The matrix for this Markov partition is of course flow equivalent to A. We will assume that we have started from this point and not rename the matrix. There is a nonsingular Smale flow ψ on S 3 where the basic sets are e + attracting closed orbits, e repelling closed obits and a saddle set topologically equivalent to B (which we will also denote by B) where e + Z(A) e Z(A) and Z(A) + 1 e + + e. This follows from Lemma 4.1 of [5]. However, we shall redo much the construction Frank s uses as we will need these details for the final step of the proof. Let α = e + + e Z(A) 1. Suppose α 2. Let D be a closed disk and let F D be the union of α 1 nearly vertical closed strips and α horizontal closed blocks arranged as in Figure 3(a). Define an embedding ρ : F D so as to have a fixed saddle point in each strip and that the image of each block in pushed down and shrunken slightly as show in Figure 3(b). (While this description is imprecise the motivated reader can construct such a map with concrete coordinates if desired.) Thus there are exactly α 1 chain recurrent points. Label these points p 1,..., p α 1 and their corresponding strips h 1,..., h α 1. Let W s,u i be the component of W s,u (p i ) h i containing p i for s and u respectively. For α = 1 there are no closed saddle orbits. Still we can, following [5], define F using just one horizontal block, ρ : F D will push the block down and have no fixed points. The flow on S 3 will just be a Hopf flow with D a cross sectional disk near the attractor. The isotopy moves described below go through just the same. The case for α = 0 is done at the end of the proof. Let U be a neighborhood of F bounded away from D. We can extend ρ to U without adding any additional chain recurrent points if U is small enough. Assume this has been done. We can define a vector field on U for which ρ is an ǫ-time map of the induced flow. If we rotate U about the axis show in Figure 3(a) we can generate a flow on a sub 3-manifold of S 3 that has α 1 loops of fixed points. A small perturbation near these loops can create a nonsingular flow with α 1 saddle orbits as its chain recurrent set. In [6, Chapter 10] it is shown that one can glue in 2α solid tori, half with flows that have single repelling orbits and half with flows that have a single attracting orbits, so as to form a non-singular Morse-Smale flow

8 MICHAEL C. SULLIVAN on S 3 whose chain recurrent set consists of α attracting closed orbits, α repelling closed orbits and α 1 saddle closed orbits. This will be used later when we want to put the pieces together and construct a flow of S 3. See also [5, page 277]. Figure 3. (a) F and (b) ρ(f) Proposition 3.5 allows us to replace A with a flow equivalent matrix that is strictly positive and has all non-diagonal entries even. But we want to insure that our designated orbit γ still has period one. The proof of Proposition 3.5 in [5] uses (2.5) in [7] which uses (2.1)- (2.4). An examination of the proofs shows that the transformations in Proposition 3.5 primarily consist of subdivisions and amalgamations, which do not affect orbit periods. Only in (2.4) is an identification moved used; this can only lower periods but a period one orbit cannot have its period lowered. The duplication move is never used. Thus the new handle set corresponding to the new matrix with even nondiagonal entries, that we will still call A, intersects γ in exactly one point which we will still call g. Let n be the size of A. We next describe a series of isotopies φ t : D D supported in U such that φ 0 = id and φ 1 ρ that will have various useful features.

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS 9 Sink and saddle move. This is an isotopy φ t in a small disk so that φ 1 ρ has a canceling saddle-source pair. In Figure 4 p is a saddle and q is a source. A strip in the disk containing p will be used as a new handle. Source and saddle move. This is an isotopy φ t in a small disk so that φ 1 ρ has a canceling saddle-sink pair. In Figure 4 r is a saddle and s is a sink. A strip in the disk containing p will be used as a new handle. Both of these moves can be thought of a saddle node bifurcations. Nilpotent handle move. We add a nilpotent handle in U adjoining F that for now contains new chain recurrent points are introduced as in Figure 4. Snake move. Given i and j select a curve λ ij : [0, 1] (U ρ(u)) h j such that λ ij (0) ρ(wj u), λ ij(1) Wi s, λ ij([0, 1]) misses all fixed points of ρ and is transverse to each Wk s that it meets. Then there is a isotopy that pushes an interval of ρ(wj u ) along the curve λ ij so that it intersects Wi s in two points and meets any other Wk s transversely an even number of times. See Figure 5. p r q s Figure 4. Handle moves: nilpotent, source-saddle and sink-saddle. Next we apply Z(A)+1 e + sink-saddle isotopies, the same number of source-saddle isotopies and n Z(A) nilpotent isotopies. (It is clear

10 MICHAEL C. SULLIVAN h j h i W s i λ ij W u j Figure 5. Snake move from the definition of Z(A) that it is not greater than n.) The geometric intersection matrix, P, for the restriction of ρ to the handles is an n n with Z(A) ones along the diagonal and zeros everywhere else. Snake isotopies can only change the entries of a geometric intersection matrix by the addition of positive even integers. Thus the Zeeman number of the geometric intersection matrix does not change. Franks shows how to use a series of snake isotopies and some wiggles, to transform ρ so that its geometric intersection matrix is A - actually it may be necessary to one again apply Proposition 3.5 to enlarge to entries of A. See the last two paragraphs of Section 4 in [5] for details. Proposition 3.3 allows us to apply a further isotopy so that ρ is hyperbolic on the handle set and has the same geometric intersection matrix. We can now produce a Smale flow on S 3 that has e attracting closed orbits, e + repelling closed orbits and a single saddle set that is the suspension of the shift of finite type for A. Now consider B. We can use these same steps except that we will perform an additional sink-saddle or source-saddle isotopy in a small disk centered at g to create a flow on S 3 with one extra attracting or repelling closed orbit and has a saddle set that is topologically equivalent to B. The corresponding geometric intersection matrix, call it A, will have mod 2 reduction the same has A except for an extra one on the diagonal. But then (I A ) 2 will have one fewer ones on its diagonal than (I A) 2. Hence Z(A ) = Z(A) + 1. We now consider the case when α = 0. See [5, page 279]. Asimov constructed a non-singular Morse-Smale flow on S 3 with one attractor, one repeller and one untwisted saddle orbit. See [6, (10.1)] or [1]. One

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS 11 starts with the map on a disk with a saddle sink shown in Figure 6. This a rotated and perturbed to a flow with a saddle and an attracting closed orbits. A tube is attached to the exit set and to a portion of the entrance and the flow extended in the manner shown. Finally a torus whose core is a repelling closed orbit is added to make the flow on S 3. The disk F is the same as for the α = 2 case the only difference being that the two horizontal blocks will move toward the same attractor in the final flow; see Figure 7. The isotopy moves used before can be applied in the same manner. Figure 6 Figure 7 5. Application to non-anomalous Anosov Flows Theorem 5.1. Let φ be a non-anomalous Anosov flow on a 3-manifold M. Suppose φ and φ are Smale flows derived from φ by splitting along

12 MICHAEL C. SULLIVAN two closed orbits. Then the nontrivial basic set in φ and the nontrivial basic set in φ will have the same Zeeman number. Proof. Let B and B be the saddle basic sets of φ and φ respectively. Let γ a and γ r be the orbits of φ that were split to produce φ ; we also use γ a and γ r to denote the attracting and repelling closed orbits of φ respectively. Define γ a and γ r similarly. Step 1: B and B are not affected by the order in which the surgeries is performed or by switching splitting type (sink-saddle or sourcesaddle) used on the two orbits. Step 2: Suppose γ a, γ r, γ a and γ r are distinct. The basic set B formed by producing all four surgeries is independent of the order in which they are done. Thus Z(B ) = Z(B ) = Z(B ) 2. Step 3: If two of the splitting orbits are the same than Z(B ) = Z(B ) = Z(B ) 1. 6. Examples Example 1 (Lonez Templates). Consider the templates in Figure 8. The one of the left is denoted [ ] L(0, 0) and the other is denoted L(0, 1). 1 1 They both have A = as an incidence matrix. We split them 1 1 along the orbits indicated in the figure and call the new templates L (0, 0) and L (0, 1). The new incidence matrices are 1 1 1 0 B = 0 0 0 1 1 0 0 0 and C = 1 1 1 0 0 1, 1 1 0 0 1 1 1 respectively. One can compute Z(A) = 0 and Z(B) = Z(C) = 1. The Parry-Sullivan number of a matrix M is defined by PS(M) = det(i M). We get PS(A) = 1, PS(B) = 0 and PS(C) = 2. Figure 8. Two Templates

ZEEMAN NUMBERS AND ORBIT SPLITTING IN FLOWS 13 Example 2 (Birman-Williams). In [2] Birman and Williams studied an Anosov flow in the complement of the figure 8 knot. They produced two template models for this flow. These are shown in Figures 1.2 and A.16 in their paper. Using the branch lines as a natural choice for Markov partitions we can find incidence matrices which we will call B and W respectively. 0 1 0 1 B = 1 0 1 0 0 1 0 1. W = 1 1 1 1 0 0. 0 1 1 1 0 1 0 We find that PS(B) = 3 while PS(W) = 1 so they are not flow equivalent. However, they both have Zeeman number zero. Acknowledgment. John Franks suggested the strategy for the proof of Theorem 5.1. References [1] Asimov, Daniel. Flaccidity of geometric index for nonsingular vector fields. Comment. Math. Helv. 52 (1977), no. 2, 161 175. [2] Birman, Joan S.; Williams, R. F. Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots. Low-dimensional topology (San Francisco, Calif., 1981), 1 60, Contemp. Math., 20, Amer. Math. Soc., Providence, RI, 1983. [3] Bowen, Rufus; Franks, John Homology for zero-dimensional nonwandering sets. Ann. Math. (2) 106 (1977), no. 1, 73 92. [4] Ghrist, R., Holmes, P. & Sullivan, M. Knots and Links in Three-Dimensional Flows, Lecture Notes in Mathematics, Volume 1654, Springer-Verlag, 1996. [5] Franks, John M. Nonsingular Smale Flows on S 3, Topology Vol 24, No 3, 1984, 265 282. [6] Franks, John M. Homology and Dynamical Systems, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, number 49, American Mathematical Society, 1982 & 1989. [7] Franks, John M. Flow equivalence of subshifts of finite type. Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53 66. [8] Franks, John M. & Williams, Bob. Anomalous Anosov Flows. Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 158 174, Lecture Notes in Math., 819, Springer, Berlin, 1980. [9] Robinson, Clark. Dynamical systems: Stability, Symbolic Dynamics, and Chaos. Second edition. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1999. [10] Smale, Stephen. Diffeomorphisms with many periodic points. 1965 Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 63 80 Princeton Univ. Press, Princeton, N.J. [11] Sullivan, Michael. Visually building Smale flows in S 3. Topology & Its Applications, 106 (2000), no. 1, 1 19.

14 MICHAEL C. SULLIVAN [12] Zeeman, E. C. Morse inequalities for diffeomorphisms with shoes and flows with solenoids, Dynamical Systems-Warwick 1974, Springer Lecture Notes, Vol. 468. Department of Mathematics (4408), Southern Illinois University, Carbondale, IL 62901, USA, msulliva@math.siu.edu, http://www.math.siu.edu/sullivan