Computing Reeb dynamics on 4d convex polytopes
CComputing Reeb dynamics on 4d convex polytopes
Julian Chaidez ∗ and Michael Hutchings † August 25, 2020
Abstract
We study the combinatorial Reeb flow on the boundary of a four-dimensionalconvex polytope. We establish a correspondence between “combinatorial Reeborbits” for a polytope, and ordinary Reeb orbits for a smoothing of the poly-tope, respecting action and Conley-Zehnder index. One can then use a com-puter to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo’sconjecture and related conjectures. In particular, we have found some newexamples of polytopes with systolic ratio 1.
Contents ∗ Partially supported by an NSF Graduate Research Fellowship. † Partially supported by NSF grant DMS-1708899, a Simons Fellowship, and a Humboldt Re-search Award. a r X i v : . [ m a t h . S G ] A ug Reeb dynamics on symplectic polytopes 26
A Rotation numbers 51
A.1 Rotation numbers of circle diffeomorphisms . . . . . . . . . . . . . . . 51A.2 A partial order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.3 Rotation numbers of symplectic matrices . . . . . . . . . . . . . . . . 53A.4 Computing products in (cid:102)
Sp(2) . . . . . . . . . . . . . . . . . . . . . . 54
This paper is about computational methods for testing Viterbo’s conjecture andrelated conjectures, via combinatorial Reeb dynamics.
We first recall two different versions of Viterbo’s conjecture. Consider R n = C n with coordinates z i = x i + √− y i for i = 1 , . . . , n . Define the standard Liouvilleform λ = 12 n (cid:88) i =1 ( x i dy i − y i dx i ) . X be a compact domain in R n with smooth boundary Y . Assume that X is“star-shaped”, by which we mean that Y is transverse to the radial vector field.Then the 1-form λ = λ | Y is a contact form on Y . Associated to λ are the contactstructure ξ = Ker( λ ) ⊂ T Y and the Reeb vector field R on Y , characterized by dλ ( R, · ) = 0 and λ ( R ) = 1. A Reeb orbit is a periodic orbit of R , i.e. a map γ : R /T Z → Y for some T > γ (cid:48) ( t ) = R ( γ ( t )), modulo reparametrization.The symplectic action of a Reeb orbit γ , denoted by A ( γ ), is the period of γ , orequivalently A ( γ ) = (cid:90) R /T Z γ ∗ λ . (1.1)Reeb orbits on Y always exist. This was first proved by Rabinowitz [20] and is aspecial case of the Weinstein conjecture; see [15] for a survey. We are interested herein the minimal period of a Reeb orbit on Y , which we denote by A min ( X ) ∈ (0 , ∞ ),and its relation to the volume of X . For this purpose, define the systolic ratio sys( X ) = A min ( X ) n n ! vol( X ) . The exponent ensures that the systolic ratio of X is invariant under scaling of X ;and the constant factor is chosen so that if X is a ball then sys( X ) = 1. Conjecture 1.1 (weak Viterbo conjecture) . Let X ⊂ R n be a compact convexdomain with smooth boundary such that ∈ int( X ) . Then sys( X ) ≤ . Conjecture 1.1 asserts that among compact convex domains with the same vol-ume, A min is largest for a ball. Although the role of the convexity hypothesis issomewhat mysterious, some hypothesis beyond the star-shaped condition is neces-sary: it is shown in [1] that there exist star-shaped domains in R with arbitrarilylarge systolic ratio . One motivation for studying Conjecture 1.1 is that it impliesthe Mahler conjecture in convex geometry [4].To put Conjecture 1.1 in more context, recall that a symplectic capacity isa function c mapping some class of 2 n -dimensional symplectic manifolds to [0 , ∞ ],such that: • (Monotonicity) If there exists a symplectic embedding ϕ : ( X, ω ) → ( X (cid:48) , ω (cid:48) ),then c ( X, ω ) ≤ c ( X (cid:48) , ω (cid:48) ). It is further shown in [2] that there are star-shaped domains in R which are dynamicallyconvex (meaning that every Reeb orbit on the boundary has rotation number greater than 1, seeProposition 1.9(a) below) and have systolic ratio 2 − ε for ε > The precise definition of “symplectic capacity” varies in the literature. For an older butextensive survey of symplectic capacities see [6]. (Conformality) If r > c ( X, rω ) = rc ( X, ω ).Of course we can regard (open) domains in R n as symplectic manifolds with therestriction of the standard symplectic form ω = (cid:80) ni =1 dx i dy i . Conformality for adomain X ⊂ R n means that c ( rX ) = r c ( X ).Following the usual convention in symplectic geometry, for r > B ( r ) = (cid:8) z ∈ C n (cid:12)(cid:12) π | z | ≤ r (cid:9) and the cylinder Z ( r ) = (cid:8) z ∈ C n (cid:12)(cid:12) π | z | ≤ r (cid:9) . We say that a symplectic capacity c is normalized if it is defined at least for allcompact convex domains in R n and if c ( B ( r )) = c ( Z ( r )) = r. An example of a normalized symplectic capacity is the
Gromov width c Gr ,where c Gr ( X, ω ) is defined to be the supremum over r such that there exists asymplectic embedding B ( r ) → ( X, ω ). It is immediate from the definition that c Gr is monotone and conformal. Since symplectomorphisms preserve volume, we have c Gr ( B ( r )) = r ; and the Gromov nonsqueezing theorem asserts that c Gr ( Z ( r )) = r .Another example of a normalized symplectic capacity is the Ekeland-Hofer-Zehnder capacity , denoted by c EHZ . If X is a compact convex domain with smoothboundary such that 0 ∈ int( X ), then c EHZ ( X ) = A min ( X ) . (1.2)This is explained in [5, Thm. 2.2], combining results from [7, 13].Any symplectic capacity which is defined for compact convex domains in R n with smooth boundary is a C continuous function of the domain (i.e., continuouswith respect to the Hausdorff distance between compact sets), and thus extendsuniquely to a C continuous function of all compact convex sets in R n . Conjecture 1.2 (strong Viterbo conjecture ) . All normalized symplectic capacitiesagree on compact convex sets in R n . The original version of Viterbo’s conjecture from [22] asserts that a normalized symplecticcapacity, restricted to convex sets in R n of a given volume, takes its maximum on a ball. (Thisfollows from what we are calling the “strong Viterbo conjecture” and implies what we are callingthe “weak Viterbo conjecture”.) Viterbo further conjectured that the maximum is achieved onlyif the interior of the convex set is symplectomorphic to an open ball; cf. Question 1.21 below. X is a compact convex domain with smooth boundary and 0 ∈ int( X ), then A min ( X ) n = c EHZ ( X ) n = c Gr ( X ) n ≤ n ! vol( X ) . Here the second equality holds by Conjecture 1.2; and the inequality on the rightholds because if there exists a symplectic embedding B ( r ) → X , then r n /n ! =vol( B ( r )) ≤ vol( X ).There are also interesting families of non-normalized symplectic capacities. Forexample, there are the Ekeland-Hofer capacities defined in [8]; more recently, andconjecturally equivalently, positive S -equivariant symplectic homology was used in[9] to define a symplectic capacity c S k for each integer k ≥
1. Each equivariantcapacity c S k ( X ) is the symplectic action of some Reeb orbit, which when X isgeneric (so that λ is nondegenerate) has Conley-Zehnder index n − k (see § S -invariantconvex domains [10], but they have not been well tested more generally. To testConjecture 1.1, and as a first step towards computing other symplectic capacitiesand testing conjectures about them, we need good methods for computing Reeborbits, their actions, and their Conley-Zehnder indices. The plan is to understandReeb orbits on a smooth convex domain in terms of “combinatorial Reeb orbits” onconvex polytopes approximating the domain. Let X be any compact convex set in R n with 0 ∈ int( X ), and let y ∈ ∂X . The tangent cone , which we denote by T + y X , is the closure of the set of vectors v such y + εv ∈ X for some ε >
0. For example, if ∂X is smooth at y , then T + y X is a closedhalf-space whose boundary is the usual tangent space T y ∂X .Also define the positive normal cone N + y X = (cid:8) v ∈ R n (cid:12)(cid:12) (cid:104) x − y, v (cid:105) ≤ ∀ x ∈ X (cid:9) . If ∂X is smooth at y , then N + y X is a one-dimensional ray and consists of the outwardpointing normal vectors to ∂X at y .Finally, define the Reeb cone R + y X = T + y X ∩ i N + y X i denotes the standard complex structure on C n = R n . If ∂X is smooth near y , then R + y X is the ray consisting of nonnegative multiples of the Reeb vector fieldon ∂X at y . Indeed, in this case we can write T y ∂X = (cid:8) v ∈ R n (cid:12)(cid:12) (cid:104) ν, v (cid:105) = 0 (cid:9) where ν is the outward unit normal vector to ∂X at y ; and the Reeb vector field at y is given by R y = 2 i ν (cid:104) ν, y (cid:105) . (1.3)Figure 1: We depict the tangent, normal and Reeb cones for two points p, q ∈ X ina polytope X ⊂ R .Suppose now that X is a convex polytope (i.e. a compact set given by theintersection of a finite set of closed half-spaces) in R n with 0 ∈ int( X ). Ourconvention is that a k -face of X is a k -dimensional subset F ⊂ ∂X which is theinterior of the intersection with ∂X of some set of the hyperplanes defining X . Fora given k -face F , the tangent cone T + y X , the positive normal cone N + y X , and theReeb cone R + y X are the same for all y ∈ F . Thus we can denote these cones by T + F X , N + F X , and R + F X respectively.We will usually restrict attention to polytopes of the following type: Definition 1.3. A symplectic polytope in R is a convex polytope X in R suchthat 0 ∈ int( X ) and no 2-face of X is Lagrangian, i.e., the standard symplectic form ω = (cid:80) i =1 dx i dy i restricts to a nonzero 2-form on each 2-face.Symplectic polytopes are generic, in the sense that in the space of polytopes in R with a given number of 3-faces, the set of non-symplectic polytopes is a propersubvariety. Proposition 1.4. (proved in § X is a symplectic polytope in R , then theReeb cone R + F X is one-dimensional for each face F . efinition 1.5. Let X be a symplectic polytope in R . A combinatorial Reeborbit for X is a finite sequence γ = (Γ , . . . , Γ k ) of oriented line segments in ∂X ,modulo cyclic permutations, such that for each i = 1 , . . . , k : • The final endpoint of Γ i agrees with the initial endpoint of Γ i +1 mod k . • There is a face F of X such that int(Γ i ) ⊂ F , the endpoints of Γ i are on theboundary of (the closure of) F , and Γ i points in the direction of R + F X .The combinatorial symplectic action of a combinatorial Reeb orbit as above isdefined by A comb ( γ ) = k (cid:88) i =1 (cid:90) Γ i λ . To give a better idea of what combinatorial Reeb orbits look like, we have thefollowing lemma.
Lemma 1.6. (proved in § X be a symplectic polytope in R . Then the Reebcones of the faces of X satisfy the following: • If E is a 3-face, then R + E X consists of all nonnegative multiples of the Reebvector field on E . • If F is a -face, then R + F X points into a 3-face E adjacent to F , and agreeswith R + E X . • If L is a -face, then one of the following possibilities holds: – R + L X points into a -face E adjacent to L and agrees with R + E X . In thiscase we say that L is a good -face. – R + L X is tangent to L , and does not agree with R + E X for any of the -faces E adjacent to L . In this case we say that L is a bad -face. • If P is a -face, then R + P X points into a -face E or bad -face L adjacent to F and agrees with R + E X or R + L X respectively. Remark 1.7.
The reason we assume that X has no Lagrangian 2-faces in Defi-nition 1.3 is that if F is a Lagrangian 2-face, then R + F X is two-dimensional andtangent to F . In fact, ∂R + F X = R + E X ∪ R + E X where E and E are the two 3-facesadjacent to F . In this case we do not have a well-posed “combinatorial Reeb flow”on ∂X . Definition 1.8.
A combinatorial Reeb orbit as above is:7
Type 1 if it does not intersect the 1-skeleton of X ; • Type 2 if it intersects the 1-skeleton of X , but only in finitely many pointswhich are some of the endpoints of the line segments Γ i ; • Type 3 if it contains a bad 1-face.Figure 2: We depict sub-trajectories of the three types of orbits, in red. Each cubeabove represents a 3-face of a hypothetical 4-polytope.It follows from the definitions that each combinatorial Reeb orbit is of one of theabove three types. Type 1 Reeb orbits are the most important for our computations.We expect that Type 2 combinatorial Reeb orbits do not exist for generic polytopes;see Conjecture 1.24 below. Type 3 combinatorial Reeb orbits generally cannot beeliminated by perturbing the polytope; but we will see in Theorem 1.11(iii) belowthat they do not contribute to the symplectic capacities that we are interested in.See Remark 5.8 for some intuition for this.
Let X be a compact star-shaped domain in R with smooth boundary Y . LetΦ t : Y → Y denote the time t flow of the Reeb vector field R . The derivative of Φ t preserves the contact form λ , and thus for each y ∈ Y defines a map d Φ t : ξ y −→ ξ Φ t ( y ) which is symplectic with respect to dλ .We say that a Reeb orbit γ : R /T Z → Y is nondegenerate if the “linearizedreturn map” d Φ T : ξ γ (0) −→ ξ γ (0) (1.4)does not have 1 as an eigenvalue. The contact form λ is called nondegenerate if allReeb orbits are nondegenerate. 8ow fix a symplectic trivialization τ : ξ → Y × R . If γ is a Reeb orbit as above,then the trivialization τ allows us to regard the map (1.4) as an element of Sp(2).Moreover, the family of maps (cid:110) R τ − −→ ξ γ (0) d Φ t −→ ξ γ ( t ) τ −→ R (cid:111) t ∈ [0 ,T ] defines a path in Sp(2) from the identity to the map (1.4). As we review in Ap-pendix A, this path has a well-defined rotation number , which we denote by ρ ( γ ) ∈ R . This rotation number does not depend on the choice of global trivialization τ .If γ is nondegenerate (which holds automatically when ρ ( γ ) is not an integer),then the Conley-Zehnder index of γ is defined byCZ( γ ) = (cid:98) ρ ( γ ) (cid:99) + (cid:100) ρ ( γ ) (cid:101) ∈ Z . (1.5) Proposition 1.9.
Let X be a compact convex domain in R with smooth boundary Y and with ∈ int( X ) . Then:(a) Every Reeb orbit γ in Y has ρ ( γ ) > . In particular, if γ is nondegenerate then CZ( γ ) ≥ .(b) There exists a Reeb orbit γ which is action minimizing, i.e. A ( γ ) = A min ( X ) ,with ρ ( γ ) ≤ . If γ is also nondegenerate then the inequality is strict, so that CZ( γ ) = 3 .Proof. (a) was proved by Hofer-Wysocki-Zehnder [12].(b) follows from the construction of the Ekeland-Hofer-Zehnder capacity and anindex calculation of Hu-Long [14]. In fact, it was recently shown by Abbondandolo-Kang [3] and Irie [18] that c EHZ ( X ) agrees with a capacity defined from symplectichomology, which by construction is the action of some Reeb orbit γ with ρ ( γ ) ≤ γ is degenerate.Suppose now that X is a symplectic polytope in R . As we explain in Defini-tion 2.23, each Type 1 combinatorial Reeb orbit γ has a well-defined combinatorialrotation number , which we denote by ρ comb ( γ ) ∈ R . There is also a combinato-rial notion of nondegeneracy for γ , which automatically holds when ρ comb ( γ ) / ∈ Z .When γ is a nondegenerate Type 1 combinatorial Reeb orbit, we can then define its combinatorial Conley-Zehnder index by analogy with (1.5) asCZ comb ( γ ) = (cid:98) ρ comb ( γ ) (cid:99) + (cid:100) ρ comb ( γ ) (cid:101) . (1.6)9he combinatorial rotation number and combinatorial Conley-Zehnder index of aType 2 combinatorial Reeb orbit are not defined; and although we do not need this,it would be natural to define the combinatorial rotation number and combinatorialConley-Zehnder index of a Type 3 combinatorial Reeb orbit to be + ∞ . Let X be a convex polytope in R n . If ε >
0, define the ε -smoothing of X by X ε = (cid:8) z ∈ R n (cid:12)(cid:12) dist( z, X ) ≤ ε (cid:9) . (1.7)The domain X ε is convex and has C -smooth boundary. The boundary is C ∞ smooth except along strata arising from the boundaries of the faces of X ; see § R , and ordinaryReeb dynamics on ε -smoothings of the polytope.There is a slight technical issue here: since ∂X ε is only C smooth, the Reebvector field on ∂X ε is only C , so that for a Reeb orbit γ , the linearized Reeb flow(1.4) might not be defined. If γ is transverse to the strata where ∂X ε is not C ∞ (which is presumably true for all γ if X and ε are generic), then the Reeb flow in aneighborhood of γ has a well-defined linearization; we call such orbits linearizable .It turns out that a non-linearizable Reeb orbit γ on ∂X ε still has a well-definedrotation number ρ ( γ ), defined in § Theorem 1.10. (proved in § X be a symplectic polytope in R , and let γ be a nondegenerate Type 1 combinatorial Reeb orbit for X . Then for all ε > sufficiently small, there is a distinguished Reeb orbit γ ε on ∂X ε such that:(i) γ ε converges in C to γ as ε → .(ii) lim ε → A ( γ ε ) = A comb ( γ ) .(iii) γ ε is linearizable and nondegenerate, ρ ( γ ε ) = ρ comb ( γ ) , and CZ( γ ε ) = CZ comb ( γ ) . The following theorem describes how Reeb orbits on smoothings give rise tocombinatorial Reeb orbits.
Theorem 1.11. (proved in § X be a symplectic polytope in R . Then thereare constants c F > for each -, -, or -face F of X with the following property. et { ( ε i , γ i ) } i =1 ,... be a sequence of pairs such that ε i > ; γ i is a Reeb orbit on ∂X ε i ; and ε i → as i → ∞ . Suppose that ρ ( γ i ) < R where R does not depend on i . Then after passing to a subsequence, there is a combinatorial Reeb orbit γ for X such that:(i) γ i converges in C to γ as i → ∞ .(ii) lim i →∞ A ( γ i ) = A comb ( γ ) .(iii) γ is either Type 1 or Type 2.(iv) If γ is Type 1, then for i sufficiently large, γ i is linearizable and ρ ( γ i ) = ρ comb ( γ ) . If γ is also nondegenerate, then for i sufficiently large, γ i is nonde-generate and CZ( γ i ) = CZ comb ( γ ) .(v) Let F , . . . , F k denote the faces containing the endpoints of the segments of thecombinatorial Reeb orbit γ . Then k (cid:88) i =1 c F i ≤ R. (1.8) Remark 1.12.
One can compute explicit constants c F – see § Corollary 1.13.
Let X be a symplectic polytope in R . Then c EHZ ( X ) = min {A comb ( γ ) } (1.9) where the minimum is over combinatorial Reeb orbits γ with (cid:80) i c F i ≤ which areeither Type 1 with ρ comb ( γ ) ≤ or Type 2. Remark 1.14.
If the coordinates of the vertices of X are rational, then the com-binatorial action of every combinatorial Reeb orbit is rational. It follows fromTheorem 1.11 that in this case, c EHZ ( X ), as well as the other symplectic capacitiesmentioned in § Definition 1.15. If X is any compact convex set in R n with 0 ∈ int( X ), a gener-alized Reeb orbit for X is a map γ : R /T Z → ∂X for some T > γ iscontinuous and has left and right derivatives at every point, which agree for almostevery t , and the left and right derivatives at t are in R + γ ( t ) X . If γ is a generalizedReeb orbit, define its symplectic action by (1.1). Proposition 1.16. [5, Prop. 2.7] If X is a compact convex set in R n with ∈ int( X ) , then c EHZ ( X ) = min {A ( γ ) } where the minimum is taken over all generalized Reeb orbits.Proof of Corollary 1.13. Pick a sequence of positive numbers ε i with lim i →∞ ε i = 0.For each i , by equation (1.2), we can find a Reeb orbit γ i on ∂X ε i with A ( γ i ) = c EHZ ( X ε i ). By Proposition 1.9(b), we can assume that ρ ( γ i ) ≤
2. By Theorem 1.11,it follows that after passing to a subsequence, there is a combinatorial Reeb orbit γ for X , satisying the conditions in Corollary 1.13, such that A comb ( γ ) = lim i →∞ A ( γ i ) = lim k →∞ c EHZ ( X ε i ) = c EHZ ( X ) . Here the last equality holds by the C continuity of c EHZ . We conclude that c EHZ ( X ) ≥ min {A comb ( γ ) } where the minimum is over combinatorial Reeb orbits γ satisfying the conditions inCorollary 1.13.The reverse inequality follows from Proposition 1.16, because by Definitions 1.5and 1.15, every combinatorial Reeb orbit is a generalized Reeb orbit. (For a sym-plectic polytope in R , a “generalized Reeb orbit” is equivalent to a generalizationof a “combinatorial Reeb orbit” in which there may be infinitely many line seg-ments.) Remark 1.17.
Haim-Kislev [11, Thm. 1.1] gives a different formula for c EHZ of aconvex polytope, which is valid in R n for all n . That formula implies that in theminimum (1.9), we can also assume that γ has at most one segment in each 3-face. If X is a convex polytope in R n , define its systolic ratio bysys( X ) = c EHZ ( X ) n n ! vol( X ) . c EHZ is translation invariant, so we can make this definition withoutassuming that 0 ∈ int( X ).Since every compact convex domain in R n can be C approximated by convexpolytopes, it follows that the weak version of Viterbo’s conjecture, namely Conjec-ture 1.1, is true if and only if every convex polytope X has systolic ratio sys( X ) ≤ n = 2. In particular, we ran optimizationalgorithms over the space of k -vertex convex polytopes in R to find local max-ima of the systolic ratio . In the results below, when listing the vertices of specificpolytopes, we use Lagrangian coordinates ( x , x , y , y ). Experimentally , every 4-simplex X has sys-tolic ratio sys( X ) ≤ / . The apparent maximum of 3 / , , , , (1 , , , , (0 , , , , (0 , , , , (0 , , , . Remark 1.18.
Corollary 1.13 does not directly apply to (a translate of) this poly-tope because it has some Lagrangian 2-faces. For examples like these, we findnumerically that a slight perturbation of the polytope to a symplectic polytope (towhich Corollary 1.13 does apply) has systolic ratio very close to the claimed value.One can compute the systolic ratio of a polytope with Lagrangian 2-faces rigorouslyusing a generalization of Corollary 1.13. For the particular example above, one canalso compute the systolic ratio by hand using [11, Thm. 1.1].We have found families of other examples of 4-simplices with systolic ratio 3 / , , , , (1 , − / , , , (0 , − / , , , ( − / , − , / , , (0 , , , . We found families of 6-vertex polytopes with systolic ratioequal to 1. An example is the polytope with vertices(0 , , , , (1 , , , , (0 , , , , (0 , , , , (0 , − , , , ( − , − , , . This is a somewhat involved process; convergence to a local maximum becomes very slow onceone is close. It helps to mod out the space of polytopes by the 15-dimensional symmetry groupgenerated by translations, linear symplectomorphisms, and scaling. To find exact local maxima,one can look at symplectic invariants, such as areas of 2-faces, and guess what these are convergingto. Perhaps this could be proved analytically using the formula in [11, Thm. 1.1].
We also found families of 7-vertex polytopes with systolicratio 1. One example has vertices(0 , , , , (1 , , , , (0 , , , , (0 , , , , (1 / , − / , / , , ( − , − , , / , (0 , , / , − / . Presumably there exist k -vertex polytopes in R with systolic ratio equal to 1 forevery k ≥ The 24-cell.
We also found a special example of a polytope with systolic ratio 1:a rotation of the 24-cell (one of the six regular polytopes in four dimensions). See § Towards a proof of the weak Viterbo conjecture?
Let X be a star-shapeddomain in R with smooth boundary Y . Following [1], we say that X is Zoll ifevery point on Y is contained in a Reeb orbit with minimal action. Note that:(a) If X is strictly convex and a local maximizer for the systolic ratio of convexdomains in the C topology, then X is Zoll.(b) If X is Zoll, then X has systolic ratio sys( X ) = 1.Part (a) holds because if X is strictly convex and if y ∈ Y is not on an actionmimizing Reeb orbit, then one can shave some volume off of X near y withoutcreating any new Reeb orbits of small action. Part (b) holds by a topologicalargument going back to [23]. Of course, these observations are not enough to proveConjecture 1.1, since we do not know that the systolic ratio for convex domainstakes a maximum, let alone on a strictly convex domain. But this does suggest thefollowing strategy for proving Conjecture 1.1 via convex polytopes.14 efinition 1.19. Let X be a convex polytope in R with 0 ∈ int( X ). We say that X is combinatorially Zoll if there is an open dense subset U of ∂X such that everypoint in U is contained in a combinatorial Reeb orbit (avoiding any Lagrangian2-faces of X ) with combinatorial action equal to c EHZ ( X ).We have checked by hand that the above examples of polytopes with systolicratio equal to 1 are combinatorially Zoll. This suggests: Conjecture 1.20.
Let X be a convex polytope in R with ∈ int( X ) . Then:(a) If X is combinatorially Zoll, then sys( X ) = 1 .(b) If k is sufficiently large ( k ≥ might suffice) and if X maximizes systolic ratioover convex polytopes with ≤ k vertices, then X is combinatorially Zoll. Part (a) of this conjecture can probably be proved following the argument inthe smooth case. Part (b) might be much harder. But both parts of the conjecturetogether would imply the weak Viterbo conjecture (using a compactness argumentto show that for each k the systolic ratio takes a maximum on the space of convexpolytopes with ≤ k vertices). Question 1.21.
If a convex polytope X in R is combinatorially Zoll, then is int( X ) symplectomorphic to an open ball? One can also use Theorems 1.10 and 1.11 to test conjectures about Reeb orbits thatdo not have minimal action. For example, if X is a convex domain with smoothboundary and 0 ∈ int( X ) such that λ | ∂X is nondegenerate, and if k is a positiveinteger, define A k ( X ) = min {A ( γ ) | CZ( γ ) = 2 k + 1 } , (1.10)where the minimum is over Reeb orbits γ on ∂X . In particular A ( X ) = A min ( X )by Proposition 1.9(b). Conjecture 1.22.
For X as above we have A ( X ) ≤ A ( X ) . This conjecture has nontrivial content when every action-minimizing Reeb orbithas rotation number at least 3 /
2. (If an action-minimizing Reeb orbit has rotationnumber less than 3 /
2, then its double cover has Conley-Zehnder index 5 and thusverifies the conjectured inequality.) To explain how to test this, we need the followingdefinitions.
Definition 1.23.
Let X be a symplectic polytope in R . Let L >
0. We say that X is L -nondegenerate if: 15 X does not have any Type 2 combinatorial Reeb orbit γ with A comb ( γ ) ≤ L . • Every Type 1 combinatorial Reeb orbit γ with A comb ( γ ) ≤ L is nondegenerate,see Definition 2.23.It follows from Theorem 1.11 that if a symplectic polytope X is L -nondegenerate,then for all ε > ∂X ε with action less than L are nondegenerate. Conjecture 1.24.
For any integer k and any real number L , the set of L -nondegeneratesymplectic polytopes with k vertices is dense in the set of all k -vertex convex polytopescontaining , topologized as an open subset of R k . Definition 1.25.
Let k be a positive integer and let X be a symplectic polytope in R . Suppose that X is L -nondegenerate and has a combinatorial Reeb orbit γ with A ( γ ) < L and CZ comb ( γ ) = 2 k + 1. By analogy with (1.10), define A comb k ( X ) = min {A comb ( γ ) | CZ comb ( γ ) = 2 k + 1 } where the minimum is over combinatorial Reeb orbits γ with combinatorial actionless than L .Conjecture 1.22 is now equivalent to the following: Conjecture 1.26.
Let X be a symplectic polytope in R . Assume that A comb1 ( X ) and A comb2 ( X ) are defined. Then A comb2 ( X ) ≤ A comb1 ( X ) . One can use Theorems 1.10 and 1.11 to compute A comb k ( X ). One can thentest Conjecture 1.26 by using optimization algorithms to try to maximize the ratio A comb2 ( X ) / (2 A comb1 ( X )). So far we have not found any example where this ratio isgreater than 1. In §
2, we investigate Type 1 combinatorial Reeb orbits in detail, we define thecombinatorial rotation number, and we work out the example of the 24-cell. In § More precisely, by Theorem 1.10, if X is a polytope as above for which A comb1 ( X ) and A comb2 ( X )are defined, and if A comb2 ( X ) > A comb1 ( X ), then Conjecture 1.22 fails for (nondegenerate C ∞ perturbations of) ε -smoothings of X for ε sufficiently small. Thus Conjecture 1.22 implies Con-jecture 1.26. If Conjecture 1.24 is true, then one can conversely show, by approximating smoothdomains by L -nondegenerate symplectic polytopes, that Conjecture 1.26 implies Conjecture 1.22.
16e establish foundational facts about the combinatorial Reeb flow on a symplecticpolytope. In § R defined using the quaternions. We explain a keycurvature identity due to Hryniewicz and Salom˜ao which implies that in the convexcase, the rotation number of a Reeb trajectory increases monotonically as it evolves.In § § Acknowledgments.
We thank A. Abbondandolo, P. Haim-Kislev, U. Hryniewicz,and Y. Ostrover for helpful conversations.
Let X be a symplectic polytope in R . In this section we give what amounts to analgorithm for finding the Type 1 combinatorial Reeb orbits and their combinatorialsymplectic actions, see Proposition 2.14. (Our actual computer implementation usesvarious optimizations not discussed here.) We also define combinatorial rotationnumbers and work out the example of the 24-cell. We start by defining “symplectic flow graphs”, which keep track of the combinatoricsneeded to find Type 1 Reeb orbits.
Definition 2.1. A linear domain is an intersection of a finite number of open orclosed half-spaces in an affine space, or an affine space itself. Definition 2.2.
The tangent space
T A of a linear domain A is the tangent space T x A for any x ∈ A ; the tangent spaces for different x are canonically isomorphic toeach other via translations. Definition 2.3.
Let A and B be linear domains. An affine map φ : A → B is therestriction of an affine map between affine spaces containing A and B . Such a mapinduces a map on tangent spaces which we denote by T φ : T A → T B . Definition 2.4.
Let A and B be linear domains. A linear flow from A to B is atriple Φ = ( D, φ, f ) consisting of: • the domain of definition : a linear domain D ⊂ A . • the flow map : an affine map φ : D → B .17 the action function : an affine function f : D → R .We sometimes write Φ : A → B . In the examples of interest for us, φ is injective,and f ≥ Definition 2.5.
Let Φ = (
D, φ, f ) be a linear flow from A to B and let Ψ = ( E, ψ, g )be a linear flow from B to C . Their composition is the linear flow Ψ ◦ Φ : A → C defined by Ψ ◦ Φ = ( φ − ( E ) , ψ ◦ φ, f + g ◦ φ ) . Remark 2.6.
Composition of linear flows is associative, and there is an identitylinear flow ι A : A → A given by ι A = ( A, id A , i = ( D i , φ i , f i ) is a linear flowfrom A i − to A i for i = 1 , . . . , k , and if Φ = ( D, φ, f ) is the composition Φ k ◦ · · · ◦ Φ ,then for x ∈ D , we have f ( x ) = k (cid:88) i =1 f i (( φ i − ◦ · · · ◦ φ )( x )) . (2.1) Definition 2.7. A linear flow graph G is a triple G = (Γ , A, Φ) consisting of: • A directed graph Γ with vertex set V (Γ) and edge set E (Γ). • For each vertex v of Γ, an open linear domain A v . • For each edge e of Γ from u to v , a linear flow Φ e = ( D e , φ e , f e ) : A u → A v .Let G = (Γ , A, Φ) be a linear flow graph. If p = e . . . e k is a path in Γ from u to v , we define an associated linear flowΦ p = ( D p , φ p , f p ) : A u −→ A v by Φ p = Φ e k ◦ · · · ◦ Φ e . Definition 2.8. A trajectory γ of G is a pair γ = ( p, x ), where p is a path in Γand x ∈ D p . Definition 2.9. A periodic orbit of G is an equivalence class of trajectories γ =( p, x ) where p is a cycle in Γ and x is a fixed point of φ p , i.e. φ p ( x ) = x . Two suchtrajectories γ = ( p, x ) and η = ( q, y ) are equivalent if there are paths r and s in Γsuch that p = rs , q = sr , and φ r ( x ) = y . We often abuse notation and denote theperiodic orbit by γ = ( p, x ), instead of by the equivalence class thereof. Definition 2.10.
The action of a periodic orbit γ = ( p, x ) is defined by f ( γ ) = f p ( x ). 18igure 3: An example of a flow graph with 4 nodes and 4 edges. The linear domainsand flows are depicted above their corresponding nodes and edges. Definition 2.11.
A periodic orbit γ = ( p, x ), where p is a cycle based at u , is degenerate if the induced map on tangent spaces T φ p : T D u → T D u has 1 as aneigenvalue. Otherwise we say that γ is nondegenerate . Definition 2.12.
An 2 n -dimensional symplectic flow graph G is a quadruple G = (Γ , A, ω, Φ) where: • (Γ , A, Φ) is a linear flow graph in which each linear domain A v has dimension2 n . • ω assigns to each vertex v of Γ a linear symplectic form ω v on T A v .We require that if e is an edge from u to v , then φ ∗ e ω v = ω u . Definition 2.13.
Let X be a symplectic polytope in R . We associate to X thetwo-dimensional symplectic flow graph G ( X ) = (Γ , A, ω, Φ) defined as follows: • The vertex set of Γ is the set of 2-faces of X . The linear domain associatedto a vertex is simply the corresponding 2-face, regarded as a linear domain in R . If F is a 2-face, then the symplectic form ω F on T F is the restriction ofthe standard symplectic form ω on R .19 If F and F are 2-faces, then there is an edge e in Γ from F to F if andonly if there is a 3-face E adjacent to F and F , and a trajectory of the Reebvector field R E on E from some point in F to some point in F . In this case,the linear flow Φ e = ( D e , φ e , f e ) : F −→ F is defined as follows: – The domain D e is the set of x ∈ F such that there exists a trajectory of R E from x to some point y ∈ F . – For x as above, φ e ( x ) = y , and f e ( x ) is the time it takes to flow alongthe vector field R E from x to y , or equivalently the integral of λ alongthe line segment from x to y .In the above definition, note that φ e and f e are affine, because the vector field R E on E is constant by equation (1.3). A simple calculation as in [12, Eq. (5.10)]shows that the map φ e is symplectic. Proposition 2.14.
Let X be a symplectic polytope in R . Then there is a canonicalbijection { periodic orbits of G ( X ) } ←→ { Type combinatorial Reeb orbits of X } . If ( p, x ) is a periodic orbit of G ( X ) , and if γ is the corresponding combinatorialReeb orbit, then f ( p, x ) = A comb ( γ ) . (2.2) Proof.
Suppose ( p = e · · · e k , x ) is a periodic orbit of G ( X ). Let E i denote the 3-faceof X associated to e i . There is then a combinatorial Reeb orbit γ = ( L , . . . , L k ),where L i is the line segment in E i from φ e − ◦ · · · ◦ φ e ( x ) to φ e i ◦ · · · ◦ φ e ( x ). Itfollows from Definitions 1.5 and 2.13 that this construction defines a bijection fromperiodic orbits of G ( X ) to combinatorial Reeb orbits of X . The identification ofactions (2.2) follows from equation (2.1).By Proposition 2.14, to find the Type 1 Reeb orbits of X , one can computethe symplectic flow graph G ( X ) = (Γ , A, ω, Φ), enumerate the cycles in the graphΓ, and for each cycle p , compute the fixed points of the map φ p in the domain D p .In order to avoid searching for arbitrarily long cycles in the graph Γ in the cases ofinterest, we now need to discuss combinatorial rotation numbers. When testing Viterbo’s conjecture and related conjectures, although all Type 1 orbits of X aredetected by the flow graph G ( X ), in view of Corollary 1.13 we must also account for Type 2 orbits.One can do this by either (1) extending G ( X ) to a flow graph that includes the lower-dimensionalfaces of X or (2) working with a flow graph G ( X ) whose linear domains A F are the closures of the2-faces, rather than 2-faces themselves. We use the first strategy in our computer program. .3 Combinatorial rotation numbers Definition 2.15. A trivialization of a 2 n -dimensional symplectic flow graph G =(Γ , A, ω, Φ) is a pair ( τ, (cid:101) φ ) consisting of: • For each vertex u of Γ, an isomorphism of symplectic vector spaces τ u : ( T A u , ω u ) (cid:39) −→ ( R n , ω ) . • For each edge e in Γ from u to v , a lift (cid:101) φ e,τ ∈ (cid:102) Sp(2 n ) of the symplectic matrix τ v ◦ T φ e ◦ τ − u ∈ Sp(2 n ) . Here ω denotes the standard symplectic form on R n , and (cid:102) Sp(2 n ) denotes theuniversal cover of the symplectic group Sp(2 n ). We sometimes abuse notation anddenote the trivialization ( τ, (cid:101) φ ) simply by τ .If p = e . . . e n is a path in Γ from u to v , we define (cid:101) φ p,τ = (cid:101) φ e n ,τ ◦ · · · ◦ (cid:101) φ e ,τ ∈ (cid:102) Sp(2 n ) . Definition 2.16.
Let G = (Γ , A, ω, Φ) be a 2-dimensional symplectic flow graph,let τ be a trivialization of G , and let p be a path in Γ. Define the rotation number of p with respect to τ by ρ τ ( p ) = ρ ( (cid:101) φ p,τ ) ∈ R , where the right hand side is the rotation number on (cid:102) Sp(2) reviewed in Appendix A.Suppose now that X is a symplectic polytope in R . We now define a canonicaltrivialization τ of the symplectic flow graph G ( X ) which has the useful propertythat if ( p, x ) is a periodic orbit of G ( X ), and if γ is the corresponding combinatorialReeb orbit on X from Proposition 2.14, then the rotation number ρ τ ( p ) is the limitof the rotation numbers of Reeb orbits on smoothings of X that converge to γ .Fix matrices i , j , k ∈ SO(4) which represent the quaternion algebra, such that i is the standard almost complex structure. It follows from the formula ω ( V, W ) = (cid:104) i V, W (cid:105) , together with the quaternion relations, that the matrices i , j , and k aresymplectic. In examples below, in the coordinates x , x , y , y , we use the choice i = − −
11 1 , j = −
11 1 − , k = − − . efinition 2.17. Let X be a symplectic polytope in R . We define the quater-nionic trivialization ( τ, (cid:101) φ ) of the symplectic flow graph G ( X ) as follows. • Let F be a 2-face of X . We define the isomorphism τ F : T F (cid:39) −→ R as follows. By Lemma 1.6, there is a unique 3-face E adjacent to F such thatthe Reeb cone R + F consists of the nonnegative multiples of the Reeb vectorfield R E , and the latter points into E from F . Let ν denote the outward unitnormal vector to E . If V ∈ T F , define τ F ( V ) = ( (cid:104) V, j ν (cid:105) , (cid:104) V, k ν (cid:105) ) . (2.3) • If e is an edge from F to F , define (cid:101) φ e,τ ∈ (cid:102) Sp(2) to be the unique lift of thesymplectic matrix τ F ◦ T φ e ◦ τ − F ∈ Sp(2) (2.4)that has rotation number in the interval ( − / , / Lemma 2.18.
Let X be a symplectic polytope in R . If F is a -face of X , thenthe linear map τ F in (2.3) is an isomorphism of symplectic vector spaces.Proof. Let E and ν be as in the definition of τ F . Then { i ν, j ν, k ν } is an orthonormalbasis for T E . We have ω ( i ν, j ν ) = ω ( i ν, k ν ) = 0 and ω ( j ν, k ν ) = 1. If V and W are any two vectors in T F ⊂ T E , then expanding them in this basis, we find that ω ( V, W ) = ω ( τ F ( V ) , τ F ( W )). Remark 2.19.
An alternate convention for the quaternionic trivialization wouldbe to define an isomorphism τ (cid:48) F : T F (cid:39) −→ R as follows. Let E (cid:48) be the other 3-face adjacent to F (so that the Reeb vector field R E (cid:48) points out of E along F ), and let ν (cid:48) denote the outward unit normal vector to E (cid:48) . Define τ (cid:48) F ( V ) = ( (cid:104) V, j ν (cid:48) (cid:105) , (cid:104) V, k ν (cid:48) (cid:105) ) . This is also an isomorphism of symplectic vector spaces by the same argument asin Lemma 2.18.
Definition 2.20. If X is a symplectic polytope in R and F is a 2-face of X , definethe transition matrix ψ F = τ F ◦ ( τ (cid:48) F ) − ∈ Sp(2) . emma 2.21. If X is a symplectic polytope in R and F is a -face of X , then thetransition matrix ψ F is positive elliptic (see Definition A.7).Proof. We compute that( τ (cid:48) F ) − = (cid:18) j ν (cid:48) − (cid:104) j ν (cid:48) , ν (cid:105)(cid:104) i ν (cid:48) , ν (cid:105) i ν (cid:48) , k ν (cid:48) − (cid:104) k ν (cid:48) , ν (cid:105)(cid:104) i ν (cid:48) , ν (cid:105) i ν (cid:48) (cid:19) . (2.5)To simplify notation, write a = (cid:104) ν (cid:48) , ν (cid:105) , a = (cid:104) i ν (cid:48) , ν (cid:105) , a = (cid:104) j ν (cid:48) , ν (cid:105) , and a = (cid:104) k ν (cid:48) , ν (cid:105) .It then follows from (2.3) and (2.5) that ψ F = 1 a (cid:18) a a − a a − a − a a + a a a + a a (cid:19) Then Tr( ψ F ) = 2 (cid:104) ν (cid:48) , ν (cid:105) ∈ ( − , ψ F is elliptic. Moreover a > ψ F is positive elliptic. Corollary 2.22. If E is a -face of X , if F and F are -faces of X , and if thereis a trajectory of the Reeb vector field on E from some point in F to some point in F , then (cid:101) φ e,τ has rotation number in the interval (0 , / .Proof. It follows from the definitions that the map (2.4) agrees with the transitionmatrix ψ F . By Lemma 2.21, this matrix is positive elliptic. It then follows fromLemma A.8 that its mod Z rotation number is in the interval (0 , / Definition 2.23.
Let X be a symplectic polytope in R . Let γ be a Type 1 com-binatorial Reeb orbit for X . • We define the combinatorial rotation number of γ by ρ comb ( γ ) = ρ τ ( p ) , where ( p, x ) is the periodic orbit of G ( X ) corresponding to γ in Proposi-tion 2.14, and τ is the quaternionic trivialization of X . • We say that γ is nondegenerate if the periodic orbit ( p, x ) is nondegenerateas in Definition 2.11. In this case we define the combinatorial Conley-Zehnder index of γ by equation (1.6). Remark 2.24.
By Corollary 2.22, the combinatorial rotation number is the rotationnumber of a product of elements of (cid:102)
Sp(2) each with rotation number in the interval(0 , / .4 Example: the 24-cell We now compute the symplectic flow graph G ( X ) = (Γ , A, ω, Φ) and the quater-nionic trivialization τ for the example where X is the 24-cell with vertices( ± , , , , (0 , ± , , , (0 , , ± , , (0 , , , ± , ( ± / , ± / , ± / , ± / . The polytope X has 24 three-faces, each of which is an octahedron. The 3-facesare contained in the hyperplaces ± x ± x = 1 , ± x ± y = 1 , ± x ± y = 1 , ± x ± y = 1 , ± x ± y = 1 , ± y ± y = 1 . There are 96 two-faces, each of which is a triangle; thus the graph Γ has 96 vertices.It follows from the calculations below that none of the 2-faces is Lagrangian, so that X is a symplectic polytope.To understand the edges of the graph Γ, consider for example the 3-face E contained in the hyperplane x + y = 1. The vertices of this 3-face are(1 , , , , (1 / , ± / , / , ± / , (0 , , , . The unit normal vector to this face is ν = 1 √ , , , . The Reeb vector field on E is R E = 2 (cid:18) − ∂∂x + ∂∂y (cid:19) . Thus the Reeb flow on E flows from the vertex (1 , , ,
0) to the vertex (0 , , , /
2. Each of the four 2-faces of E adjacent to (1 , , ,
0) flows to one of thefour 2-faces of E adjacent to (0 , , , F be the 2-face with vertices (1 , , , / , / , / , ± / F be the 2-face with vertices (0 , , , / , / , / , ± / F flowsto F , so there is an edge e in the graph Γ from F to F . More explicitly, we canparametrize F as (cid:18) − t + t , t + t , t + t , t − t (cid:19) , t , t > , t + t < , and we can parametrize F as (cid:18) t + t , t + t , − t + t , t − t (cid:19) , t , t > , t + t < . φ e is simply φ e ( t , t ) = ( t , t ) . The domain D e of φ e is all of F , and the action function is f e ( t , t ) = 1 − t − t . It turns out that for every other 3-face E (cid:48) , there is a linear symplectomorphism A of R such that AX = X and AE = E (cid:48) . In fact, we can take A to be rightmultiplication by an appropriate unit quaternion. It follows from this symplecticsymmetry that the Reeb flow on each 3-face behaves analogously. Putting theseReeb flows together, one finds that the graph Γ consists of 8 disjoint 12-cycles.(This example is highly non-generic!) Further calculations show that for each 12-cycle p , the map φ p is the identity, so that every point in the interior of a 2-face ison a Type 1 combinatorial Reeb orbit. Moreover, the action of each such orbit isequal to 2. In particular, X is “combinatorially Zoll” in the sense of Definition 1.19.Also, the volume of X is 2, so X has systolic ratio 1.To see how the quaternionic trivialization works, let us compute (cid:101) φ e,τ for the edge e above. For the 2-face F above, the isomorphism τ F is given in terms of the unitnormal vector ν to E . We compute that j ν = 1 √ , , , − , k ν = 1 √ , , , . It follows that in terms of the basis ( ∂ t , ∂ t ) for T F , we have τ F = 1 √ (cid:18) (cid:19) . For the 2-face F above, the isomorphism τ F is given in terms of the unit normalvector to the other F . This other 3-face is in the hyperplane x + y = 1 and so has unit normal vector ν (cid:48) = 1 √ , , , . We then similarly compute that in terms of the basis ( ∂ t , ∂ t ) for T F , we have τ F = 1 √ (cid:18) − (cid:19) Therefore the matrix (2.4) for the edge e is τ F ◦ T φ e ◦ τ − F = (cid:18) − (cid:19) (cid:18) (cid:19) − = (cid:18) −
11 1 (cid:19) . e ± iπ/ . It follows that its lift (cid:101) φ e,τ in (cid:102) Sp(2) has rotation number 1 / E , the matrix (2.4) is the same asabove, and for the other two edges associated to E , the matrix is (cid:18) −
11 0 (cid:19) , whoselift also has rotation number 1 /
6. It then follows from the quaternionic symmetry of X mentioned earlier that for every edge e (cid:48) of the graph Γ, the lift (cid:101) φ e (cid:48) ,τ is one of theabove two matrices with rotation number 1 /
6. One can further check that for each12-cycle in the graph, one obtains just one of the above two matrices repeated 12times, so each corresponding Type 1 combinatorial Reeb orbit has rotation numberequal to 2.
The goal of this section is to Proposition 1.4 and Lemma 1.6, describing the Reebdynamics on the boundary of a symplectic polytope in R . We now prove some lemmas about tangent and normal cones which we will need;see § C is a cone in R m , its polar dual is defined by C o = { y ∈ R m | (cid:104) x, y (cid:105) ≤ ∀ x ∈ X } . Lemma 3.1.
Let X be a convex set in R m and let y ∈ ∂X . Then N + y X = ( T + y X ) o , T + y X = ( N + y X ) o . Proof. If C is a closed cone then ( C o ) o = C , so it suffices to prove that N + y X =( T + y X ) o .To show that N + y X ⊂ ( T + y X ) ◦ , let v ∈ N + y X and w ∈ T + y X ; we need to showthat (cid:104) v, w (cid:105) ≤
0. By the definition of T + y X , there exist a sequence of vectors { w i } and a sequence of positive real numbers { ε i } such that y + ε i w i ∈ X for each i andlim i →∞ w i = w . By the definition of N + y X we have (cid:104) v, w i (cid:105) ≤
0, and so (cid:104) v, w (cid:105) ≤ v ∈ ( T + x X ) o , then for any x ∈ X we have x − y ∈ T + y X , so (cid:104) v, x − y (cid:105) ≤
0. It follows that v ∈ N + y X .If X is a convex polytope in R m and if E is an ( m − X , let ν E denotethe outward unit normal vector to E . 26 emma 3.2. Let X be a convex polytope in R m and let F be a face of X . Let E , . . . , E k denote the ( m − -faces whose closures contain F . Then T + F X = { w ∈ R m | (cid:104) w, ν E i (cid:105) ≤ ∀ i = 1 , . . . , k } , (3.1) N + F X = Cone ( ν E , . . . , ν E k ) . (3.2) Proof.
Let y ∈ F , and let B be a small ball around y . Then B ∩ X = ∩ i ( B ∩ H i )where { H i } is the set of all defining half-spaces for X whose boundaries contain F .The boundaries of the half-spaces H i are the hyperplanes that contain the ( m − E , . . . , E k . It follows that B ∩ X is the set of x ∈ B such that (cid:104) x − y, ν E i (cid:105) ≤ i = 1 , . . . , k . Equation (3.1) follows. Taking polar duals and using Lemma 3.1then proves (3.2). Lemma 3.3.
Let X be a convex polytope in R m and let F be a face of X . Let v ∈ N + F X \ { } and let w ∈ T + F X \ { } . Then (cid:104) v, w (cid:105) = 0 if and only if there is aface E of X with F ⊂ E such that v ∈ N + E X and w ∈ T + F E . Here if E (cid:54) = F then T + F E denotes the tangent cone of the polytope E at the face F of E ; if E = F , then we interpret T + F E = T F . Proof of Lemma 3.3.
As in Lemma 3.2, let E , . . . , E k denote the ( m − F .( ⇒ ) By the definitions of N + F X and T + F X , if v ∈ N + F X and w ∈ T + F X then (cid:104) v, w (cid:105) ≤
0. Assume also that v and w are both nonzero and (cid:104) v, w (cid:105) = 0. Then wemust have v ∈ ∂N + F X and w ∈ ∂T + F X ; otherwise we could perturb v or w to makethe inner product positive, which would be a contradiction.Since w ∈ ∂T + F X , it follows from (3.1) that (cid:104) w, ν E i (cid:105) = 0 for some i . By renum-bering we can arrange that (cid:104) w, ν E i (cid:105) = 0 if and only if i ≤ l where 1 ≤ l ≤ k . Let E = ∩ li =1 E i . Then E is a face of X adjacent to F , and w ∈ T + F E .We now want to show that v ∈ N + E X . By (3.2), we can write v = (cid:80) ki =1 a i ν E i with a i ≥
0. Since (cid:104) v, w (cid:105) = 0 and (cid:104) w, ν E i (cid:105) = 0 for i ≤ l and (cid:104) w, ν E i (cid:105) < i > l ,we must have a i = 0 for i > l . Thus v ∈ Cone( ν E , . . . , ν E l ), so by (3.2) again, v ∈ N + F X .( ⇐ ) Assume that there is a face E adjacent to X such that v ∈ N + E X and w ∈ T + F E . We can renumber so that E = ∩ li =1 E i where 1 ≤ l ≤ k . Then v ∈ Cone( ν E , . . . , ν E l ), and (cid:104) w, ν E i (cid:105) = 0 for i ≤ l , so (cid:104) v, w (cid:105) = 0. We now prove Proposition 1.4, asserting that the “combinatorial Reeb flow” on theboundary of a symplectic polytope in R is locally well-posed. This is a consequenceof the following two lemmas: 27 emma 3.4. Let X be a convex polytope in R , and let F be a face of X . Then theReeb cone R + F X = i N + F X ∩ T + F X has dimension at least . Note that there is no need to assume that 0 ∈ int( X ) in the above lemma,because the Reeb cone is invariant under translation of X . Lemma 3.5.
Let X be a symplectic polytope in R and let F be a face of X . Thenthe Reeb cone R + F X has dimension at most .Proof of Lemma 3.4. The proof has four steps.
Step 1.
We need to show that there exists a unit vector in R + F X . We firstrephrase this statement in a way that can be studied topologically.Define B = (cid:8) ( v, w ) ∈ N + F X × T + F X (cid:12)(cid:12) (cid:107) v (cid:107) = (cid:107) w (cid:107) = 1 , (cid:104) v, w (cid:105) = 0 (cid:9) . Define a fiber bundle π : Z → B with fiber S by setting Z ( v,w ) = (cid:8) u ∈ R (cid:12)(cid:12) (cid:107) u (cid:107) = 1 , (cid:104) u, v (cid:105) = 0 (cid:9) . Define two sections s , s : B −→ Z by s ( v, w ) = i v,s ( v, w ) = w. To show that there exists a unit vector in R + F X , we need to show that there existsa point ( v, w ) ∈ B with s ( v, w ) = s ( v, w ). Step 2.
Let B = (cid:8) w ∈ ∂T + F X (cid:12)(cid:12) (cid:107) w (cid:107) = 1 (cid:9) . The space B is the set of unit vectors on the boundary of a nondegenerate cone, andthus is homeomorphic to S . Recall from the proof of Lemma 3.3 that if ( v, w ) ∈ B then w ∈ B . We now show that the projection B → B sending ( v, w ) (cid:55)→ w is ahomotopy equivalence.To do so, observe that by Lemma 3.3, we have B = (cid:91) F ⊂ E (cid:8) v ∈ N + E X (cid:12)(cid:12) (cid:107) v (cid:107) = 1 (cid:9) × (cid:8) w ∈ T + F E (cid:12)(cid:12) (cid:107) w (cid:107) = 1 (cid:9) . (3.3)28f F is a 3-face, then in the union (3.3), we only have E = F ; there is a uniqueunit vector v ∈ N + E X , and so the projection B → B is a homeomorphism.If F is a 2-face, then in (3.3), E can be either F itself, or one of the two three-faces adjacent to F , call them E and E . The contribution from E = F is acylinder, while the contributions from E = E and E are disks which are glued tothe cylinder along its boundary. The projection B → B collapses the cylinder to acircle, which again is a homotopy equivalence.If F is a 1-face, with k adjacent 3-faces, then the contribution to (3.3) from E = F consists of two disjoint closed k -gons. Each 2-face E adjacent to F contributes asquare with opposite edges glued to one edge of each k -gon. Each 3-face E adjacentto F contributes a bigon filling in the gap between two consecutive squares. Theprojection B → B collapses each k -gon to a point and each bigon to an interval,which again is a homotopy equivalence.Finally, suppose that F is a 0-face. Then E = F makes no contribution to(3.3), since T F = { } contains no unit vectors. Now B has a cell decompositionconsisting of a k -cell for each ( k + 1)-face adjacent to F . The space B is obtainedfrom B by thickening each 0-cell to a closed polygon, and thickening each 1-cell toa square. Again, this is a homotopy equivalence. Step 3.
The S -bundle Z → B is trivial. To see this, observe that Z is thepullback of a bundle over N + F X \ { } , whose fiber over v is the set of unit vectorsorthogonal to v . Since N + F X \{ } is contractible, the latter bundle is trivial, and thusso is Z . In particular, the bundle Z has two homotopy classes of trivialization, whichdiffer only in the orientation of the fiber. We now show that, using a trivializationto regard s and s as maps B → S , the mod 2 degrees of these maps are given bydeg( s ) = 0 and deg( s ) = 1.It follows from the triviality of the bundle Z that deg( s ) = 0.To prove that deg( s ) = 1, we need to pick an explicit trivialization of Z . To doso, fix a vector v ∈ int( T + F X ). Let S denote the set of unit vectors in the orthogonalcomplement v ⊥ . Let P : R → v ⊥ denote the orthogonal projection. We then havea trivialization Z (cid:39) −→ B × S sending (( v, w ) , u ) (cid:55)−→ (( v, w ) , P u/ (cid:107) P u (cid:107) ) . Note here that for every ( v, w ) ∈ B , the restriction of P to v ⊥ is an isomorphism,because otherwise v would be orthogonal to v , but in fact we have (cid:104) v, v (cid:105) < s is a map B → S which is thecomposition of the projection B → B with the map B → S sending w (cid:55)−→ P w/ (cid:107)
P w (cid:107) . v is not parallel to any vector in ∂T + F X . Thus deg( s ) = 1. Step 4.
We now complete the proof of the lemma. Suppose to get a contradictionthat there does not exist a point p ∈ B with s ( p ) = s ( p ). It follows, using atrivialization of Z to regard s and s as maps B → S , that s is homotopic tothe composition of s with the antipodal map. Then deg( s ) = − deg( s ). Thiscontradicts Step 3. Remark 3.6.
It might be possible to generalize Lemma 3.4 to show that if X isany convex set in R n with nonempty interior and if z ∈ ∂X , then the Reeb cone R + z X is at least one dimensional.We now prepare for the proof of Lemma 3.5. Lemma 3.7.
Let X be a convex polytope in R n . Then for every face F of X , thereexists a face E with F ⊂ E such that R + F X ⊂ T + F ¯ E. Proof.
Let { E i } Ni =1 denote the set of faces whose closures contain F . By Lemma 3.3,we have R + F X ⊂ N (cid:91) i =1 T + F ¯ E i . (3.4)Let V denote the subspace of R n spanned by R + F X . Note that since the latterset is a cone, it has a nonempty interior in V . We claim now that V ⊂ T E i for some i . If not, then V ∩ T E i is a proper subspace of V for each i . But by (3.4), we have R + F X = (cid:0) ∪ i T + F ¯ E i (cid:1) ∩ R + F X ⊂ ( ∪ i T E i ) ∩ V. This is a contradiction, since the left hand side has a nonempty interior in V , whilethe right hand side is a union of proper subspaces of V .Since V ⊂ T E i , it follows that R + F X ⊂ T + F ¯ E i , because by (3.4) again, R + F X = R + F X ∩ V = R + F X ∩ T E i ⊂ T E i ∩ (cid:18) (cid:91) j T + F ¯ E j (cid:19) = T F ¯ E i , (cid:50) Lemma 3.8.
Let X be a convex polytope in R n , and let F be a face of X . Let v ∈ R + F X . Suppose that v ∈ int( T + F E ) for some (2 n − -face E whose closurecontains F . Then v is a positive multiple of i ν E . roof. Let E = E , . . . , E N denote the (2 n − F , andlet ν i denote the outward unit normal vector to E . Since v ∈ int( T + F E ), we have (cid:104) v, ν (cid:105) = 0 and (cid:104) v, ν i (cid:105) < i >
1. Since − i v ∈ N + F X , it follows from Lemma 3.2that we can write − i v = N (cid:88) i =1 a i ν i with a i ≥
0. Since (cid:104) v, i v (cid:105) = 0, we conclude that a i = 0 for i >
1. Thus − i v = a ν ,and a > Proof of Lemma 3.5.
Suppose v , v are distinct unit vectors in R + F X . By Lemma 3.7,there is a 3-face E such that v and v are both in T + F ¯ E . In particular, v and v are linearly independent.Since v and v are both in the cone R + F X , it follows that if t ∈ [0 ,
1] then theaffine linear combination (1 − t ) v + tv is also in this cone. Since v and v are linearlyindependent, these affine linear combinations cannot be in the interior of T + F E , orelse this would contradict the projective uniqueness in Lemma 3.8. Consequently v and v are both contained in T + F E (cid:48) for some 2-face E (cid:48) on the boundary of E .We now have ω ( v , v ) = (cid:104) v , − i v (cid:105) ≤ , where the inequality holds since v ∈ T + F X and − i v ∈ N + F X . By a symmetriccalculation, ω ( v , v ) ≤
0. It follows that ω ( v , v ) = 0. Since v and v are linearlyindependent vectors in T E (cid:48) , this contradicts the hypothesis that ω | T E (cid:48) is nondegen-erate.
We now prove Lemma 1.6, describing the possibilities for the Reeb cone of a face ofa symplectic polytope in R . Lemma 3.9.
Let X be a convex polytope in R and let F be a -face of X . Let E and E denote the -faces adjacent to F , and let ν i denote the outward unit normalvector to E i .(a) If (cid:104) i ν , ν (cid:105) < , then every nonzero vector w in the Reeb cone R + E points into E from F , that is w ∈ int( T + F E ) .(b) If (cid:104) i ν , ν (cid:105) > , then every nonzero vector w in the Reeb cone R + E points outof E from F , that is w ∈ int( − T + F E ) .(c) If (cid:104) i ν , ν (cid:105) = 0 , then F is Lagrangian. roof. Let η denote the unit normal vector to F in T E pointing into E . Thevector η must be a linear combination of ν and ν (since it is normal to F ), itmust be orthogonal to ν (since it is tangent to E ), and it must have negative innerproduct with ν (since it points into E ). It follows that η = − ν + (cid:104) ν , ν (cid:105) ν (cid:107) − ν + (cid:104) ν , ν (cid:105) ν (cid:107) . (3.5)The vector w points into E if and only if (cid:104) η, w (cid:105) >
0, and the vector w pointsout of E if and only if (cid:104) η, w (cid:105) <
0. For w in the Reeb cone of E , we know that w is a positive multiple of i ν . By equation (3.5), we have (cid:104) η, i ν (cid:105) = −(cid:104) i ν , ν (cid:105)(cid:107) − ν + (cid:104) ν , ν (cid:105) ν (cid:107) . Thus if (cid:104) i ν , ν (cid:105) is nonzero, then it has opposite sign from (cid:104) η, w (cid:105) . This proves (a)and (b).If (cid:104) i ν , ν (cid:105) = 0, then ω ( i ν , i ν ) = 0, but i ν and i ν are linearly independenttangent vectors to F , so F is Lagrangian. This proves (c). Lemma 3.10.
Let X be a convex polytope in R and let F be a 2-face of X . If T F ∩ R + F X (cid:54) = { } , then F is Lagrangian.Proof. If w ∈ T F ∩ R + F X , then for any other vector u ∈ T F , we have ω ( w, u ) = (cid:104) i w, u (cid:105) = 0since − i w ∈ N + F X . If we also have w (cid:54) = 0, then it follows that F is Lagrangian. Proof of Lemma 1.6. If F is a 3-face, then by the definition of the Reeb cone, R + F X consists of all nonnegative multiples of i ν F ; and i ν F is a positive multiple of theReeb vector field on F by equation (1.3).Suppose now that F is a k -face with k <
3, and that w is a nonzero vector inthe Reeb cone R + F X . Applying Lemma 3.3 to v = − i w and w , we deduce that thereis a face E of X with F ⊂ E such that − i w ∈ N + E X and w ∈ T + F E . In particular, w ∈ T E ∩ R + E X. (3.6)By Lemma 3.10 and our hypothesis that X is a symplectic polytope, E is not a2-face.If F is a 2-face, we conclude that w is in the Reeb cone R + E X for one of the3-faces E adjacent to F . By Lemma 3.9, w must point into E .If F is a 1-face, then E is either a 3-face adjacent to F , or F itself. In the casewhen E = F , the vector w cannot be in the Reeb cone of any 3-face F adjacent32o F . The reason is that if F is one of the two 2-faces with F ⊂ F ⊂ F , thenby Lemma 3.9, the Reeb cone of F is not tangent to F , so it certainly cannot betangent to F .If F is a 0-face, then E is adjacent to F and is either a 3-face or a 1-face. If E is a 1-face, then it is a bad 1-face by (3.6). In this section let Y ⊂ R be a smooth star-shaped hypersurface with the contactform λ = λ | Y and contact structure ξ = Ker( λ ). We now define a special trivializa-tion τ of the contact structure ξ , and we prove a key property of this trivialization. The following definition is a smooth analogue of Definition 2.17.
Definition 4.1.
Define the quaternionic trivialization τ : ξ (cid:39) −→ Y × R (4.1)as follows. If y ∈ Y and V ∈ T y Y , let ν denote the outward unit normal to Y at y ,and define τ ( V ) = ( y, (cid:104) V, j ν (cid:105) , (cid:104) V, k ν (cid:105) ) . By abuse of notation, for fixed y ∈ Y we write τ : ξ y (cid:39) −→ R to denote the restrictionof (4.1) to ξ y followed by projection to R .From now on we always use the quaternionic trivialization τ for smooth star-shaped hypersurfaces in R . Lemma 4.2.
The quaternionic trivialization τ is a symplectic trivialization of ξ .Proof. Same calculation as the proof of Lemma 2.18(a).
Remark 4.3.
The inverse τ − : Y × R (cid:39) −→ ξ is described as follows. Recall from (1.3) that the Reeb vector field at y is a positivemultiple of i ν . Then τ − ( y, (1 , j ν to ξ y along the Reebvector field, while τ − ( y, (0 , k ν to ξ y along the Reebvector field. 33 .2 Linearized Reeb flow We now make some definitions which we will need in order to bound the rotationnumbers of Reeb orbits and Reeb trajectories.
Definition 4.4. If y ∈ Y and t ≥
0, define the linearized Reeb flow φ ( y, t ) ∈ Sp(2) to be the composition R τ − −→ ξ y d Φ t −→ ξ Φ t ( y ) τ −→ R (4.2)where Φ t : Y → Y denotes the time t flow of the Reeb vector field, and τ is thequaternionic trivialization. Define the lifted linearized Reeb flow (cid:101) φ ( y, t ) ∈ (cid:102) Sp(2)to be the arc (cid:101) φ ( y, t ) = { φ ( y, s ) } s ∈ [0 ,t ] . (4.3)Note that we have the composition property (cid:101) φ ( y, t + t ) = (cid:101) φ ( φ t ( y ) , t ) ◦ (cid:101) φ ( y, t ) . Next, let P ξ denote the “projectivized” contact structure P ξ = ( ξ \ Z ) / ∼ where Z denotes the zero section, and two vectors are declared equivalent if theydiffer by multiplication by a positive scalar. Thus P ξ is an S -bundle over Y . TheReeb vector field R on Y canonically lifts, via the linearized Reeb flow, to a vectorfield (cid:101) R on P ξ .The quaternionic trivialization τ defines a diffeomorphism τ : P ξ (cid:39) −→ Y × S . Let σ : P ξ −→ S denote the composition of τ with the projection Y × S → S . Definition 4.5.
Define the rotation rate r : P ξ −→ R to be the derivative of σ with respect to the lifted linearized Reeb flow, r = (cid:101) Rσ.
Define the minimum rotation rate r min : Y −→ R by r min ( y ) = min (cid:101) y ∈ P ξ y r ( (cid:101) y ) .
34t follows from (A.6) and (A.7) that we have the following lower bound on therotation number of the lifted linearized flow of a Reeb trajectory.
Lemma 4.6.
Let y be a smooth star-shaped hypersurface in R , let y ∈ Y , and let t ≥ . Then ρ ( (cid:101) φ ( y, t )) ≥ (cid:90) t r min (Φ s ( y )) ds. We now prove a key identity which relates the linearized Reeb flow, with respect tothe quaternionic trivialization τ , to the curvature of Y . This identity (in differentnotation) is due to U. Hryniewicz and P. Salom˜ao [17]. Below, let S : T Y ⊗ T Y → R denote the second fundamental form defined by S ( u, w ) = (cid:104)∇ u ν, w (cid:105) , where ν denotes the outward unit normal vector to Y , and ∇ denotes the trivialconnection on the restriction of T R to Y . Also write S ( u ) = S ( u, u ). Proposition 4.7.
Let Y be a smooth star-shaped hypersurface in R , let y ∈ Y , let θ ∈ R / π Z , and write σ = θ/ π ∈ R / Z . Then at the point τ − ( y, σ ) ∈ P ξ , we have (cid:101) Rσ = 1 π (cid:104) ν, y (cid:105) ( S ( i ν ) + S (cos( θ ) j ν + sin( θ ) k ν )) . (4.4) Proof.
It follows from the definitions that2 π (cid:101) Rσ = (cid:104)L R ((cos θ ) j ν + (sin θ ) k ν ) , (sin θ ) j ν − (cos θ ) k ν (cid:105) = − (cos θ ) (cid:104)L R j ν, k ν (cid:105) + (sin θ ) (cid:104)L R k ν, j ν (cid:105) + (sin θ cos θ )( (cid:104)L R j ν, j ν (cid:105) − (cid:104)L R k ν, k ν (cid:105) ) . (4.5)We compute (cid:104)L R j ν, k ν (cid:105) = (cid:104)∇ R j ν − ∇ j ν R, k ν (cid:105) = 2 (cid:104) ν, y (cid:105) ( (cid:104)∇ i ν j ν, k ν (cid:105) − (cid:104)∇ j ν i ν, k ν (cid:105) )= 2 (cid:104) ν, y (cid:105) ( −(cid:104)∇ i ν ν, i ν (cid:105) − (cid:104)∇ j ν ν, j ν (cid:105) )= 2 (cid:104) ν, y (cid:105) ( − S ( i ν ) − S ( j ν )) . (4.6)35ere in the second to third lines we have used the fact that multiplication on theleft by a constant unit quaternion is an isometry. Similar calculations show that (cid:104)L R k ν, j ν (cid:105) = 2 (cid:104) ν, y (cid:105) ( S ( i ν ) + S ( k ν )) , (4.7) (cid:104)L R j ν, j ν (cid:105) = −(cid:104)L R k ν, k ν (cid:105) = 2 (cid:104) ν, y (cid:105) S ( j ν, k ν ) . (4.8)Plugging (4.6), (4.7) and (4.8) into (4.5) proves the curvature identity (4.5). Remark 4.8.
Since the second fundamental form is positive definite when Y isstrictly convex, and positive semidefinite when Y is convex, by Lemma 4.6 we obtainthe following corollary: If Y is a convex star-shaped hypersurface in R then (cid:101) Rσ ≥ everywhere, so (cid:101) φ ( y, t ) has nonnegative rotation number for all y ∈ Y and t ≥ . If Y is a strictly convex star-shaped hypersurface in R then (cid:101) Rσ > everywhere, so (cid:101) φ ( y, t ) has positive rotation number for all y ∈ Y and t > . In § § R . In § § C , and in particular how to makesense of the “rotation number” of Reeb trajectories. In § If X ⊂ R m is a compact convex set and ε >
0, define the ε -smoothing X ε of X byequation (1.7). Observe that X ε is convex. Denote its boundary by Y ε = ∂X ε . Wenow describe Y ε more explicitly, in a way which mostly does not depend on ε . Wefirst have: Lemma 5.1. If X is a compact convex set then Y ε = { y ∈ R m | dist( y, X ) = ε } . Proof.
The left hand side is contained in the right hand side because distance to X is a continuous function on R m . The reverse inclusion holds because given y ∈ R m with dist( y, X ) = ε , since X is compact and convex, there is a unique point x ∈ X which is closest to y . By convexity again, X is contained in the closed half-space { z ∈ R m | (cid:104) z, y − x (cid:105) ≤ } . It follows that dist( t ( y − x ) , X ) = εt for t >
0, so that y ∈ ∂X ε . 36 efinition 5.2. If X ⊂ R m is a compact convex set, define the “blown-up bound-ary” Y = (cid:8) ( y, v ) (cid:12)(cid:12) y ∈ ∂X, v ∈ N + y X, | v | = 1 (cid:9) ⊂ ∂X × S m − . We then have the following lemma, which is proved by similar arguments toLemma 5.1:
Lemma 5.3.
Let X ⊂ R m be a compact convex set and let ε > . Then:(a) There is a homeomorphism Y (cid:39) −→ Y ε sending ( y, v ) (cid:55)→ y + εv .(b) The inverse homeomorphism sends y (cid:55)→ ( x, ε − ( y − x )) where x is the uniqueclosest point in X to y .(c) For y ∈ Y ε , if x is the closest point in X to y , then the positive normal cone N + y X ε is the ray consisting of nonnegative multiples of y − x . Suppose now that X ⊂ R m is a convex polytope and ε > Definition 5.4. If F is a face of X , define the ε -smoothed face F ε = { x ∈ Y ε | dist( x, F ) = ε } . By Lemma 5.3, we have Y ε = (cid:71) F F ε and F ε = F + { v ∈ N + F X | | v | = ε } . In particular, it follows that Y ε is a C smooth hypersurface, and it is C ∞ exceptalong strata of the form ∂F + { v ∈ N + F X | | v | = ε } . Suppose now that X is a symplectic polytope in R and ε >
0. As noted above, Y ε = ∂X ε is a C convex hypersurface, and as such it has a well-defined C Reebvector field, which is smooth except along the strata of Y ε arising from the boundariesof the faces of X . We now investigate the Reeb flow on Y ε in more detail, as well asthe lifted linearized Reeb flow (cid:101) φ from Definition 4.4. We do not also need to mention strata of the form F + ∂ { v ∈ N + F X | | v | = ε } , because anypoint in ∂N + F X is contained in N + E X where E is a face with F ⊂ ∂E . eneral remarks. By Lemma 5.3, a point in Y ε lives in an ε -smoothed face F ε for a unique face F of X , and thus has the form y + εv where y ∈ F and v ∈ N + F X isa unit vector. By equation (1.3) and Lemma 5.3(c), the Reeb vector field at y + εv is given by R y + εv = 2 i v (cid:104) v, y (cid:105) + ε . (5.1) Lemma 5.5.
The Reeb vector field (5.1) on the ε -smoothed face F ε , regarded as amap F ε → R , depends only v ∈ N + F X and not on the choice of y ∈ F .Proof. This follows from equation (5.1), because for fixed v ∈ N + F X and for twopoints y, y (cid:48) ∈ F , by the definition of positive normal cone we have (cid:104) v, y − y (cid:48) (cid:105) = 0. Smoothed 3-faces.
The Reeb flow on a smoothed 3-face is very simple.
Lemma 5.6.
Let X ⊂ R be a symplectic polytope, let ε > , and let E be a -faceof X with outward unit normal vector ν .(a) The Reeb vector field on E ε , regarded as a map E ε → R , agrees with the Reebvector field on E , up to rescaling by a positive constant which limits to as ε → .(b) If γ : [0 , t ] → E ε is a Reeb trajectory, then (cid:101) φ ( γ (0) , t ) = 1 ∈ (cid:102) Sp(2) .(c) If y ∈ ∂E , then at the point y + εν ∈ Y ε , the Reeb vector field on Y ε is nottangent to ∂E ε .Proof. (a) This follows from equation (5.1).(b) For s ∈ [0 , t ], the Reeb flow Φ s : Y ε → Y ε is a translation on a neighborhoodof γ (0). Consequently the linearized Reeb flow d Φ s : ξ γ (0) → ξ γ ( s ) is the identity,if we regard ξ γ (0) and ξ γ ( s ) as (identical) two-dimensional subspaces of R . Thequaternionic trivialization τ : R → ξ γ ( s ) likewise does not depend on s ∈ [0 , t ].Consequently φ ( y, s ) = 1 for all s ∈ [0 , t ]. Thus (cid:101) φ ( y, t ) is the constant path at theidentity in Sp(2).(c) It is equivalent to show that the Reeb vector field on E at y is not tangent to ∂E . If the Reeb vector field on E at y is tangent to ∂E , then it is tangent to some2-face F ⊂ ∂E . By Lemma 3.10, the face 2-face F is Lagrangian, contradicting ourhypothesis that the polytope X is symplectic.38 moothed 2-faces. Let F be a 2-face. Let E and E be the 3-faces adjacentto F . By Lemma 1.6, we can choose these so that R E points out of F ; and asimilar argument shows that then R E points into F . Let ν and ν denote theoutward unit normal vectors to E and E respectively. By Lemma 3.2, the normalcone N + F consists of nonnegative linear combinations of ν and ν . Let { v, w } bean orthonormal basis for F ⊥ , such that the orientation given by ( v, w ) agrees withthe orientation given by ( ν , ν ). For i = 1 , ν i = (cos θ i ) v + (sin θ i ) w where 0 < θ − θ < π . We then have a homeomorphism F × [ θ , θ ] (cid:39) −→ F ε , ( y, θ ) (cid:55)−→ y + ε ((cos θ ) v + (sin θ ) w ) . (5.2)In the coordinates ( y, θ ), the Reeb vector field R on F ε depends only on θ byLemma 5.5, and has positive ∂ θ coordinate for both θ = θ and θ = θ by our choiceof labeling of E and E . By equation (5.1), Lemma 3.10, and our hypothesis thatthe polytope X is symplectic, the ∂ θ component of the Reeb vector field is positiveon all of F ε .Let U F,ε ⊂ F denote the set of y ∈ F such that the Reeb flow on Y ε startingat ( y, θ ) ∈ F ε stays in F ε until reaching a point in F × { θ } , which we denote by( φ F,ε ( y ) , θ ). Thus we have a well-defined “flow map” φ F,ε : U F,ε → F . Lemma 5.7.
Let F be a two-face of a symplectic polytope X ⊂ R . Then:(a) The flow map φ F,ε : U F,ε → F above is translation by a vector V F,ε ∈ T F .(b) | V F,ε | = O ( ε ) and lim ε → U F,ε = F .(c) Let y ∈ U F,ε and let t be the Reeb flow time on F ε from y + εν to φ F,ε ( y ) + εν .Then φ ( y, t ) ∈ Sp(2) agrees with the transition matrix ψ F in Definition 2.20,and (cid:101) φ ( y, t ) ∈ (cid:102) Sp(2) is the unique lift of ψ F with rotation number in the interval (0 , / .Proof. (a) If y, y (cid:48) ∈ U F,ε , then it follows from the translation invariance in Lemma 5.5that φ F,ε ( y ) − y = φ F,ε ( y (cid:48) ) − y (cid:48) , so φ F,ε is a translation.(b) It follows from equation (5.1) that for each v , the Reeb vector field R y + εv ,regarded as a vector in R , has a well-defined limit as ε →
0, which by Lemma 3.10is not tangent to F . Since ∂ θ , regarded as a vector in R , has length ε , it followsthat the flow time of the Reeb vector field on F ε from F × { θ } to F × { θ } is O ( ε ). Consequently the translation vector V F,ε has length O ( ε ), and the complement F \ U F,ε of the domain of the flow map is contained within distance O ( ε ) of ∂F .(c) Write y = y + εν and y = φ F,ε ( y ) + εν . By part (a) and the translationinvariance in Lemma 5.5, the time t Reeb flow Φ t on Y ε restricted to U F,ε + εν is a39ranslation in R . Hence the derivative of Φ t on the full tangent space of Y ε , namely d Φ t : T y Y ε −→ T y Y ε , restricts to the identity on T F . We now have a commutative diagram ξ y −−−→ T F τ (cid:48) F −−−→ R d Φ t (cid:121) (cid:121) (cid:121) ψ F ξ y −−−→ T F τ F −−−→ R . Here the upper left horizontal arrow is projection along the Reeb vector field in T y Y ε , and the lower left horizontal arrow is projection along the Reeb vector field in T y Y ε . The right horizontal arrows were defined in Definition 2.17 and Remark 2.19.The left square commutes because d Φ t preserves the Reeb vector field. The rightsquare commutes by Definition 2.20. The composition of the arrows in the top rowis the quaternionic trivialization τ on ξ y , and the composition of the arrows in thebottom row is the quaternionic trivialization τ on ξ y . Going around the outside ofthe diagram then shows that φ ( y, t ) = ψ F .To determine the lift (cid:101) φ ( y, t ), note that this is actually defined for, and dependscontinuously on, any ε > E and E that do notcontain the origin and that intersect in a non-Lagrangian 2-plane F . Thus we candenote this lift by (cid:101) φ ( E , E , ε ) ∈ (cid:102) Sp(2). Now fixing E , F , and ε , we can interpolatefrom E and E via a 1-parameter family of hyperplanes { E s } s ∈ [1 , such that 0 / ∈ E s and E ∩ E s = F for 1 < s ≤
2. The rotation number ρ : (cid:102) Sp(2) → R then gives usa continuous map f : (1 , −→ R ,s (cid:55)−→ ρ (cid:16) (cid:101) φ ( E , E s , ε ) (cid:17) We have lim τ (cid:38) (cid:101) φ ( E , E s , ε ) = 1, so lim s (cid:38) f ( s ) = 0. On the other hand, for each s ∈ (1 , f ( s ) is in the interval (0 , /
2) by Lemma 2.21. Itfollows by continuity that f ( s ) ∈ (0 , /
2) for all s ∈ (1 , f (2) ∈ (0 , / Smoothed 1-faces.
The Reeb flow on a smoothed 1-face is more complicated,but we will not need to analyze this in detail. We just remark that one can see thedifference between good and bad 1-faces in the Reeb dynamics on their smoothings.Namely: 40 emark 5.8. If L is a bad 1-face, then by definition, there is a unique unit vector v ∈ N + L X such that i v is tangent to L . The line segment L + εv ⊂ L ε is then a Reebtrajectory. On the complement of this line in L ε , the Reeb vector field spirals aroundthe line, with the number of times that it spirals around going to infinity as ε → L is a good 1-face, then the Reeb vector field on L ε always has anonzero component in the N + L X direction. Smoothed 0-faces. If P is a 0-face, then by Lemma 5.3, P ε is identified with adomain in S . By equation (5.1), the Reeb vector field on this domain agrees, upto reparametrization, with the standard Reeb vector field on the unit sphere in R . We now investigate in more detail how Reeb trajectories on Y ε intersect the stratawhere Y ε is not C ∞ .Let Σ denote the subset of Y ε where Y ε is not locally C ∞ . By the discussion atthe end of § (cid:116) Σ (cid:116) Σ where: • Σ is the disjoint union of sets P + { v ∈ N + L X | | v | = ε } (5.3)where P is a vertex of X , and L is a 1-face adjacent to P . • Σ is the disjoint union of sets L + { v ∈ N + F X | | v | = ε } (5.4)where L is a 1-face, and F is a 2-face adjacent to L . • Σ is the disjoint union of sets F + εν where F is a 2-face, and ν is the outward unit normal vector to one of the two3-faces E adjacent to F . 41 emma 5.9. Let X ⊂ R be a symplectic polytope, let ε > , and let γ : [ a, b ] → Y ε be a Reeb trajectory. Then there exist a nonnegative integer k and real numbers a ≤ t < t < · · · < t k ≤ b with the following properties:(a) γ ( t i ) ∈ Σ for each i .(b) For each i = 0 , . . . , k , one of the following possibilities holds:(i) γ maps ( t i , t i +1 ) to Y ε \ Σ . (Here we interpret t = a and t k +1 = b .)(ii) γ maps ( t i , t i +1 ) to a Reeb trajectory in a component of Σ . (Each com-ponent of Σ contains at most one Reeb trajectory of positive length.)(iii) γ maps ( t i , t i +1 ) to a Reeb trajectory in a component of Σ . (This canonly happen when the corresponding -face F is complex linear, and inthis case the component of Σ is foliated by Reeb trajectories.)Proof. We need to show that a Reeb trajectory intersects Σ in isolated points, or inReeb trajectories of the types described in (ii) and (iii).We have seen in § . Thusthe Reeb trajectory γ intersects Σ only in isolated points.Next let us consider the Reeb vector field on a component of Σ of the form(5.4). As in § E and E denote the 3-faces adjacent to F , with outward unitnormal vectors ν and ν respectively. The smoothing F ε is parametrized by (5.2).This parametrization extends by the same formula to a parametrization of F ε by F × [ θ , θ ]. The latter parametrization includes the component (5.4) of Σ as therestriction to L × [ θ , θ ]. By equation (5.1), at the point corresponding to ( y, θ ) in(5.2), the Reeb vector is given by R = 2 (cid:104) (cos θ ) v + (sin θ ) w, y (cid:105) + ε i ((cos θ ) v + (sin θ ) w ) . (5.5)This vector is tangent to the component (5.4) if and only if the orthogonal projectionof i ((cos θ ) v + (sin θ ) w ) to F is parallel to L .If the projections of i v and i w to F are not parallel, then this tangency will onlyhappen for isolated values of θ , and since the Reeb vector field on F ε always has apositive ∂ θ component, a Reeb trajectory will only intersect the component (5.4) inisolated points.If on the other hand the projections of i v and i w to F are parallel, then there is anontrivial linear combination of i v and i w whose projection to F is zero. This meansthat there is a nonzero vector ν perpendicular to F such that i ν is also perpendicularto F . This means that F ⊥ is complex linear, and thus F is also complex linear.Then i v and i w are both perpendicular to F , so in the parametrization (5.2), theReeb vector field vector field (5.5) is a just a positive multiple of ∂ θ .42he conclusion is that a Reeb trajectory will intersect each component (5.4)of Σ either in isolated points, or (when F is complex linear) in Reeb trajectorieswhich, in the parametrization (5.2), start on L × { θ } and end on L × { θ } , keepingthe L component constant.Finally we consider the Reeb vector field on a component (5.3) of Σ . The set ofvectors v that arise in (5.3) is a domain D in the intersection of the sphere | v | = ε with the hyperplane L ⊥ . As we have seen at the end of § Y ε at a point in (5.3) agrees, up to scaling, with the standard Reeb vector fieldon the sphere | v | = ε , whose Reeb orbits are Hopf circles. There is a unique Hopfcircle C contained entirely in L ⊥ . All other Hopf circles intersect L ⊥ transversely.Thus any Reeb trajectory in Y ε intersects the component (5.3) in isolated pointsand/or the arc corresponding to C ∩ D , if the latter intersection is nonempty. Suppose γ : [ a, b ] → Y ε is a Reeb trajectory. Let D ⊂ Y ε be a disk through γ ( a )tranverse to γ , and let D (cid:48) ⊂ Y ε be a disk through γ ( b ) transverse to γ . We canidentify D with a neighborhood of 0 in ξ γ ( a ) , and D (cid:48) with a neighborhood of 0 in ξ γ ( b ) , via orthogonal projection in R . If D is small enough, then there is a well-defined map continuous map φ : D → D (cid:48) with φ ( γ ( a )) = γ ( b ), such that for each x ∈ D , there is a unique Reeb trajectory near γ starting at x and ending at φ ( x ). Lemma 5.10.
Let X be a symplectic polytope in R , let ε > , and let γ : [ a, b ] → Y ε be a Reeb trajectory. Then there is a unique (independent of the choice of D and D (cid:48) ) homeomorphism P γ : ξ γ ( a ) −→ ξ γ ( b ) such that:(a) lim x → φ ( x ) − P γ ( x ) (cid:107) x (cid:107) = 0 . (5.6) (b) P γ is linear along rays, i.e. if x ∈ ξ γ ( a ) and c > then P γ ( cx ) = cP γ ( x ) .This map P γ has the following additional properties:(c) If γ does not include any arcs as in Lemma 5.9(ii)-(iii), and in particular if γ does not intersect any smoothed -face or smoothed -face, then P γ is linear.(d) For t ∈ ( a, b ) we have the composition property P γ = P γ | [ t,b ] ◦ P γ | [ a,t ] . e) For t ∈ [ a, b ] , the homeomorphism R → R given by the composition R τ − −→ ξ γ ( a ) P γ | [ a,b ] −→ ξ γ ( t ) τ −→ R is a continuous, piecewise smooth function of t .Proof. Uniqueness of the homeomorphism P γ follows from properties (a) and (b).Independence of the choice of D and D (cid:48) follows from properties (a) and (b) togetherwith continuity of the Reeb vector field. Assuming existence of the homeomorphism P γ , the composition property (d) follows from uniqueness.We now need to prove existence of the homeomorphism satisfying properties (a),(b), (c), and (e). Let a ≤ t < t < · · · < t k ≤ b be the subdivision of the inteveral[ a, b ] given by Lemma 5.9. For i = 0 , . . . , k , let γ i denote the restriction of γ to[ t i , t i +1 ], where we interpret t = a and t k = b . It is enough to prove existence of ahomeomorphism P γ i : ξ γ ( t i ) −→ ξ γ ( t i +1 ) with the required properties for each i . The desired homeomorphism P γ is thengiven by the composition P k · · · P .For case (i) in Lemma 5.9, a homeomorphism P γ i with properties (a), (b), and(e) is given by the usual linearized return map on the smooth hypersurface Y ε \ Σfrom t i + δ to t i +1 − δ , in the limit as δ →
0. Since P γ i is linear, we also obtainproperty (c).For case (ii) or (iii) in Lemma 5.9, the existence of P γ i with the desired propertiesfollows from the fact that γ i is on a smooth hypersurface separating two regions of Y ε , on each of which the Reeb vector field is C ∞ . Remark 5.11.
In case (ii) or (iii) above, the description of the Reeb flow in § P γ i quite explicitly. Namely, for a suitable trivi-alization, P γ i is given by the flow for some positive time of a continuous, piecewisesmooth vector field V on R , which is the derivative of a shear on one half of R ,and which is the derivative of a rotation or the identity on the other half of R . Forcase (ii), the vector field has the form V ( x, y ) = (cid:26) − y∂ x , x ≥ ,x∂ y − y∂ x , x ≤ . (5.7)For case (iii), the vector field has the form V ( x, y ) = (cid:26) x∂ y , x ≥ , , x ≤ . (5.8)44ince the map P γ : ξ γ ( a ) → ξ γ ( b ) sends rays to rays, it induces a well-defined map P ξ γ ( a ) → P ξ γ ( b ) . It follows from Lemma 5.10(c),(d) and equations (5.7) and (5.8)that the latter map is C . Similarly to (4.2), we obtain a C diffeomorphism of S given by the composition S τ − −→ P ξ γ ( a ) P γ −→ P ξ γ ( b ) τ −→ S . Stealing the notation from Definition 4.4, let us denote this map by φ ( y, t ) where y = γ ( a ) and t = b − a . By analogy with (4.3), we define (cid:101) φ ( y, t ) = { φ ( y, s ) } s ∈ [0 ,t ] ∈ (cid:103) Diff( S ) . This then has a well-defined rotation number, see Appendix A, which we denote by ρ ( γ ) = ρ ( (cid:101) φ ( y, t )) ∈ R . We now prove the following lower bound on the rotation number.
Lemma 5.12.
Let X be a symplectic polytope in R . Then there exists a constant C > , depending only on X , such that if ε > is small, then the following holds.Let γ : [ a, b ] → Y ε be a Reeb trajectory, and assume that if t ∈ ( a, b ) and E is a -face then γ ( t ) / ∈ E ε . Then ρ ( γ ) ≥ Cε − ( b − a ) . Proof.
Define a function r min ε : Y ε −→ R as follows. A point Y ε can by uniquely written as y + εv where y ∈ Y and v is aunit vector in N + y X . Then define r min ε ( y + εv ) = min θ ∈ R / π Z π ( (cid:104) v, y (cid:105) + ε ) ( S ( i v ) + S (cos( θ ) j v + sin( θ ) k v )) . (5.9)Here S : T Y ε → R is the single-argument version of the second fundamental form,which is well-defined, even though along the non-smooth strata of Y ε there is nocorresponding bilinear form.More explicitly, T y + εv Y ε , regarded as a subspace of R , does not depend on ε . Atangent vector V ∈ T y + εv Y ε can be uniquely decomposed as V = V T + V N (5.10)45here V T ∈ T y ∂X is tangent to a face F such that y ∈ F and v ∈ N + F X , and V N ∈ T v N + y X is perpendicular to v . We then have S ( V ) = ε − | V N | . (5.11)Lemma 4.6 and Proposition 4.7 carry over to the present situation to show that ρ ( γ ) ≥ (cid:90) ba r min ε ( γ ( s )) ds. (5.12)In (5.9), by compactness, there is a uniform upper bound on (cid:104) v, y (cid:105) for y ∈ ∂X and v ∈ N + y X a unit vector. Thus by (5.11) and (5.12), to complete the proof of thelemma, it is enough to show that there is a constant C > | ( i v ) N | + | (cos( θ ) j v + sin( θ ) k v ) N | ≥ C (5.13)whenever y ∈ ∂X , v ∈ N + y X is a unit vector, θ ∈ R / π Z , and y + εv is not in theclosure of E ε where E is a 3-face. To prove this, it is enough to show that for each k -face F with k <
3, there is a uniform positive lower bound on the left hand sideof (5.13) for all y ∈ F , all unit vectors v in N + F X that are not normal to a 3-faceadjacent to F , and all θ .If k = 2, then we have a positive lower bound on | ( i v ) N | by the discussion ofsmoothed 2-faces in § k = 1, denote the 1-face F by L . If v is on the boundary of N + L X , then wehave a positive lower bound on | ( i v ) N | as in the case k = 2 above. Suppose nowthat v is in the interior of N + L X . We have a positive lower bound on | ( i v ) N | when i v N is away from the Reeb cone of L . This is sufficient when L is a good 1-face.If L is a bad 1-face, then we have to consider the case where i v is on or near theReeb cone R + L X . If i v is in the Reeb cone, then all vectors in V ∈ T y + εv Y ε thatare not in the real span of the Reeb cone R + L X have V N (cid:54) = 0. Since the vectorscos( θ ) j v + sin( θ ) k v are all unit length and orthogonal to i v , we get a positive lowerbound on | (cos( θ ) j v + sin( θ ) k v ) N | for all θ when i v is on or near the Reeb cone.Suppose now that k = 0. If v is on the boundary of N + L X , then the desired lowerbound follows as in the cases k = 1 and k = 2 above. If v is in the interior of N + F X ,then we have | ( i v ) N | = 1.We now deduce a related rotation number bound. Let γ : [ a, b ] → Y ε be a Reebtrajectory. By Lemma 5.3, we can write γ ( t ) = y ( t ) + εv ( t )where y ( t ) ∈ ∂X and v ( t ) is a unit vector in N + y ( t ) X for each t .46 emma 5.13. Let X be a symplectic polytope in R . Then there exists a constant C > , depending only on X , such that if ε > is small and γ : [ a, b ] → Y ε is aReeb trajectory as above, then ρ ( γ ) ≥ C (cid:90) ba | v (cid:48) ( s ) | ds. Proof.
By Lemma 5.12, it is enough to show that there is a constant C such that | v (cid:48) ( s ) | ≤ Cε − . To prove this last statement, observe that by equation (5.1), in the notation (5.10)we have v (cid:48) ( s ) = 2 ε − (cid:104) v ( s ) , y ( s ) (cid:105) + ε ( i v ( s )) N . Thus | v (cid:48) ( s ) | ≤ ε − (cid:104) v ( s ) , y ( s ) (cid:105) + ε . If y ∈ ∂X and v ∈ N + y X is a unit vector, then (cid:104) v, y (cid:105) > X is convex and0 ∈ int( X ). By compactness, there is then a uniform lower bound on (cid:104) v, y (cid:105) for allsuch pairs ( y, v ). We now prove Theorems 1.10 and 1.11.
We first prove Theorem 1.10. In fact we will prove a slightly more precise statementin Lemma 6.1 below.Let X be a symplectic polytope in R and let γ = ( L , . . . , L k ) be a Type 1combinatorial Reeb orbit. This means that there are 3-faces E , . . . , E k and 2-faces F , . . . , F k such that F i is adjacent to E i − and E i , and L i is an oriented line segmentin E i from a point in F i to a point in F i +1 which is parallel to the Reeb vector fieldon E i . Here the subscripts i − i + 1 are understood to be mod k . Belowwe will regard γ as a piecewise smooth parametrized loop γ : R /T Z → X , where T = A comb ( γ ), which traverses the successive line segments L i as Reeb trajectories. Lemma 6.1.
Let X be a symplectic polytope in R , and let γ = ( L , . . . , L k ) be anondegenerate Type 1 combinatorial Reeb orbit. Then there exists δ > such thatfor all ε > sufficiently small: a) There is a unique Reeb orbit γ ε on the smoothed boundary Y ε such that | γ ε − γ | C < δ. (b) γ ε converges in C to γ as ε → .(c) γ ε does not intersect F ε where F is a -face or -face.(d) γ ε is linearizable, i.e. has a well-defined linearized return map.(e) A ( γ ε ) − A comb ( γ ) = O ( ε ) .(f ) γ ε is nondegenerate, ρ ( γ ε ) = ρ comb ( γ ) , and CZ( γ ε ) = CZ comb ( γ ) .Proof. Setup. For i = 1 , . . . , k , let p i denote the initial point of the segment L i .Using the notation E i , F i above, let D i denote the set of points y ∈ F i such thatReeb flow along E i starting at y reaches a point in F i +1 , which we denote by φ i ( y ).Thus we have a well-defined affine linear map φ i : D i −→ F i +1 . and by definition φ i ( p i ) = p i +1 . In particular, the composition φ k ◦ · · · ◦ φ : F −→ F is an affine linear map defined in a neighborhood of p sending p to itself. For V ∈ T F small, this composition sends p + V (cid:55)−→ p + AV, where A is a linear map T F → T F . Since the combinatorial Reeb orbit γ isassumed nondegenerate, the linear map A does not have 1 as an eigenvalue.By Lemma 5.7(a), the Reeb flow along the smoothed 2-face ( F i ) ε induces awell-defined map φ F i ,ε : U F i ,ε −→ F i (6.1)which is translation by a vector V F i ,ε . Proof of (a). If ε > p i is in the domain U F i ,ε for each i , and Reeb orbits on Y ε that are C close to γ correspond to fixed points of thecomposition φ F ,ε ◦ φ k ◦ · · · ◦ φ ◦ φ F ,ε ◦ φ : F −→ F . (6.2)It follows from the above that for V ∈ T F small, the composition (6.2) sends p + V (cid:55)−→ p + AV + W ε (6.3)48here W ε ∈ T F has length O ( ε ). Since the linear map A − p + V for some V ∈ T F . If ε is sufficiently small, this fixed point will also be in the domain of the composition(6.2), and thus will correspond to the desired Reeb orbit γ ε . Proof of (b).
This holds because for the above fixed point, V has length O ( ε ). Proof of (c).
The Reeb orbit γ ε does not intersect F ε where F is a 0-face or1-face, by the definition of the domain of the map (6.1). Proof of (d).
This follows from Lemma 5.10(c).
Proof of (e).
The symplectic action of the Reeb orbit γ ε is the sum of itsflow times over the smoothed 2-faces ( F i ) ε , plus the sum of its flow times overthe smoothed 3-faces ( E i ) ε . The former sum is O ( ε ) as explained in the proof ofLemma 5.7(b). The latter sum is (1 + O ( ε )) times the sum of the correspondingflow times over the 3-faces E i , and the latter differs from A comb ( γ ) by O ( ε ), becausethe fixed point of (6.3) has distance O ( ε ) from p . Proof of (f ).
Let T ε denote the period of γ ε , and let y ε be a point on the image of γ ε in E k . If F is a 2-face, let (cid:101) ψ F ∈ (cid:102) Sp(2) denote the lift of the transition matrix ψ F in Definition 2.20 with rotation number in the interval (0 , / (cid:101) φ ( y ε , T ε ) is given by (cid:101) φ ( y ε , T ε ) = (cid:101) ψ F k ◦ · · · ◦ (cid:101) ψ F . (6.4)Nondegeneracy of the combinatorial Reeb orbit γ means that the projection φ ( y ε , T ε ) = ψ F k ◦ · · · ◦ ψ F ∈ Sp(2)does not have 1 as an eigenvalue, so γ ε is nondegenerate. Moreover, it follows from(6.4) and the definition of combinatorial rotation number in Definition 2.23 that ρ comb ( γ ) = ρ ( γ ε ). This implies that CZ comb ( γ ) = CZ( γ ε ). Proof of Theorem 1.11.
We proceed in four steps.
Step 1.
We claim that for each i , the Reeb orbit γ i can be expressed as aconcatenation of a finite number, k i , of arcs such that:(a) Each endpoint of an arc maps to the boundary of E ε i where E is a 3-face.(b) For each arc, either:(i) There is a 3-face E such that the interior of the arc maps to E ε i , or(ii) No point in the interior of the arc maps to E ε i where E is a 3-face.49he above decomposition follows from parts (a) and (b)(i) of Lemma 5.9, be-cause the boundary of E ε i where E is a 3-face is contained in the singular set Σ.(Note that the decomposition into arcs in Lemma 5.9 is a subdivision of the abovedecomposition into arcs. Moreover, if k i >
1, then k i is even and the arcs alternatebetween types (i) and (ii).) Step 2.
We claim now that there is a constant
C >
0, not depending on i , suchthat if γ : [ a.b ] → Y ε i is an arc of type (ii) above, then if we write γ ( t ) = y ( t ) + ε i v ( t )for y ( t ) ∈ ∂X and v ( t ) ∈ N + y ( t ) X a unit vector, then we have (cid:90) ba | v (cid:48) ( s ) ds | ≥ C. (6.5)To see this, note that by (a) above, there are 3-faces E and E (cid:48) such that γ ( a ) ∈ E ε i and γ ( b ) ∈ E (cid:48) ε i . Then v ( a ) = ν E , where ν E denotes the outward unit normalvector to E , and likewise v ( b ) = ν E (cid:48) . If E (cid:54) = E (cid:48) , then the integral in (6.5) isbounded from below by the distance in S between ν E and ν E (cid:48) , and this distancehas a uniform positive lower bound because X has only finitely many 3-faces, eachwith distinct outward unit normal vectors.We now consider the case where E = E (cid:48) . The proof of Lemma 5.13 shows thatthere is a neighborhood U of ν E in S , and a constant C >
0, such that for anypoint y + ε i v ∈ Y ε i \ E ε i with v ∈ U , with respect to the decomposition (5.10), wehave | ( i v ) N | ≥ C . By shrinking the the neighborhood U , we can replace this lastinequalty with (cid:104) ( i v ) N , ν E (cid:105) >
0. Since v (cid:48) ( t ) is a positive multiple of ( i v ( t )) N , it followsthat the path [ a, b ] → S sending t (cid:55)→ v ( t ) must initially exit the neighborhood U before returning to ν E . So in this case, we can take the constant C in (6.5) to betwice the distance in S from ν E to ∂U . Step 3.
We now show that we can pass to a subsequence so that the sequenceof Reeb orbits γ i on Y ε i converges in C to a Type 1 or Type 2 combinatorial Reeborbit γ for X .By Lemma 5.12 and our hypothesis that ρ ( γ i ) < R , we must have k i > i is sufficiently large. Then, by Lemma 5.13 and Step 2, there is an i -independentupper bound on k i . We can then pass to a subsequence such that k i is equal to aneven constant k .By compactness, we can pass to a further subsequence such that the endpointsof the k arcs from Step 1 for γ i converge to k points in the 2-skeleton of X . ByLemma 5.6, the k/ X . On the other hand, by Lemma 5.12, for each arc of type (ii), the length of itsparametrizing interval converges to 0. A compactness argument also shows thatthere is an upper bound on the length of the Reeb vector field on Y ε i . It followsthat each arc of type (ii) is converging in C to a point. Then γ i converges in C
50o a Type 1 or Type 2 combinatorial Reeb orbit consisting of the line segments on3-faces given by the limits of the k/ Step 4.
To complete the proof, we now prove that the subsequence and limitingorbit constructed above satisfy all of the requirements (i)-(v) of the theorem.We have proved assertions (i) and (iii). Assertion (ii) follows from the proof ofLemma 6.1(e). Assertion (iv) follows from the proof of Lemma 6.1(d),(f). Assertion(v) follows from Lemma 5.13 and Step 2. (To get explicit constants C F , one onlyneeds to consider the case E (cid:54) = E (cid:48) in Step 2.) A Rotation numbers
Let (cid:102)
Sp(2) denote the universal cover of the group Sp(2) of 2 × S ) denote the group of orientation-preserving C diffeomorphisms of S = R / Z , and let (cid:103) Diff( S ) denote its universal cover. In this appendix, wereview two invariants of elements of (cid:102) Sp(2), and more generally (cid:103)
Diff( S ): the rota-tion number ρ and the “minimum rotation number” r . The former is a standardnotion in dynamics and is a key ingredient in Theorem 1.11; and we use the latterto bound the former. We also explain how to use rotation numbers to efficientlycompute certain products in (cid:102) Sp(2), which is needed for our algorithms.
A.1 Rotation numbers of circle diffeomorphisms
We can identify the universal cover (cid:103)
Diff( S ) with the group of C diffeomorphismsΦ : R → R which are Z -equivariant in the sense that Φ( t +1) = Φ( t )+1 for all t ∈ R .Such a diffeomorphism of R descends to an orientation-preserving diffeomorphismof S , and this defines the covering map (cid:103) Diff( S ) → Diff( S ). Definition A.1.
Given σ ∈ S , we define the rotation number with respect to σ , denoted by r σ : (cid:103) Diff( S ) −→ R , as follows. Let Φ be a Z -equivariant diffeomorphism of R as above. Let t ∈ R be alift of σ ∈ R / Z . We then define r σ (Φ) = Φ( t ) − t. (A.1) Definition A.2.
Given Φ ∈ (cid:103) Diff( S ), we define the rotation number ρ (Φ) = lim n →∞ r σ (Φ n ) n ∈ R (A.2) For the most part we could work more generally with orientation-preserving homeomorphisms. σ ∈ S . This limit does not depend on the choice of σ . Equivalently, ρ (Φ) = lim n →∞ Φ n ( t ) − tn (A.3)where t ∈ R .Note that we have the Z -equivariance property ρ (Φ + 1) = ρ (Φ) + 1 . (A.4)We can bound the rotation number as follows. Definition A.3.
We define the minimum rotation number r : (cid:103) Diff( S ) → R by r (Φ) = min σ ∈ S r σ (Φ) . (A.5)Alternatively, if Φ ∈ (cid:103) Diff( S ) is presented as a piecewise smooth path { φ t } t ∈ [0 , in Diff( S ) with φ = id S , then r (Φ) = min σ ∈ S (cid:90) dds φ s ( σ ) ds. In particular, it follows that r (Φ) ≥ (cid:90) min σ ∈ S (cid:18) dds φ s ( σ ) (cid:19) ds. (A.6)It follows from the definitions that ρ (Φ) ≥ r (Φ) . (A.7) A.2 A partial order
Definition A.4.
We define a partial order ≥ on (cid:103) Diff( S ) as follows:Φ ≥ Ψ if and only if r s (Φ) ≥ r s (Ψ) for all s ∈ S . (A.8)Equivalently, Φ( t ) ≥ Ψ( t ) for all t ∈ R . Lemma A.5.
The partial order ≥ on (cid:103) Diff ( S ) is left and right invariant. roof. Let Φ , Ψ , Θ ∈ (cid:103) Diff( S ), and suppose that Φ ≥ Ψ, i.e.Φ( t ) ≥ Ψ( t ) (A.9)for every t ∈ R . We need to show that ΦΘ ≥ ΨΘ and ΘΦ ≥ ΘΨ.Since Θ : R → R is an orientation preserving diffeomorphism, it preserves theorder on R , so it follows from (A.9) thatΘ(Φ( t )) ≥ Θ(Ψ( t ))for every t ∈ R , so ΘΦ ≥ ΘΨ.On the other hand, replacing t by Θ( t ) in the inequality (A.9), we deduce thatΦ(Θ( t )) ≥ Ψ(Θ( t ))for every t ∈ R , so ΦΘ ≥ ΨΘ.
Lemma A.6. If Φ , Ψ ∈ (cid:103) Diff ( S ) and Φ ≥ Ψ , then ρ (Φ) ≥ ρ (Ψ) .Proof. By (A.3), it is enough to show that given t ∈ R , we have Φ n ( t ) ≥ Ψ n ( t )for each positive integer n . This follows by induction on n , using the fact that Φpreserves the order on R . A.3 Rotation numbers of symplectic matrices
There is a natural homomorphism Sp(2) → Diff( S ), sending a symplectic linearmap A : R → R to its action on the set of positive rays (identified with R / Z by the map sending t ∈ R / Z to the ray through e πit ). This lifts to a canonicalhomomorphism (cid:102) Sp(2) → (cid:103) Diff( S ). Under this homomorphism, the invariants r s , r ,and ρ defined above pull back to functions (cid:102) Sp(2) → R , which by abuse of notationwe denote using the same symbols.We can describe the rotation number ρ : (cid:102) Sp(2) → R more explicitly in terms ofthe following classification of elements of the symplectic group Sp(2). Definition A.7.
Let A ∈ Sp(2). We say that A is • positive hyperbolic if Tr( A ) > negative hyperbolic if Tr( A ) < − • a positive shear if Tr( A ) = 2 and a negative shear if Tr( A ) = − • positive elliptic if − < Tr( A ) < v, Av ]) > v ∈ R \ { } . • negative elliptic if − < Tr( A ) < v, Av ]) < v ∈ R \ { } .53y the equivariance property (A.4), the rotation number ρ : (cid:102) Sp(2) → R descendsto a “mod Z rotation number” ¯ ρ : Sp(2) → R / Z . Lemma A.8.
The mod Z rotation number ¯ ρ : Sp (2) → R / Z can be computed asfollows: ¯ ρ ( A ) = if A is positive hyperbolic or a positive shear, if A is negative hyperbolic or a negative shear, θ if A is positive elliptic with eigenvalues e ± πiθ for θ ∈ (0 , ) , − θ if A is negative elliptic with eigenvalues e ± πiθ for θ ∈ (0 , ) . Proof.
In the first two cases, A has 1 or − s ∈ S which is fixed or sent to its antipode, and one can use this s in thedefinition (A.2).In the third case, A is conjugate to rotation by 2 πθ . One can then lift A toan element of (cid:102) Sp(2) whose image in (cid:103)
Diff( S ) is a Z -equivariant diffeomorphismΦ : R → R such that | Φ n ( t ) − t − nθ | < t ∈ R . It then follows from (A.3)that ρ (Φ) = θ . The last case is analogous. A.4 Computing products in (cid:102) Sp (2) Observe that (cid:102)
Sp(2) can be identified with the set of pairs (
A, r ), where A ∈ Sp(2)and r ∈ R is a lift of ρ ( A ) ∈ R / Z . The identification sends a lift (cid:101) A to the pair( A, ρ ( (cid:101) A )).For computational purposes, we can keep track of the lifts of A using less infor-mation, which is useful when for example we do not want to compute ρ ( A ) exactly.Namely, we can identify a lift (cid:101) A with a pair ( A, r ), where r is either an integer (when A has positive eigenvalues), an open interval ( n, n + 1 /
2) for some integer n (when A is positive elliptic), a half-integer (when A has negative eigenvalues), or an openinterval ( n − / , n ) (when A is negative elliptic).The following proposition allows us to compute products in the group (cid:102) Sp(2) interms of the above data, in the cases that we need (see Remark 2.24).
Proposition A.9.
Let (cid:101) A, (cid:101) B ∈ (cid:102) Sp (2) . Suppose that ρ ( (cid:101) A ) ∈ (0 , / . Then ρ ( (cid:101) B ) ≤ ρ ( (cid:101) A (cid:101) B ) ≤ ρ ( (cid:101) B ) + 12 . To apply this proposition, if for example (cid:101) B is described by the pair ( B, ( m, m +1 / (cid:101) A (cid:101) B is described by either ( AB, ( m, m + 1 / AB, m +1 / AB, ( m + 1 / , m + 1)). To decide which of these three possibilities holds,by Lemma A.8 it is enough to check whether AB is positive elliptic, has negativeeigenvalues, or is negative elliptic. 54 roof of Proposition A.9. Let Φ and Ψ denote the elements of (cid:103)
Diff( S ) determinedby (cid:101) A and (cid:101) B respectively. Let Θ : R → R denote translation by 1 /
2. By Lemma A.8, (cid:101) A projects to a positive elliptic element of Sp(2). It follows that with respect to thepartial order on (cid:103) Diff( S ), we have id R ≤ Φ ≤ Θ . By Lemma A.5, we can multiply on the right by Ψ to obtainΨ ≤ ΦΨ ≤ ΘΨ . Using Lemma A.6, we deduce that ρ (Ψ) ≤ ρ (ΦΨ) ≤ ρ (ΘΨ) . Since Ψ comes from a linear map, it commutes with Θ, so we have ρ (ΘΨ) = ρ (Ψ) + 12 . Combining the above two lines completes the proof.
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