3d Convex Contact Forms And The Ruelle Invariant
33D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT
J. CHAIDEZ AND O. EDTMAIRAbstract. Let ๐ ฤ (cid:82) be a convex domain with smooth boundary ๐ . We use a relation betweenthe extrinsic curvature of ๐ and the Ruelle invariant Ru p ๐ q of the natural Reeb ๏ฌow on ๐ to provethat there exist constants ๐ถ ฤ ๐ ฤ ๐ such that ๐ ฤ Ru p ๐ q vol p ๐ q ยจ sys p ๐ q ฤ ๐ถ Here sys p ๐ q is the systolic ratio of ๐ , i.e. the square of the minimal period of a closed Reeb orbit of ๐ divided by twice the volume of ๐ . We then construct dynamically convex contact forms on ๐ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomรฃo. Theseare the ๏ฌrst examples of dynamically convex contact 3-spheres that are not strictly contactomorphicto a convex boundary ๐ .
1. IntroductionA contact manifold p ๐, ๐ q is an odd dimensional manifold equipped with a hyperplane ๏ฌeld ๐ ฤ ๐๐ , called the contact structure, that is the kernel of a 1-form ๐ผ such thatker p ๐ ๐ผ q ฤ ๐๐ is rank 1 and ๐ผ | ker p ๐ ๐ผ q ฤ
0A 1-form satisfying this condition is called a contact form on p ๐, ๐ q . Every contact form comesequipped with a natural Reeb vector ๏ฌeld ๐ , de๏ฌned by ๐ผ p ๐ q โ ๐ ๐ ๐ ๐ผ โ Example 1.1.
We say that a domain ๐ ฤ (cid:82) ๐ with smooth boundary ๐ is star-shaped if0 P int p ๐ q and B ๐ is transverse to ๐ Let ๐ and ๐ denote the standard symplectic form and Liouville vector ๏ฌeld on (cid:82) ๐ . That is ๐ โ ๐ รฟ ๐ โ ๐๐ฅ ๐ ^ ๐๐ฆ ๐ ๐ โ รฟ ๐ ๐ฅ ๐ B ๐ฅ ๐ ` ๐ฆ ๐ B ๐ฆ ๐ โ ๐ B ๐ Then the restriction ๐ | ๐ of the Liouville 1-form ๐ โ ๐ ๐ ๐ is a contact form. Example 1.2.
The standard contact structure ๐ on ๐ ๐ ยด ฤ (cid:82) ๐ is given by ๐ โ ker p ๐ | ๐ ๐ ยด q .Every contact form on the standard contact sphere arises as the pullback of ๐ | ๐ via a di๏ฌeomor-phism to some star-shaped boundary ๐ . Moreover, every star-shaped boundary ๐ admits such amap from the sphere. Thus, from the perspective of contact geometry, the study of star-shapedboundaries is equivalent to the study of contact forms on the standard contact sphere. a r X i v : . [ m a t h . S G ] D ec J. CHAIDEZ AND O. EDTMAIR
Convexity.
In this paper, we are primarily interested in studying contact forms arising asboundaries of convex domains.
De๏ฌnition 1.3.
A contact form ๐ผ on ๐ ๐ ยด is convex if there is a convex star-shaped domain ๐ ฤ (cid:82) ๐ with boundary ๐ and a strict contactomorphism p ๐ , ๐ผ q ยป p ๐, ๐ | ๐ q .In contrast to the star-shaped case, not every contact form on ๐ ๐ ยด is convex, and the Reeb ๏ฌowsof convex contact forms possess many special dynamical properties, both proven and conjectural.In [20], Viterbo proposed a particularly remarkable systolic inequality for Reeb ๏ฌows on convexboundaries. To state it, let p ๐, ๐ผ q be a closed contact manifold with contact form of dimension2 ๐ ยด
1, and recall that the volume vol p ๐, ๐ผ q and systolic ratio sys p ๐, ๐ผ q are given by(1.1) vol p ๐, ๐ผ q โ ลผ ๐ ๐ผ ^ ๐ ๐ผ ๐ ยด and sys p ๐, ๐ผ q โ min t period ๐ of an orbit u ๐ p ๐ ยด q ! vol p ๐, ๐ผ q The weak Viterbo conjecture that originally appeared in [20] can be stated as follows.
Conjecture 1.4. [20] Let ๐ผ be a convex contact form on ๐ ๐ ยด . Then the systolic ratio is bounded by . sys p ๐ ๐ ยด , ๐ผ q ฤ Problem 1.5.
Give an intrinsic characterization of convexity that does not reference a map to (cid:82) ๐ .1.2. Dynamical Convexity.
In the seminal paper [11], Hofer-Wysocki-Zehnder provided a can-didate answer to Problem 1.5.
De๏ฌnition 1.6 (Def. 3.6, [11]) . A contact form ๐ผ on ๐ is dynamically convex if the Conley-Zehnderindex CZ p ๐พ q of any closed Reeb orbit ๐พ is greater than or equal to 3.The Conley-Zehnder index of a Reeb orbit plays the role of the Morse index in symplectic๏ฌeld theory and other types of Floer homology (see ยง2.2 for a review). Thus, on a naive level,dynamical convexity may be viewed as a type of โFloer-theoreticโ convexity. If ๐ is a convexdomain whose boundary ๐ has positive de๏ฌnite second fundamental form, then ๐ is dynamicallyconvex [11, Thm 3.7]. Note that this condition is open and generic among convex boundaries.In [11], Hofer-Wysocki-Zehnder proved that the Reeb ๏ฌow of a dynamically convex contactform always admits a surface of section. In the decades since, dynamical convexity has been usedas a key hypothesis in many signi๏ฌcant works on Reeb dynamics and other topics in contact andsymplectic geometry. See the papers of Hryniewicz [12], Zhou [22, 23], Abreu-Macarini [2, 3],Ginzburg-Gรผrel [7], Fraunfelder-Van Koert [6] and Hutchings-Nelson [14] for just a few examples.However, the following question has remained stubbornly open (c.f. [6, p. 5]). Question 1.7.
Is every dynamically convex contact form on ๐ also convex?The recent paper [1] of Abbondandolo-Bramham-Hryniewicz-Salomรฃo (ABHS) has suggestedthat the answer to Question 1.7 should be no. They construct dynamically convex contact formson ๐ with systolic ratio close to 2. There is substantial evidence for the weak Viterbo conjecture(cf. [5]), and so these contact forms are likely not convex. However, this was not proven in [1].Even more recently, Ginzburg-Macarini [8] addressed a version of Question 1.7 in higherdimensions that incorporates the assumption of symmetry under the antipod map ๐ ๐ ยด ร ๐ ๐ ยด . Their work did not address the general case of Question 1.7. D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 3
Main Result.
The main purpose of this paper is to resolve Question 1.7.
Theorem 1.8.
There exist dynamically convex contact forms ๐ผ on ๐ that are not convex. Theorem 1.8 is an immediate application of Proposition 1.9 and 1.12, which we will now describe.1.4.
Ruelle Bound.
For our ๏ฌrst result, recall that any closed contact 3-manifold p ๐, ๐ q withcontact form ๐ผ that satis๏ฌes ๐ p ๐ q โ ๐ป p ๐ ; (cid:90) q โ Ruelle invariant [18]Ru p ๐, ๐ผ q P (cid:82) Roughly speaking, the Ruelle invariant is the integral over ๐ of a time-averaged rotation numberthat measures the degree to which di๏ฌerent Reeb trajectories twist counter-clockwise aroundeach other (see ยง2.4 for a detailed review). Our result is stated most elegantly using the quantityru p ๐, ๐ผ q โ Ru p ๐, ๐ผ q vol p ๐, ๐ผ q This
Ruelle ratio is invariant under scaling of the contact form, unlike the Ruelle invariant itself.In recent work [13] motivated by embedded contact homology, Hutchings investigated theRuelle invariant of toric domains in (cid:67) . In that paper, the Ruelle invariant of the standardellipsoid ๐ธ โ ๐ธ p ๐, ๐ q ฤ (cid:67) with symplectic radii 0 ฤ ๐ ฤ ๐ (see ยง3.1) was computed as(1.2) Ru p ๐ธ q โ ๐ ` ๐ The systolic ratio and volume of ๐ธ are well-known to be ๐ { ๐ and ๐๐ { p ๐ธ q โ p sys p ๐ธ q ` q sys p ๐ธ q and thus 1 ฤ ru p ๐ธ q ยจ sys p ๐ธ q โ p ๐ ` ๐ q ๐ ฤ ๐ . Proposition 1.9 (Prop 3.1) . There are constants ๐ถ ฤ ๐ ฤ such that, for any convex contact form ๐ผ on ๐ , the following inequality holds. (1.3) ๐ ฤ ru p ๐ , ๐ผ q ยจ sys p ๐ , ๐ผ q ฤ ๐ถ Note that a result of Viterbo [20, Thm 5.1] states that there exists a constant ๐พ such thatsys p ๐ , ๐ผ q ฤ ๐พ for any convex contact form. Thus, Proposition 1.9 also implies that Corollary 1.10.
There is a constant ๐ ฤ such that, for any convex contact form ๐ผ on ๐ , we have (1.4) ๐ ฤ ru p ๐ , ๐ผ q We have included a helpful visualization of Proposition 1.9 in the sys ยด ru plane in Figure 1.Let us explain the idea of the proof of Proposition 1.9. First, as explained above, the resultholds for ellipsoids. By Johnโs ellipsoid theorem, we can always sandwich a convex domain ๐ between a standard ellipsoid and its scaling, after applying a linear symplectomorphism. ๐ธ p ๐, ๐ q ฤ ๐ ฤ ยจ ๐ธ p ๐, ๐ q Now note that the volume and minimum closed orbit length are monotonic under inclusion ofconvex domains. In particular, ๐ satis๏ฌes(1.5) ๐๐ ฤ vol p ๐ q ฤ ยจ ๐๐ ยด ยจ ๐๐ ฤ sys p ๐ q ฤ ยจ ๐๐ If the Ruelle invariant were also monotonic, then one could immediately acquire Proposition 1.9from (1.5) and (1.2). Unfortunately, this is not evidently the case.
J. CHAIDEZ AND O. EDTMAIR
Figure 1. A plot of the region of the sys ยด ru plane containing convex contactforms, depicted in light red. The blue arc is the region occupied by ellipsoids,and the green lines represent the sys โ โ ๐พ bound.The resolution of this issue comes from a beautiful formula (Proposition 3.10) relating thesecond fundmantal form and local rotation of the Reeb ๏ฌow on a contact hypersurface ๐ in (cid:82) . This is due originally to Ragazzo-Salomรฃo [17], albeit in di๏ฌerent language from this paper.Using this relation (ยง3.2), we derive estimates for the Ruelle invariant in terms of diameter, areaand total mean curvature. By standard convexity theory (i.e. the theory of mixed volumes),these quantities are monotonic under inclusion of convex domains. This allows us to comparethe Ruelle invariant of ๐ to that of its sandwiching ellipsoids, and thus prove the result. Remark 1.11 (Enhancing Prop 1.9) . In future work, we plan to investigate optimal constants ๐ and ๐ถ for Proposition 1.9, and to generalize the result to higher dimensions.1.5. A Counterexample.
In order to prove Theorem 1.8 using Proposition 1.9, we explicitly ๏ฌnda dynamically convex contact form that violates the estimate (1.4). This is the subject of oursecond new result.
Proposition 1.12 (Prop 4.1) . For every ๐ ฤ , there is a dynamically convex contact form ๐ผ on ๐ with vol ` ๐ , ๐ผ ห โ p ๐ , ๐ผ q ฤ ยด ๐ Ru p ๐ , ๐ผ q ฤ ๐ The construction of these examples follows the open book methods of Abbondandolo-Bramham-Hryniewicz-Salomรฃo in [1]. Namely, we develop a detailed correspondence between the prop-erties of a Hamiltonian disk map ๐ : (cid:68) ร (cid:68) and the properties of a contact form ๐ผ on ๐ constructed using ๐ via the open book construction (see Proposition 4.10). This includes a newformula relating the Ruelle invariant of ๐ in the sense of [18] and the Ruelle invariant of p ๐ , ๐ผ q .We then construct a Hamiltonian disk map ๐ with all of the appropriate properties to producea dynamically convex contact form on ๐ satisfying the conditions in Proposition 1.12. The map ๐ is acquired by composing two maps ๐ ๐ป and ๐ ๐บ . The map ๐ ๐ป is a counter-clockwise rotationby angle 2 ๐ p ` { ๐ q for large ๐ . The map ๐ ๐บ is compactly supported on a disjoint union ๐ ofdisks ๐ท , and rotates (most of) each disk ๐ท clockwise about its center by angle slightly less than4 ๐ . See Figure 2 for an illustration of this map.Applying Proposition 4.10, we can show that the volume and Ruelle invariant of p ๐ , ๐ผ q are(up to negligible error) proportional to the following quantities.vol ` ๐ , ๐ผ ห โ ๐ ยด รฟ ๐ท area p ๐ท q Ru p ๐ , ๐ผ q โ ๐ ยด รฟ ๐ท area p ๐ท q D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 5
Figure 2. The map ๐ โ ๐ ๐บ ห ๐ ๐ป for ๐ โ
4. Here ๐ ๐ป rotates (cid:68) counter-clockwiseby 45 degrees and ๐ ๐บ twists each disk ๐ท by roughly 720 degrees clockwise.By choosing ๐ to ๏ฌll most of (cid:68) and choosing all of the disks in ๐ to be very small, we can makethe Ruelle invariant very small relative to the volume. This process preserves the minimal actionof a closed orbit (up to a small error) and dynamical convexity, producing the desired example. Remark 1.13.
Our examples do not coincide with the ABHS examples in [1]. However, we believethat improvements of Proposition 1.12 may make our analysis applicable to those examples.
Outline.
This concludes the introduction ยง1 . The rest of the paper is organized as follows.In ยง2 , we cover basic preliminaries needed in later sections: the rotation number (ยง2.1), theConley-Zehnder index (ยง2.2), invariants of Reeb orbits (ยง2.3) the Ruelle invariant (ยง2.4).In ยง3 , we prove Proposition 1.9. We start by discussing the curvature-rotation formula andsome consequences (ยง3.2). We then derive a lower bound for a relevant curvature integral (ยง3.3).We conclude by proving the main bound (ยง3.4).In ยง4 , we prove Proposition 1.12. We ๏ฌrst discuss general preliminaries on Hamiltonian diskmaps (ยง4.1), open books (ยง4.2) and radial Hamiltonians (ยง4.3). We then construct a Hamiltonian๏ฌow on the disk (ยง4.4) before concluding with the main proof (ยง4.5). Acknowledgements.
We are deeply indepted to Alberto Abbondandolo, Umberto Hryniewiczand Michael Hutchings, who explained a number of the ideas and arguments in ยง2 to JC invarious discussions and private communications. JC was supported by the NSF GraduateResearch Fellowship under Grant No. 1752814.2. Rotation Numbers And Ruelle InvariantIn this section, we review some preliminaries on rotation numbers, Conley-Zehnder indicesand the Ruelle invariant, which we will need in later parts of the paper.2.1.
Rotation Number.
Consider the universal cover ฤ Sp p q of the symplectic group Sp p q . Wewill view a group element ฮฆ as a homotopy class of paths with ๏ฌxed endpoints ฮฆ : r , s ร Sp p q with ฮฆ p q โ IdRecall that a quasimorphism ๐ : ๐บ ร (cid:82) from a group ๐บ to the real line is a map such that thereexists a ๐ถ ฤ | ๐ p ๐ โ q ยด ๐ p ๐ q ยด ๐ p โ q| ฤ ๐ถ for all ๐, โ P ๐บ A quasimorphism is homogeneous if ๐ p ๐ ๐ q โ ๐ ยจ ๐ p ๐ q for any ๐ P ๐บ . Finally, two quasimorphisms ๐ and ๐ are called equivalent if the function | ๐ ยด ๐ | on ๐บ is bounded.The universal cover of the symplectic group possesses a canonical homogeneous quasimor-phism, due to the following result of Salamon-Simon [19]. J. CHAIDEZ AND O. EDTMAIR
Theorem 2.1 ( [19], Thm 1) . There exists a unique homogeneous quasimorphism ๐ : ฤ Sp p q ร (cid:82) that restricts to the standard homomorphism ๐ : r U p q ร (cid:82) on the universal cover of the unitary group (2.2) ๐ p ๐พ q โ ๐ฟ on the path ๐พ : r , s ร U p q with ๐พ p ๐ก q โ exp p ๐ ๐๐ฟ๐ก q De๏ฌnition 2.2.
The rotation number ๐ : ฤ Sp p q ร (cid:82) is the quasimorphism in Theorem 2.1.The rotation number is often characterized more explicitly in the literature as a lift of a mapto the circle. More precisely, it is characterized as the unique lift(2.3) r ๐ : ฤ Sp p q ร (cid:82) of ๐ : Sp p q ร (cid:82) { (cid:90) such that r ๐ p Id q โ ๐ is de๏ฌned as follows. Let ฮฆ P Sp p q have real eigenvalues ๐ , ๐ ยด and let ฮจ P Sp p q havecomplex (unit) eigenvalues exp pห ๐ ๐ ๐ q for ๐ P p , { q . Also ๏ฌx an arbitrary ๐ฃ P (cid:82) z
0. Then(2.4) ๐ p ฮฆ q โ ๐ ฤ { ๐ ฤ ๐ p ฮจ q โ ๐ if x ๐๐ฃ, ฮฆ ๐ฃ y ฤ ยด ๐ if x ๐๐ฃ, ฮฆ ๐ฃ y ฤ p q fall into one of the two categories above, and so ๐ is determinedeverywhere by (2.4). Lemma 2.3.
The rotation number ๐ : ฤ Sp p q ร (cid:82) is the unique lift of ๐ : Sp p q ร (cid:82) { (cid:90) with ๐ p Id q โ .Proof. We verify the properties in Theorem 2.1. The lift r ๐ is a quasimorphism by Lemmas 2.5and 2.6 below. It is homogeneous since ๐ p ฮฆ ๐ q โ ๐ ยจ ๐ p ฮฆ q mod 1, implying the same identity onthe lift. Finally, if ๐พ : r , s ร Sp p q is given by ๐พ p ๐ก q โ exp p ๐ ๐๐ฟ๐ก q then ๐ ห ๐พ : r , s ร (cid:82) { (cid:90) is given by ๐ ห ๐พ p ๐ก q โ ๐ฟ๐ก mod 1 P (cid:82) { (cid:90) This implies that the lift is ๐ก รร ๐ฟ๐ก , so that r ๐ p ๐พ q โ ๐ฟ , and we have proven the needed criteria. (cid:3) We will also need to utilize several inhomogeneous versions of the rotation number dependingon a choice of unit vector. These are de๏ฌned a follows.
De๏ฌnition 2.4.
The rotation number ๐ ๐ : ฤ Sp p q ร (cid:82) relative to ๐ P ๐ is the lift of the map ๐ ๐ : Sp p q ร ๐ ฮฆ รร | ฮฆ ๐ | ยด ยจ ฮฆ ๐ P ๐ ฤ (cid:82) via the covering map (cid:82) ร ๐ ฤ (cid:67) given by ๐ รร ๐ ๐ ๐ ๐ ยจ ๐ .The rotation numbers relative to ๐ P ๐ and the lift of ๐ all agree up to a constant factor. Lemma 2.5.
The maps ๐ ๐ : ฤ Sp p q ร (cid:82) and the lift r ๐ : ฤ Sp p q ร (cid:82) of ๐ have bounded di๏ฌerence. Moreprecisely, we have the following bounds. (2.5) | ๐ ๐ ยด r ๐ | ฤ and | ๐ ๐ ยด ๐ ๐ก | ฤ for any pair ๐ , ๐ก P ๐ Proof.
First, assume that ฮฆ : r , s ร Sp p q is a path such that ฮฆ p ๐ก q has no negative real eigenvaluesfor any ๐ก P r , s . Then r ๐ ห ฮฆ p ๐ก q โฐ { ๐ ๐ ห ฮฆ p ๐ก q โฐ ยด ๐ P ๐ for any ๐ P ๐ and ๐ก P r , s It follows that the relevant lifts of ๐ ห ฮฆ and ๐ ๐ ห ฮฆ to maps r , s ร (cid:82) remain in the interval pยด { , { q for all ๐ก . Thus r ๐ p ฮฆ q P pยด { , { q and ๐ ๐ p ฮฆ q P pยด { , { q This clearly implies (2.5). Since ๐ induces an isomorphism ๐ p Sp p qq ร ๐ p ๐ q , we know thatfor any pair ฮฆ , ฮฆ P ฤ Sp p q lifting the same element of Sp p q , we have r ๐ p ฮฆ q โ r ๐ p ฮฆ q implies ฮฆ โ ฮฆ D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 7
In particular, the above analysis extends to any ฮฆ with r ๐ p ฮฆ q P pยด { , { q . In the general case,note that the path ๐พ : r , s ร ๐ given by ๐พ p ๐ก q โ exp p ๐ ๐ ยจ ๐๐ก q for an integer ๐ P (cid:90) satis๏ฌes r ๐ p ๐พ q โ ๐ ๐ p ๐พ q โ ๐ { r ๐ p ฮฆ ๐พ q โ r ๐ p ฮฆ q ` r ๐ p ๐พ q ๐ ๐ p ฮฆ ๐พ q โ ๐ ๐ p ฮฆ q ` ๐ ๐ p ๐พ q Any path ฮจ can be decomposed (up to homotopy) as ฮฆ ๐พ where ๐พ is as above and ฮฆ : r , s ร Sp p q is a path with r ๐ p ฮฆ q P pยด { , { q . This reduces to the special case. (cid:3) This can be used to demonstrate that ๐ ๐ is a quasimorphism. As noted in the proof of Lemma2.3, this implies that r ๐ is a quasimorphism as well. Lemma 2.6.
The map ๐ ๐ : ฤ Sp p q ร (cid:82) is a quasimorphism for any ๐ P ๐ . In fact, we have (2.6) | ๐ ๐ p ฮจฮฆ q ยด ๐ ๐ p ฮจ q ยด ๐ ๐ p ฮฆ q| ฤ for any ๐ P ๐ Proof.
Let ฮฆ : r , s ร Sp p q and ฮจ : r , s ร Sp p q be two elements of ฤ Sp p q viewed as paths inSp p q . Consider the product ฮจฮฆ in the universal cover of Sp p q , represented by the path ฮฆ p ๐ก q for ๐ก P r , { s and ฮจ p ๐ก ยด q ฮฆ p q for ๐ก P r { , s By examining the path ๐ ๐ ห ฮจฮฆ : r , s ร ๐ and the lift to (cid:82) , we deduce the following property.(2.7) ๐ ๐ p ฮจฮฆ q โ ๐ ฮฆ p ๐ q p ฮจ q ` ๐ ๐ p ฮฆ q Here ฮฆ p ๐ q is shorthand for the unit vector ฮฆ p ๐ q{| ฮฆ p ๐ q| . Applying Lemma 2.5, we have | ๐ ๐ p ฮจฮฆ q ยด ๐ ๐ p ฮจ q ยด ๐ ๐ p ฮฆ q| ฤ | ๐ ฮฆ p ๐ q p ฮจ q ยด ๐ ๐ p ฮจ q| ฤ (cid:3) Conley-Zehnder Index.
Let Sp โน p q ฤ Sp p q denote the subset of ฮฆ P Sp p q such that ฮฆ ยด Idis invertible. The
Conley-Zehnder index is a continuous mapCZ : ฤ Sp โน p q ร (cid:90) Here ฤ Sp โน p q is the inverse image of Sp โน p q under ๐ : ฤ Sp p q ร Sp p q . The Conley-Zehnder indexcan be written using the rotation number as follows.(2.8) CZ p ฮฆ q โ t ๐ p ฮฆ q u ` r ๐ p ฮฆ q s There are several inequivalent ways to extend the Conley-Zehnder index to the entire symplecticgroup. We will follow [11, ยง3] and [1, ยง2.2], and use the following extension.
Convention 2.7.
In this paper, the
Conley-Zehnder index
CZ : ฤ Sp p q ร (cid:90) will be the maximallower semi-continuous extension of the ordinary Conley-Zehnder index.The extension in Convention 2.7 can be bounded below in terms of the rotation number. Lemma 2.8.
Let ฮฆ P ฤ Sp p q . Then (2.9) CZ p ฮฆ q ฤ ยจ r ๐ p ฮฆ q s ยด Proof.
For ฮฆ P ฤ Sp โน p q , (2.9) is an immediate consequence of (2.8). In the other case, note that themaximal lower semicontinuous extension is de๏ฌned by the property thatCZ p ฮฆ q โ inf lim ๐ ร8 CZ p ฮฆ ๐ q for any ฮฆ R ฤ Sp โน p q Here the in๏ฌmum is over all sequences ฮฆ ๐ P ฤ Sp โน p q with ฮฆ ๐ ร ฮฆ . Any ฮฆ R ฤ Sp โน p q has eigenvalue1, and so Lemma 2.3 implies that ๐ p ฮฆ q P (cid:90) . Since ๐ is continuous, we ๏ฌnd thatCZ p ฮฆ q โ inf lim ๐ ร8 t ๐ p ฮฆ ๐ q u ` r ๐ p ฮฆ ๐ q s ฤ t ๐ p ฮฆ q ยด { u ` r ๐ p ฮฆ q ยด { s โ ยจ r ๐ p ฮฆ q s ยด (cid:3) J. CHAIDEZ AND O. EDTMAIR
Invariants Of Reeb Orbits.
Let p ๐, ๐ q be a closed contact 3-manifold with ๐ p ๐ q โ ๐ผ be a contact 1-form on ๐ .Under this hypothesis on the Chern class, ๐ is isomorphic as a symplectic vector-bundle to thetrivial bundle (cid:82) . A trivialization ๐ of ๐ is a bundle isomorphism ๐ : ๐ ยป (cid:82) denoted by ๐ p ๐ฆ q : ๐ ๐ฆ ยป (cid:82) satisfying ๐ p ๐ฆ q ห ๐ โ ๐ ๐ผ | ๐ Two trivializations are homotopic if they are connected by a 1-parameter family of bundle isomor-phisms. Given a trivialization ๐ , we may associate a linearized Reeb ๏ฌow(2.10) ฮฆ ๐ : (cid:82) ห ๐ ร Sp p q given by ฮฆ ๐ p ๐ , ๐ฆ q โ ๐ p ๐ p ๐ , ๐ฆ qq ห ๐ ๐ p ๐ , ๐ฆ q ห ๐ ยด p ๐ฆ q Here ๐ : (cid:82) ห ๐ ร ๐ is the Reeb ๏ฌow, i.e. the ๏ฌow generated by the Reeb vector ๏ฌeld ๐ . Thelinearized ๏ฌow lifts uniquely to a map r ฮฆ ๐ : (cid:82) ห ๐ ร ฤ Sp p q with r ฮฆ ๐ | ห ๐ โ Id P ฤ Sp p q We will refer to r ฮฆ ๐ as the lifted linearized Reeb ๏ฌow. Explicitly, it maps p ๐ฆ, ๐ q to the homotopyclass of the path ฮฆ ๐ pยจ , ๐ฆ q| r ,๐ s . Note that this lift satis๏ฌes the cocyle property(2.11) r ฮฆ ๐ p ๐ ` ๐ , ๐ฆ q โ r ฮฆ ๐ p ๐ , ๐ ๐ p ๐ฆ qq ยจ r ฮฆ ๐ p ๐, ๐ฆ q De๏ฌnition 2.9.
Let ๐พ : (cid:82) { ๐ฟ (cid:90) ร ๐ be a closed Reeb orbit of ๐ . The action of ๐พ is given by(2.12) A p ๐พ q โ ลผ ๐พ ห ๐ผ โ ๐ฟ Likewise, the rotation number and
Conley-Zehnder index of ๐พ with respect to ๐ are given by(2.13) ๐ p ๐พ , ๐ q : โ ๐ ห r ฮฆ ๐ p ๐ฟ, ๐ฆ q CZ p ๐พ , ๐ q : โ CZ p r ฮฆ ๐ p ๐ฟ, ๐ฆ qq where ๐ฆ โ ๐พ p q These invariants depend only on the homotopy class of ๐ , and if ๐ป p ๐ ; (cid:90) q โ ๐ is the3-sphere) there is a unique trivialization up to homotopy. In this case, we let(2.14) ๐ p ๐พ q : โ ๐ p ๐พ , ๐ q and CZ p ๐พ q : โ CZ p ๐พ , ๐ q for any ๐ In ยง4, we will need the following easy observation, which follows immediately from Lemma2.8 and our way of de๏ฌning CZ (see Convention 2.7).
Lemma 2.10.
Let ๐ผ be a contact form on ๐ with ๐ p ๐พ q ฤ for every closed Reeb orbit. Then ๐ผ isdynamically convex. Ruelle Invariant.
Let p ๐, ๐ q be a closed contact 3-manifold with ๐ p ๐ q โ ๐ผ and a homotopy class of trivialization r ๐ s of ๐ . Here we discuss the Ruelle invariant Ru p ๐, ๐ผ , r ๐ sq P (cid:82) associated to the data of ๐, ๐ผ and r ๐ s . This invariant was originally introduced by Ruelle in [18]It will be helpful to describe a more general construction that subsumes that of the Ruelleinvariant. For this purpose, we also ๏ฌx a uniformly continuous quasimorphism ๐ : ฤ Sp p q ร (cid:82) Pick a representative trivialization ๐ of r ๐ s and let r ฮฆ ๐ : ๐ ห (cid:82) ร ฤ Sp p q be the lifted linearizedReeb ๏ฌow. We can associate a time-averaged version of ๐ over the space ๐ , as follows. Proposition 2.11.
The -parameter family of functions ๐ ๐ : ๐ ร (cid:82) given by the formula (2.15) ๐ ๐ p ๐ฆ q : โ ๐ ห r ฮฆ ๐ p ๐ , ๐ฆ q ๐ converges in ๐ฟ p ๐ ; (cid:82) q to a function ๐ p ๐ผ , ๐, ๐ q : ๐ ร (cid:82) with the following properties. D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 9 (a) (Quasimorphism) If ๐ and ๐ are equivalent quasimorphisms, i.e. | ๐ ยด ๐ | is bounded, then ๐ p ๐ผ , ๐, ๐ q โ ๐ p ๐ผ , ๐, ๐ q (b) (Trivialization) If ๐ and ๐ are homotopic trivializations of ๐ , then ๐ p ๐ผ , ๐, ๐ q โ ๐ p ๐ผ , ๐, ๐ q (c) (Contact Form) The integral ๐น p ๐ผ q of ๐ p ๐ผ , ๐, ๐ q over ๐ is continuous in the ๐ถ -topology on ฮฉ p ๐ q .Proof. We prove the existence of the limit and the properties (a)-(c) separately.
Limit Exists.
We apply Kingmanโs ergodic theorem [15]. Fix a constant ๐ถ ฤ ๐ satisfying (2.1). Let ๐ ๐ denote the function on ๐ given by ๐ ๐ : โ ๐ ๐ ๐ ` ๐ถ โ ๐ ห r ฮฆ ๐ pยด , ๐ q ` ๐ถ Note that ๐ ๐ de๏ฌnes a sub-additive process, as described in [15, ยง1.3]. First, due to the cocycleproperty (2.11) we have(2.16) ๐ ๐ ` ๐ โ ๐ ห r ฮฆ ๐ p ๐ ` ๐ , ยดq ` ๐ถ ฤ ๐ ห r ฮฆ ๐ p ๐, ยดq ` ๐ ห r ฮฆ ๐ p ๐ , ๐ ๐ pยดqq ` ๐ถ โ ๐ ๐ ` ๐ ห ๐ ๐ ๐ We can analogously show that ๐ ๐ ` ๐ ฤ ๐ ๐ ` ๐ ห ๐ ๐ ๐ ยด ๐ถ . In particular, if ๐ ฤ ๐ โ ๐ ` ๐ and ๐ P r , s , then(2.17) ลผ ๐ ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ ฤ ๐ ยด รฟ ๐ โ ลผ ๐ ๐ ห ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ ` ลผ ๐ ๐ ห ๐ ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ ยด ๐ถ๐ ฤ ยด ๐ด๐ Here ๐ด is any number larger than 2 ๐ถ and larger than the quantity ยด min "ลผ ๐ ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ : ๐ P r , s * Since ๐ ๐ satsi๏ฌes (2.16) and (2.17), we may apply Kingmanโs subadditive ergodic theorem [15,Thm 4] to conclude that there is a limiting function in ๐ฟ . ๐ ๐ ๐ ๐ฟ p ๐ ; (cid:82) q รรรรร ๐ p ๐ผ , ๐, ๐ q P ๐ฟ p ๐ ; (cid:82) q On the other hand, ๐ ๐ ๐ is Cauchy if and only if ๐ ๐ is Cauchy, and they have the same limit, since } ๐ ๐ ยด ๐ ๐ ๐ } ๐ฟ ฤ ๐ถ๐ ยจ vol p ๐, ๐ผ q This proves that ๐ ๐ converges in ๐ฟ p ๐ ; (cid:82) q to ๐ p ๐ผ , ๐, ๐ q . Quasimorphisms.
Let ๐ and ๐ be equivalent quasimorphisms, and pick ๐ถ ฤ | ๐ ยด ๐ | ฤ ๐ถ everywhere. Then } ๐ ห r ฮฆ ๐ ๐ ยด ๐ ห r ฮฆ ๐ ๐ } ๐ฟ ฤ ๐ถ ยจ vol p ๐, ๐ผ q ๐ Taking the limit as ๐ ร 8 shows that the limiting functions ๐ p ๐ผ , ๐, ๐ q and ๐ p ๐ผ , ๐, ๐ q are equal. Trivializations.
Let ๐ and ๐ be two trivializations of ๐ in the homotopy class r ๐ s . Then thereis a transition map ฮจ : ๐ ร Sp p q given by ฮจ p ๐ฆ q : (cid:82) ร (cid:82) with ฮจ p ๐ฆ q โ ๐ p ๐ฆ q ยจ ๐ p ๐ฆ q ยด The linearized ๏ฌows of ๐ and ๐ are related via this transition map, by the following formula. ฮฆ ๐ p ๐ , ๐ฆ q โ ฮจ p ๐ p ๐ , ๐ฆ qq ยจ ฮฆ ๐ p ๐ , ๐ฆ q ยจ ฮจ ยด p ๐ฆ q The homotopy equivalence of ๐ and ๐ is equivalent to the fact that ฮจ is null-homotopic, and inparticular lifts to the universal cover of Sp p q . Thus we may write r ฮฆ ๐ p ๐ , ๐ฆ q โ r ฮจ p ๐ p ๐ , ๐ฆ qq ยจ r ฮฆ ๐ p ๐ , ๐ฆ q ยจ r ฮจ ยด p ๐ฆ q Here r ฮจ : ๐ ร ฤ Sp p q is any lift of ฮจ . The quasimorphism property of ๐ now implies that } ๐ ห r ฮฆ ๐ p ๐ , ๐ฆ q ๐ ยด ๐ ห r ฮฆ ๐ p ๐ , ๐ฆ q ๐ } ๐ฟ ฤ ๐ถ ` sup | ๐ ห r ฮจ | ` sup | ๐ ห r ฮจ ยด | ๐ ยจ vol p ๐, ๐ผ q Taking the limit as ๐ ร 8 shows that ๐ p ๐ผ , ๐, ๐ q โ ๐ p ๐ผ , ๐, ๐ q . Contact Form.
Fix a contact form ๐ผ and an ๐ ฤ
0. Since ๐ is a quasimorphism, there exists a ๐ถ ฤ ๐ such that | ๐ ห r ฮฆ ๐ p ๐๐ , ๐ฆ q ยด ๐ ยด รฟ ๐ โ ๐ ห r ฮฆ ๐ p ๐ , ๐ ๐๐ p ๐ฆ qq| ฤ ๐ถ๐ for any ๐, ๐ ฤ ๐๐ and rewrite this estimate in terms of ๐ ๐ to see that | ๐ ๐๐ ยด ๐ ๐ ยด รฟ ๐ โ ๐ ๐ ห ๐ ๐๐ | ฤ ๐ถ๐ for any ๐, ๐ ฤ ๐ and take the limit as ๐ ร 8 to acquire(2.18) | ๐น p ๐ผ q ยด ลผ ๐ ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ | โ lim ๐ ร8 | ลผ ๐ p ๐ ๐๐ ยด ๐ ๐ q ยจ ๐ผ ^ ๐ ๐ผ |ฤ lim ๐ ร8 | ลผ ๐ p ๐ ๐๐ ยด ๐ ๐ ยด รฟ ๐ โ ๐ ๐ ห ๐ ๐๐ q ยจ ๐ผ ^ ๐ ๐ผ | ฤ ๐ถ ยจ vol p ๐, ๐ผ q ๐ Next, ๏ฌx a di๏ฌerent contact form ๐ฝ . Let r ฮจ ๐ be the lifted linearized ๏ฌow for ๐ฝ , and let ๐ ๐ : ๐ ร (cid:82) where ๐ ๐ p ๐ฆ q โ ๐ ห r ฮจ ๐ p ๐ , ยดq ๐ Due to (2.18), we can ๏ฌx a ๐ ฤ ๐ฝ su๏ฌciently ๐ถ -close to ๐ผ , we have(2.19) | ๐น p ๐ผ q ยด ลผ ๐ ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ | ฤ ๐ | ๐น p ๐ฝ q ยด ลผ ๐ ๐ ๐ ยจ ๐ฝ ^ ๐ ๐ฝ | ฤ ๐ถ vol p ๐, ๐ผ q ๐ ฤ ๐ ๐ฝ su๏ฌciently ๐ถ -close to ๐ผ so that r ฮฆ ๐ and r ฮจ ๐ are ๐ถ -close on ๐ ห r , ๐ s for any ๏ฌxed ๐ ฤ
0. Thus, for ๐ฝ su๏ฌciently close to ๐ผ in ๐ถ , we have(2.20) | ลผ ๐ ๐ ๐ ยจ ๐ผ ^ ๐ ๐ผ ยด ลผ ๐ ๐ ๐ ยจ ๐ฝ ^ ๐ ๐ฝ | ฤ } ๐ ๐ ยด ๐ ๐ } ๐ถ ยจ vol p ๐, ๐ผ q ` } ๐ ๐ } ๐ถ ยจ | vol p ๐, ๐ผ q ยด vol p ๐, ๐ฝ q|ฤ ๐ } r ฮฆ ๐ ยด r ฮจ ๐ } ๐ถ ๐ ยจ vol p ๐, ๐ผ q ` } ๐ ๐ } ๐ถ ยจ | vol p ๐, ๐ผ q ยด vol p ๐, ๐ฝ q| ฤ ๐ ๐ฝ su๏ฌciently ๐ถ -close to ๐ผ , we have | ๐น p ๐ผ q ยด ๐น p ๐ฝ q| ฤ ๐ ,which proves continuity.This concludes the proof of the existence and properties of ๐ p ๐ผ , ๐, ๐ q , and of Proposition 2.11. (cid:3) Proposition 2.11 allows us to introduce the Ruelle invariant as an integral quantity, as follows.
De๏ฌnition 2.12 (Ruelle Invariant) . The local rotation number rot ๐ of a closed contact manifold p ๐, ๐ผ q equipped with a (homotopy class of) trivialization ๐ is the following limit in ๐ฟ .(2.21) rot ๐ : ๐ ร (cid:82) given by rot ๐ : โ lim ๐ ร8 ๐ ห r ฮฆ ๐ p ๐ , ยดq ๐ Similarly, the
Ruelle invariant Ru p ๐, ๐ผ , ๐ q is the integral of the local rotation number over ๐ , i.e.(2.22) Ru p ๐, ๐ผ , ๐ q โ ลผ ๐ rot ๐ ยจ ๐ผ ^ ๐ ๐ผ โ lim ๐ ร8 ๐ ลผ ๐ ๐ ห r ฮฆ ๐ ยจ ๐ผ ^ ๐ ๐ผ D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 11
We will require an alternative expression for the Ruelle invariant in order to derive estimateslater in the paper.The Reeb ๏ฌow ๐ on ๐ preserves the contact structure, and so lifts to a ๏ฌow on the totalspace of the contact structure ๐ . Since this ๏ฌow is ๏ฌberwise linear, it descends to the (oriented)projectivization ๐ ๐ . A trivialization ๐ determines an identi๏ฌcation ๐ ๐ ยป ๐ ห (cid:82) { (cid:90) , and so a ๏ฌow(2.23) ยฏ ฮฆ : (cid:82) ห ๐ ห (cid:82) { (cid:90) ร ๐ ห (cid:82) { (cid:90) generated by a vector ๏ฌeld ยฏ ๐ on ๐ ห (cid:82) { (cid:90) Let ๐ : ๐ ห (cid:82) { (cid:90) ร (cid:82) { (cid:90) denote the tautological projection. De๏ฌnition 2.13.
The rotation density ๐ ๐ : ๐ ห (cid:82) { (cid:90) ร (cid:82) is the Lie derivative(2.24) ๐ ๐ : โ ยฏ ๐ p ๐ q Lemma 2.14.
The Ruelle invariant Ru p ๐, ๐ผ , ๐ q is written using the rotation density ๐ ๐ as Ru p ๐, ๐ผ , ๐ q โ lim ๐ ร8 ๐ ลผ ๐ ` ลผ ๐ ยฏ ฮฆ ห ๐ก ๐ ๐ pยด , ๐ q ยจ ๐ผ ^ ๐ ๐ผ ห ๐๐ก for any ๏ฌxed ๐ P (cid:82) { (cid:90) Proof.
By comparing De๏ฌnition 2.4 with the formula (2.23), one may verify that ๐ ๐ ห ฮฆ ๐ p ๐ , ๐ฆ q and ๐ ห ยฏ ฮฆ p ๐ , ๐ฆ, ๐ q ยด ๐ are equal in (cid:82) { (cid:90) Therefore, these formulas de๏ฌne a single map (cid:82) ห ๐ ห (cid:82) { (cid:90) ร (cid:82) { (cid:90) , admitting a unique lift to amap ๐น : (cid:82) ห ๐ ห (cid:82) { (cid:90) ร (cid:82) that vanishes on 0 ห ๐ ห (cid:82) { (cid:90) . The ๏ฌrst formula implies that(2.25) ๐น p ๐ , ๐ฆ, ๐ q โ ๐ ๐ ห r ฮฆ ๐ p ๐ , ๐ฆ q On the other hand, let ๐ก be the (cid:82) -variable of ๐น and ๐ ห ยฏ ฮฆ . Then the ๐ก -derivative of ๐น is ๐๐น๐๐ก | ๐ โ ๐๐๐ก p ๐ ห ยฏ ฮฆ q| ๐ โ ยฏ ฮฆ ห ๐ก p L ยฏ ๐ p ๐ qq| ๐ โ ยฏ ฮฆ ห ๐ก ๐ ๐ Integrating this identity and combining it with (2.25), we acquire the formula(2.26) ๐ ๐ ห r ฮฆ ๐ p ๐ , ๐ฆ q โ ๐น p ๐ , ๐ฆ, ๐ q โ ลผ ๐ r ยฏ ฮฆ ห ๐ก ๐ ๐ sp ๐ฆ, ๐ q ยจ ๐๐ก Now, since ๐ ๐ and ๐ are equivalent by Lemma 2.5, we can apply Proposition 2.11(a) to see that(2.27) Ru p ๐, ๐ผ , ๐ q โ lim ๐ ร8 ลผ ๐ ๐ ๐ ห r ฮฆ ๐ p ๐ , ยดq ๐ ยจ ๐ผ ^ ๐ ๐ผ We then apply (2.26) to see that the righthand side is given by(2.28) lim ๐ ร8 ๐ ลผ ๐ ลผ ๐ ยฏ ฮฆ ห ๐ ๐ pยด , ๐ q ยจ ๐ผ ^ ๐ ๐ผ โ lim ๐ ร8 ๐ ลผ ๐ ` ลผ ๐ ยฏ ฮฆ ห ๐ ๐ pยด , ๐ q ยจ ๐ผ ^ ๐ ๐ผ ห ๐๐ก Combining the formulas (2.4) and (2.28) ๏ฌnishes the proof. (cid:3)
3. Bounding The Ruelle InvariantLet ๐ ฤ (cid:82) be a convex domain containing 0 in its interior, and let p ๐, ๐ q be the contactboundary of ๐ . In this section, we derive the following estimate for the Ruelle ratio. Proposition 3.1.
There exist positive constants ๐ and ๐ถ independent of ๐ such that ๐ ฤ ru p ๐, ๐ q ยจ sys p ๐, ๐ q ฤ ๐ถ The proof follows the outline discussed in the introduction.We begin (ยง3.1) with a review of the geometry of standard ellipsoids ๐ธ p ๐, ๐ q in (cid:67) , includinga variant of Johnโs theorem (Corollary 3.6). We then present the key curvature-rotation formula(ยง3.2) and use it to bound the Ruelle invariant between two curvature integrals (Lemma 3.11).We then prove several bounds for one of these curvature integrals in terms of diameter, area andtotal mean curvature (ยง3.3). We collect this analysis together in the ๏ฌnal proof (ยง3.4). Notation 3.2.
We will require the following notation throughout this section.(a) ๐ is the standard metric on (cid:82) with connection โ , and dvol ๐ โ ๐ is the correspondingvolume form. We also use x ๐ข, ๐ฃ y to denote the inner product of two vectors ๐ข, ๐ฃ P (cid:82) .(b) ๐ is the outward normal vector ๏ฌeld to ๐ and ๐ ห is the dual 1-form with respect to ๐ .(c) ๐ is the restriction of ๐ to ๐ and dvol ๐ is the corresponding metric volume form. Thevolume form ๐ ^ ๐ ๐ and dvol ๐ are related (via the Liouville vector ๏ฌeld ๐ of (cid:82) ) by(3.1) ๐ ^ ๐ ๐ โ ๐ ๐ p ๐ | ๐ q โ ๐ ๐ p dvol ๐ | ๐ q โ ๐ ๐ p ๐ ห ^ dvol ๐ q โ x ๐, ๐ y dvol ๐ (d) ๐ is the second fundamental form of ๐ , i.e. the bilinear form given on any ๐ข, ๐ค P ๐๐ by ๐ p ๐ข, ๐ค q : โ x โ ๐ข ๐ , ๐ค y (e) ๐ป is the mean curvature of ๐ . It is given by ๐ป : โ
13 trace ๐ Standard Ellipsoids.
Recall that a standard ellipsoid ๐ธ p ๐ , . . . , ๐ ๐ q ฤ (cid:67) ๐ with parameters ๐ ๐ ฤ ๐ โ , . . . , ๐ is de๏ฌned as follows.(3.2) ๐ธ p ๐ , . . . , ๐ ๐ q : โ ! ๐ง โ p ๐ง ๐ q P (cid:67) ๐ : รฟ ๐ ๐ | ๐ง ๐ | ๐ ๐ ฤ ) For example, ๐ธ p ๐ q ฤ (cid:67) is the disk of area ๐ , and ๐ธ p ๐, . . . , ๐ q ฤ (cid:67) ๐ is the ball of radius p ๐ { ๐ q { .We beginn this section with a discussion of the Riemannian and symplectic geometry ofstandard ellipsoids in (cid:67) . All of the relevant geometric quantities for this section can be computedexplicitly in this setting. Let us record the outcome of these calculations. Lemma 3.3 (Ellipsoid Quantities) . Let ๐ธ โ ๐ธ p ๐, ๐ q be a standard ellipsoid with ฤ ๐ ฤ ๐ . Then(a) The diameter, surface area and volume of ๐ธ are given by diam p ๐ธ q โ ๐ { ยจ ๐ { area pB ๐ธ q โ ๐ { ยจ ๐ ๐ { ยด ๐ { ๐ ๐ ยด ๐ vol p ๐ธ q โ ๐๐ (b) The total mean curvature of B ๐ธ (i.e. the integral of the mean curvature over B ๐ธ ) is given by ลผ B ๐ธ ๐ป ยจ dvol ๐ โ ๐ ยจ p ๐ ` ๐ ` ๐๐๐ ยด ๐ ยจ log p ๐ { ๐ qq (c) The minimum action of a closed orbit on B ๐ธ and the systolic ratio of B ๐ธ are given by ๐ pB ๐ธ q โ ๐ sys pB ๐ธ q โ ๐๐ (d) The Ruelle invariant of B ๐ธ is given by Ru pB ๐ธ q โ ๐ ` ๐ The area, total mean curvature and volume are straightforward but tedious calculus compu-tations, which we omit. The Ruelle invariant is computed in [13, Lem 2.1 and 2.2], while theminimum period of a closed orbit is computed in [9, ยง2.1].Any convex boundary in (cid:82) ๐ can be sandwiched between a standard ellipsoid and a scalingof that ellipsoid by a factor of 2 ๐ , after the application of an a๏ฌne symplectomorphism. To seethis, ๏ฌrst recall the following well-known result of John. Theorem 3.4 (John Ellipsoid) . Let ๐ ฤ (cid:82) ๐ be a convex domain. Then there exists an ellipsoid ๐ธ centeredat some ๐ P ๐ such that ๐ธ ฤ ๐ ฤ ๐ ` ๐ p ๐ธ ยด ๐ q D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 13
Any ellipsoid ๐ธ is carried to a standard ellipsoid ๐ธ p ๐, ๐ q by some a๏ฌne symplectomorphism ๐ .Furthermore, note that we have the following elementary result, which can be demonstratedusing a Moser argument. Lemma 3.5.
Let ๐ : p ๐, ๐ q ร p ๐ , ๐ q be a di๏ฌeomorphism such that ๐ ห ๐ โ ๐ ` ๐ ๐ . Then ๐ is isotopicto a strict contactomorphism. Since (cid:82) ๐ is contractible, ๐ ห ๐ โ ๐ ` ๐ ๐ automatically on (cid:82) ๐ . Thus, ๐ carries any star-shapedhypersurface ๐ โ B ๐ to a strictly contactomorphic ๐ p ๐ q by Lemma 3.5, and we conclude thefollowing result. Corollary 3.6.
Let ๐ ฤ (cid:82) ๐ be a convex domain with boundary ๐ . Then ๐ is strictly contactomorphicto the boundary B ๐พ of a convex domain ๐พ with ๐ธ p ๐ , . . . , ๐ ๐ q ฤ ๐พ ฤ ยจ ๐ธ p ๐ , . . . , ๐ ๐ q . When a convex domain in (cid:82) is squeezed between an ellipsoid and its scaling, we can estimatemany important geometric quantities of ๐ in terms of the ellipsoid itself. Lemma 3.7.
Let ๐ ฤ (cid:82) be a convex domain with smooth boundary ๐ such that (3.3) ๐ธ p ๐, ๐ q ฤ ๐ ฤ ๐ ยจ ๐ธ p ๐, ๐ q for some ๐ ฤ ๐ ฤ and ๐ ฤ Then there is a constant ๐ถ ฤ dependent only on ๐ such that (3.4) ๐ { ฤ diam p ๐ q ฤ ๐ถ ยจ ๐ { ๐๐ { ฤ area p ๐ q ฤ ๐ถ ยจ ๐๐ { (3.5) ๐ ฤ ลผ ๐ ๐ป ยจ dvol ๐ ฤ ๐ถ ยจ ๐ ๐๐ ฤ vol p ๐ q ฤ ๐ถ ยจ ๐๐ (3.6) ๐ ฤ ๐ p ๐ q ฤ ๐ถ ยจ ๐ ๐ถ ยด ยจ ๐๐ ฤ sys p ๐ q ฤ ๐ถ ยจ ๐๐ Remark 3.8.
The optimal constants in the estimates (3.4)-(3.6) are not important to the argumentsbelow. They could be explicitly computed in the following proof.
Proof.
First, note that ๐ ยจ ๐ธ p ๐, ๐ q is also a standard ellipsoid. More precisely, we know that ๐ ยจ ๐ธ p ๐, ๐ q โ ๐ธ p ๐ ยจ ๐, ๐ ยจ ๐ q We now derive the desired estimates from Lemma 3.3 and the monotonicity of the relevantquantities under inclusion of convex domains.The diameter diam p ๐ q and volume vol p ๐ q are monotonic with respect to inclusion of arbitraryopen subsets, and so from Lemma 3.3(a) we acquire ๐ { ฤ diam p ๐ q ฤ ๐ ๐ { ยจ ๐ { and ๐๐ ฤ vol p ๐ q ฤ ๐ ยจ ๐๐ The surface area and total mean curvature are monotonic with respect to inclusion of convexdomains, since ลผ ๐ ๐ป dvol ๐ โ ยจ ๐ p ๐ q and area p ๐ q โ ยจ ๐ p ๐ q Here ๐ ๐ p ๐ q is the ๐ th cross-sectional measure [4, ยง19.3], which is monotonic with respect to inclusionsof convex domains by [4, p.138, Equation 13]. Furthermore, when 0 ฤ ๐ ฤ ๐ (and in the limit as ๐ ร ๐ ), one may verify that(3.7) ๐๐ { ฤ ๐ ๐ { ยด ๐ { ๐ ๐ ยด ๐ ฤ ๐๐ { and ๐ ฤ ๐ ` ๐ ` ๐๐๐ ยด ๐ ยจ log p ๐ { ๐ q ฤ ๐ Thus, by applying the monotonicity property, (3.7) and Lemma 3.3(a)-(b), we have ๐๐ { ฤ area p ๐ q ฤ ๐ ยจ ๐๐ { and ๐ ฤ ลผ ๐ ๐ป ยจ dvol ๐ ฤ ๐ ยจ ๐ Finally, the minimum orbit length ๐ p ๐ q coincides with the 1st Hofer-Zehnder capacity ๐ ๐ป๐ p ๐ q on convex domains, and is thus monotonic with respect to symplectic embeddings. Thus byLemma 3.3(a) and (c), we have ๐ ฤ ๐ p ๐ q ฤ ๐ ยจ ๐ and ๐ ยด ยจ ๐๐ ฤ ๐ p ๐ q p ๐ q โ sys p ๐ q ฤ ๐ ยจ ๐๐ This concludes the proof, after choosing ๐ถ larger than the constants appearing above. (cid:3) Curvature-Rotation Formula.
Identify (cid:82) with the quaternions (cid:72) via (cid:82) Q p ๐ฅ , ๐ฆ , ๐ฅ , ๐ฆ q รร ๐ฅ ` ๐ฆ ๐ผ ` ๐ฅ ๐ฝ ` ๐ฆ ๐พ P (cid:72) This equips (cid:82) with a triple of complex structures. ๐ผ : ๐ (cid:82) ร ๐ (cid:82) ๐ฝ : ๐ (cid:82) ร ๐ (cid:82) ๐พ : ๐ (cid:82) ร ๐ (cid:82) We can utilize these structures to formulate an explicit representative of the standard homotopyclass of trivialization ๐ : ๐ ยป (cid:82) . De๏ฌnition 3.9.
The quaternion trivialization ๐ : ๐ ยป ๐ ห (cid:67) is the symplectic trivialization given by ๐ : ๐ ๐ รร ๐ ๐ ยด รรร ๐ ห (cid:67) Here ๐ ฤ ๐๐ is the symplectic sub-bundle span p ๐ฝ ๐ , ๐พ ๐ q , ๐ : ๐ ร ๐ is the projection map from ๐ to ๐ along the Reeb direction, and ๐ : ๐ ห (cid:67) ร ๐ is the bundle map given on ๐ง โ ๐ ` ๐๐ by(3.8) ๐ ๐ p ๐ง q : โ ๐ง ยจ ๐ฝ ๐ ๐ โ p ๐ ` ๐ผ๐ q ยจ ๐ฝ ๐ ๐ The key property of the quaternion trivialization is the following relation of the rotationdensity (see De๏ฌnition 2.13) to extrinsic curvature, originally due to Ragazzo-Salomรฃo (c.f. [17]).
Proposition 3.10 (Curvature-Rotation) . [5, Prop 4.7] Let ๐ ฤ (cid:82) be a star-shaped domain withboundary ๐ transverse to the Liouville vector ๏ฌeld ๐ of (cid:82) and let ๐ be the quaternion trivialization. Then (3.9) ๐ ๐ p ๐ฆ, ๐ q โ ๐ ยจ x ๐ ๐ฆ , ๐ ๐ฆ y p ๐ p ๐ผ ๐ ๐ฆ , ๐ผ ๐ ๐ฆ q ` ๐ p ๐ ยจ ๐ฝ ๐ ๐ฆ , ๐ ยจ ๐ฝ ๐ ๐ฆ qq As an easy consequence of (3.9), we have the following bound on the Ruelle invariant of ๐ . Lemma 3.11.
The Ruelle invariant Ru p ๐ q is bounded by the following curvature integrals. (3.10) 12 ๐ ยจ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q dvol ๐ ฤ Ru p ๐ q ฤ ๐ ยจ ลผ ๐ ๐ป dvol ๐ Proof.
By Lemma 2.14, we have the following integral formula for the Ruelle invariant.(3.11) Ru p ๐ q โ lim ๐ ร8 ๐ ลผ ๐ หลผ ๐ r ยฏ ฮฆ ห ๐ก ๐ ๐ spยด , ๐ q ยจ ๐ ^ ๐ ๐ ห ๐๐ก By the curvature-rotation formula in Proposition 3.10, we can write the integrand as(3.12) r ยฏ ฮฆ ห ๐ก ๐ ๐ spยด , ๐ q โ ยฏ ฮฆ ห ๐ก ยด x ๐, ๐ y p ๐ p ๐ผ ๐ , ๐ผ ๐ q ` ๐ p ๐ ยจ ๐ฝ ๐ , ๐ ยจ ๐ฝ ๐ qq ยฏ To bound the righthand side of (3.12), note that ๐ผ ๐ , ๐ ยจ ๐ฝ ๐ and ๐ ยจ ๐พ ๐ form an orthonormal basisof ๐๐ with respect to the restricted metric ๐ | ๐ , so that ๐ p ๐ผ ๐ , ๐ผ ๐ q ` ๐ p ๐ ยจ ๐ฝ ๐ , ๐ ยจ ๐ฝ ๐ q ` ๐ p ๐ ยจ ๐พ ๐ , ๐ ยจ ๐พ ๐ q โ trace p ๐ q โ ๐ป Furthermore, since ๐ is convex, the second fundamental form ๐ is positive de๏ฌnite. Thereforeby (3.12), we have the following lower and upper bound.(3.13) ยฏ ฮฆ ห ๐ก ยด ๐ p ๐ผ ๐ , ๐ผ ๐ qx ๐, ๐ y ยฏ ฤ r ยฏ ฮฆ ห ๐ก ๐ ๐ spยด , ๐ q ฤ ยจ ยฏ ฮฆ ห ๐ก ยด ๐ป x ๐, ๐ y ยฏ D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 15
To simplify the two sides of (3.13), let ๐น : ๐ ห ๐ ร (cid:82) be any map pulled back from a map ๐น : ๐ ร (cid:82) . Since the ๏ฌow ยฏ ฮฆ ๐ก on ๐ ห ๐ lifts the Reeb ๏ฌow ๐ ๐ก on ๐ , and ๐ ๐ก preserves ๐ , we have ยฏ ฮฆ ห ๐ก ยด ๐น x ๐, ๐ y ยฏ ยจ ๐ ^ ๐ ๐ โ ๐ ห ๐ก ยด ๐น x ๐, ๐ y ยฏ ยจ ๐ ^ ๐ ๐ โ ๐ ห ๐ก ยด ๐น ยจ ๐ ^ ๐ ๐ x ๐, ๐ y ยฏ โ ๐ ห ๐ก ยด ๐น ยจ dvol ๐ ยฏ Since the integral of ๐ ห ๐ก p ๐น ยจ dvol ๐ q over ๐ is independent of ๐ก , we have(3.14) 1 ๐ ลผ ๐ หลผ ๐ ยฏ ฮฆ ห ๐ก ยด ๐น x ๐, ๐ y ยฏ ยจ ๐ ^ ๐ ๐ ห ๐๐ก โ ๐ ลผ ๐ หลผ ๐ ๐น ยจ dvol ๐ ห ๐๐ก โ ลผ ๐ ๐น ยจ dvol ๐ By plugging in the estimate (3.13) to the integral formula (3.11) and applying (3.14) to thefunctions ๐ p ๐ผ ๐ , ๐ผ ๐ q and ๐ป on ๐ , we acquire the desired bound (3.10). (cid:3) Bounding Curvature Integrals.
We now further simplify the lower bound of the Ruelleinvariant in Lemma 3.11 by estimating (from below) the integral ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ยจ dvol ๐ using the geometric quantities (e.g. area and diameter) appearing in ยง3.1. This will help us toleverage the sandwich estimates in Lemma 3.7 in the proof of the Ruelle invariant bound in ยง3.4.Recall that ๐ ฤ (cid:82) denotes a convex domain with smooth boundary ๐ . Let ๐ : (cid:82) ห ๐ ร ๐ bethe ๏ฌow by ๐ผ ๐ . Let ๐ ๐ and ๐ป ๐ denote the time-averaged versions of ๐ p ๐ผ ๐ , ๐ผ ๐ q and ๐ป , respectively.(3.15) ๐ ๐ : โ ๐ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ห ๐ ๐ก ๐๐ก ๐ป ๐ : โ ๐ ลผ ๐ ๐ป ห ๐ ๐ก ๐๐ก We will also need to consider a time-averaged acceleration function ๐ด ๐ on ๐ . Namely, let ๐พ : (cid:82) ร ๐ be a trajectory of ๐ผ ๐ with ๐พ p q โ ๐ฅ . Then we de๏ฌne(3.16) ๐ด ๐ : โ ๐ ลผ ๐ | โ ๐ผ ๐ ๐ผ ๐ | ห ๐ ๐ก ๐๐ก or equivalently ๐ด ๐ p ๐ฅ q โ ๐ ลผ ๐ | : ๐พ | ๐๐ก The ๏ฌrst ingredient to the bounds in this section is the following estimate relating these threetime-averaged functions.
Lemma 3.12.
For any ๐ ฤ , the functions ๐ด ๐ , ๐ป ๐ and ๐ ๐ satisfy ๐ด ๐ ฤ ยจ ๐ป ๐ ยจ ๐ ๐ pointwise.Proof. In fact, the non-time-averaged version of this estimate holds. We will now show that(3.17) | โ ๐ผ ๐ ๐ผ ๐ | ฤ ๐ป ยจ ๐ p ๐ผ ๐ , ๐ผ ๐ q To start, we need a formula for โ ๐ผ ๐ ๐ผ ๐ in terms of the second fundamental form, as follows. โ ๐ผ ๐ ๐ผ ๐ โ x ๐ , โ ๐ผ ๐ ๐ผ ๐ y ๐ ` x ๐ผ ๐ , โ ๐ผ ๐ ๐ผ ๐ y ๐ผ ๐ ` x ๐ฝ ๐ , โ ๐ผ ๐ ๐ผ ๐ y ๐ฝ ๐ ` x ๐พ ๐ , โ ๐ผ ๐ ๐ผ ๐ y ๐พ ๐ โ ยดx ๐ผ ๐ , โ ๐ผ ๐ ๐ y ๐ ยด x ๐ผ ๐ , โ ๐ผ ๐ ๐ y ๐ผ ๐ ยด x ๐ผ๐ฝ ๐ , โ ๐ผ ๐ ๐ y ๐ฝ ๐ ยด x ๐ผ๐พ ๐ , โ ๐ผ ๐ ๐ y ๐พ ๐ Applying the quaternionic relations ๐ผ โ ยด ๐ผ๐ฝ โ ๐พ and ๐ผ๐พ โ ยด ๐ฝ , we can rewrite this as ยดx ๐ผ ๐ , โ ๐ผ ๐ ๐ y ๐ ` x ๐ , โ ๐ผ ๐ ๐ y ๐ผ ๐ ยด x ๐พ ๐ , โ ๐ผ ๐ ๐ y ๐ฝ ๐ ` x ๐ฝ ๐ , โ ๐ผ ๐ ๐ y ๐พ ๐ Finally, applying the de๏ฌnition of the second fundamental form we ๏ฌnd that โ ๐ผ ๐ ๐ผ ๐ โ ยด ๐ p ๐ผ ๐ , ๐ผ ๐ q ๐ ยด ๐ p ๐ผ ๐ , ๐พ ๐ q ๐ฝ ๐ ` ๐ p ๐ผ ๐ , ๐ฝ ๐ q ๐พ ๐ To estimate the righthand side, we note that ๐ p ๐ข, ๐ฃ q ฤ ๐ p ๐ข, ๐ข q ๐ p ๐ฃ, ๐ฃ q for any vector๏ฌelds ๐ข and ๐ฃ by Cauchy-Schwarz, since ๐ is positive semi-de๏ฌnite. Thus we have | โ ๐ผ ๐ ๐ผ ๐ | ฤ ๐ p ๐ผ ๐ , ๐ผ ๐ q ` ๐ p ๐ผ ๐ , ๐ผ ๐ q ๐ p ๐ฝ ๐ , ๐ฝ ๐ q ` ๐ p ๐ผ ๐ , ๐ผ ๐ q ๐ p ๐พ ๐ , ๐พ ๐ q โ ๐ป ยจ ๐ p ๐ผ ๐ , ๐ผ ๐ q This proves (3.17) and the desired estimate follows immediately by Cauchy-Schwarz.(3.18) ๐ด ๐ โ ` ๐ ลผ ๐ | โ ๐ผ ๐ ๐ผ ๐ | ห ๐ ๐ก ๐๐ก ห ฤ ยจ ๐ ลผ ๐ ๐ป ห ๐ ๐ก ๐๐ก ยจ ๐ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ห ๐ ๐ก ๐๐ก โ ๐ป ๐ ยจ ๐ ๐ This concludes the proof of the lemma. (cid:3)
As a consequence, we get the following estimate for the curvature integral of interest in termsof area, total mean curvature and the time-averaged acceleration ๐ด ๐ . Lemma 3.13.
Let ฮฃ ฤ ๐ be an open subset of ๐ and let ๐ ฤ . Then (3.19) ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ยจ dvol ๐ ฤ area p ฮฃ q ยจ ล ๐ ๐ป dvol ๐ ยจ min ฮฃ p ๐ด ๐ q Proof.
We ๏ฌrst note that ๐ผ ๐ preserves the volume form dvol ๐ , since L ๐ผ ๐ p dvol ๐ q โ ๐ ๐ ๐ผ ๐ dvol ๐ โ ๐ ๐ ๐ p ๐ ^ ๐ ๐ q โ ๐ ๐ โ ๐ is the Reeb vector-๏ฌeld on ๐ . Thus, time-averaging leaves the integral over ๐ unchanged. ลผ ๐ ๐ป ๐ dvol ๐ โ ลผ ๐ ๐ป dvol ๐ and ลผ ๐ ๐ ๐ dvol ๐ โ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q dvol ๐ We can thus integrate the estimate ๐ด ๐ ฤ ๐ป ๐ ยจ ๐ ๐ to see thatmin p ๐ด ๐ q ยจ area p ฮฃ q ฤ ยด ลผ ฮฃ ๐ด ๐ ยจ dvol ๐ ยฏ ฤ ยด ? ยจ ลผ ฮฃ ๐ป { ๐ ยจ ๐ { ๐ ยจ dvol ๐ ยฏ ฤ ยจ ลผ ฮฃ ๐ป ๐ ยจ dvol ๐ ยจ ลผ ฮฃ ๐ ๐ ยจ dvol ๐ ฤ ยจ ลผ ๐ ๐ป ยจ dvol ๐ ยจ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ยจ dvol ๐ After some rearrangement, this is the desired estimate. (cid:3)
Every quantity on the righthand side of (3.19) can be controlled using the estimates in Lemma3.7, with the exception of the term involving the time-averaged acceleration ๐ด ๐ . However, we canbound ๐ด ๐ in terms of diam p ๐ q ยด , using the following general fact about curves of unit speed. Lemma 3.14.
Let ๐พ : r ,
8q ร ๐ be a curve with | ๐พ | โ and let ๐ถ satisfy ฤ ๐ถ ฤ . Then ๐ ลผ ๐ | : ๐พ | ๐๐ก ฤ ๐ถ diam p ๐ q for all ๐ " Proof.
Let ๐ satisfy ๐ ฤ ๐ถ๐ ` ยจ diam p ๐ q . Then by Cauchy-Schwarz, we have(3.20) diam p ๐ q ลผ ๐ | : ๐พ | ๐๐ก ฤ ลผ ๐ | ๐พ | ยจ | : ๐พ | ๐๐ก ฤ ลผ ๐ |x : ๐พ , ๐พ y| ๐๐ก ฤ หหห ลผ ๐ x : ๐พ , ๐พ y ๐๐ก หหห On the other hand, by integration by parts we acquire(3.21) หหห ลผ ๐ x : ๐พ , ๐พ y ๐๐ก หหห ฤ หหห ลผ ๐ | ๐พ | ๐๐ก ยด x ๐พ , ๐พ y| ๐ หหห ฤ ๐ ยด p ๐ q ฤ ๐ถ๐ Combining the estimates (3.20) and (3.21) yields the claimed bound. (cid:3)
In particular, Lemma 3.14 implies that ๐ด ๐ ฤ ๐ถ ยจ diam p ๐ q ยด for all ๐ถ ฤ ๐ . Combining this with Lemma 3.13 and taking ๐ถ ร
1, we acquire the following corollary.
Corollary 3.15.
Let ๐ ฤ (cid:82) be a convex domain with smooth boundary ๐ . Then (3.22) ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q dvol ๐ ฤ area p ๐ q ยจ diam p ๐ q ยจ ล ๐ ๐ป dvol ๐ We will use Corollary 3.15 in the proof of the main Ruelle invariant bound later in ยง3.4.We will also need a less crude estimate on the time-averaged acceleration that uses the geom-etry of vector-๏ฌeld ๐ผ ๐ , but requires the hypothesis that ๐ has small systolic ratio. D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 17
Lemma 3.16.
Suppose that ๐ satis๏ฌes ๐ธ p ๐, ๐ q ฤ ๐ ฤ ยจ ๐ธ p ๐, ๐ q and let ฮฃ ฤ ๐ be the open subset ฮฃ โ ๐ X (cid:67) ห int p ๐ธ p ๐ { qq Then there is an ๐ ฤ and a ๐ถ ฤ independent of ๐, ๐ and ๐ such that, if ๐ { ๐ ฤ ๐ and ๐ โ ๐ { , then ๐ด ๐ ฤ ๐ถ ยจ ๐ ยด { on ฮฃ and area p ฮฃ q ฤ ๐ถ ยจ area p ๐ q Proof.
To bound ๐ด ๐ , the strategy is to show that the projection of ๐ผ ๐ to the 2nd (cid:67) -factor is boundedalong ฮฃ by p ๐ { ๐ q { . Thus, a length ๐ โ ๐ { trajectory ๐พ of ๐ผ ๐ stays within a ball of diameterroughly ๐ { , and a variation of Lemma 3.14 implies the desired bound.To bound area p ฮฃ q , the strategy is (essentially) to use the monotonicity of area under theinclusion ๐ธ p ๐, ๐ q ฤ ๐ to reduce to the case of an ellipsoid. We can then use the estimates inLemmas 3.3 and 3.7 to deduce the result. Projection Bound.
Let ๐ ๐ : (cid:82) ยป (cid:67) ร (cid:67) denote the projections to each (cid:67) -factor for ๐ โ , ๐ด ฤ ๐ , ๐ and ๐ such that(3.23) | ๐ ห ๐ผ ๐ p ๐ฅ q| โ | ๐ ห ๐ p ๐ฅ q| ฤ ๐ด ยจ p ๐ { ๐ q { if ๐ p ๐ฅ q P ๐ธ p ๐ { q To deduce (3.23), assume that ๐ฅ P ๐ satis๏ฌes ๐ p ๐ฅ q P ๐ธ p ๐ { q and that ๐ ห ๐ p ๐ฅ q โฐ
0. Let ๐ง P ห B ๐ธ p ๐ q be the unique vector such that ๐ p ๐ง ยด ๐ฅ q is a positive scaling of ๐ p ๐ q . Note that ๐ง P ๐ since 0 ห ๐ธ p ๐ q ฤ ๐ธ p ๐, ๐ q ฤ ๐ Furthermore, since ๐ is convex, we know that x ๐ p ๐ฅ q , ๐ค ยด ๐ฅ y ฤ ๐ค P ๐ . Therefore(3.24) 0 ฤ x ๐ p ๐ฅ q , ๐ง ยด ๐ฅ y โ | ๐ ห ๐ p ๐ฅ qq| ยจ | ๐ p ๐ง ยด ๐ฅ q| ` x ๐ ห ๐ p ๐ฅ q , ๐ p ๐ง ยด ๐ฅ qy Now note that since ๐ p ๐ฅ q P ๐ธ p ๐ { q and ๐ p ๐ง q P B ๐ธ p ๐ q , we know that | ๐ p ๐ง ยด ๐ฅ q| ฤ ยด p { q { ๐ { ยจ ๐ { Likewise, ๐ p ๐ q ฤ ยจ ๐ธ p ๐ q so that | ๐ p ๐ง ยด ๐ฅ q| ฤ ๐ { { ๐ { . Finally, | ๐ ห ๐ p ๐ฅ q| ฤ | ๐ p ๐ฅ q| โ
1. Thus,we can conclude that | ๐ ห ๐ p ๐ฅ q| ฤ | ๐ ห ๐ p ๐ฅ q| ยจ | ๐ p ๐ง ยด ๐ฅ q|| ๐ p ๐ง ยด ๐ฅ q| ฤ ยด p { q { ยจ p ๐ { ๐ q { Acceleration Bound.
Now let ๐ โ ๐ { and let ๐พ : r , ๐ s ร ๐ be a trajectory of ๐ผ ๐ with ๐พ p q P ฮฃ . Since ๐ p ๐พ p qq P ๐ธ p ๐ { q , we know that there is an interval r , ๐ s ฤ r , ๐ s where ๐ ห ๐พ pr , ๐ sq ฤ ๐ธ p ๐ { q . Thus, by (3.23), we know that for ๐ก P r , ๐ s we have(3.25) | ๐ p ๐พ p ๐ก q ยด ๐พ p qq| ฤ ลผ ๐ก | ๐ ห ๐ผ ๐ ห ๐พ | ๐๐ก ฤ ๐ด ยจ p ๐ { ๐ q { ยจ ๐ก ฤ ๐ด ยจ ๐ { By picking ๐ ฤ ๐ด๐ { ฤ p ๐ ๐ q { ยด p ๐ ๐ q { With this choice of ๐ , (3.25) implies that ๐ p ๐พ p ๐ก q ยด ๐พ p qq P ๐ธ p ๐ { q if 0 ฤ ๐ก ฤ ๐ . In fact, (3.25)implies that ๐พ is inside of a ball, i.e. ๐พ p ๐ก q P ๐ธ p ๐ q ห ๐ธ p ๐ ๐ด ยจ ๐ q ` ๐ ฤ ๐ต ยจ ๐ธ p ๐, ๐ q ` ๐ where ๐ : โ ห ๐ p ๐พ p qq Here ๐ต : โ p ` ๐ ๐ด q { . The diameter of the ball ๐ต ยจ ๐ธ p ๐, ๐ q is 2 ๐ต ยจp ๐ { ๐ q { . Therefore, by applying(3.20) and (3.21) we see that2 ๐ต๐ { ๐ { ยจ ๐ด ๐ p ๐ฅ q โ diam p ๐ต ยจ ๐ธ p ๐, ๐ qq ๐ ยจ ลผ ๐ | : ๐พ | ๐๐ก ฤ ยด p ๐ต ยจ ๐ธ p ๐, ๐ qq ๐ โ ยด ๐ต ๐ { ยจ p ๐ { ๐ q {
28 J. CHAIDEZ AND O. EDTMAIR
We now choose ๐ถ ฤ ๐ ฤ ๐, ๐ and ๐ , such that ๐ด ๐ p ๐ฅ q ฤ p ๐ { ๐ต ยด ยจ p ๐ { ๐ q { q ยจ ๐ ยด { ฤ ๐ถ ๐ ยด { if ๐ { ๐ ฤ ๐ This proves the desired bound on time-averaged acceleration.
Area Bound.
Let ๐ denote the convex domain given by the intersection ๐ X p (cid:67) ห ๐ธ p ๐ { qq .Note that we have the following inclusion. ๐ธ p ๐ { , ๐ { q ฤ ๐ธ p ๐, ๐ q X p (cid:67) ห ๐ธ p ๐ { qq ฤ ๐ Furthermore, the boundary of ๐ decomposes as follows. B ๐ โ ฮฃ Y ฮฃ where ฮฃ : โ ๐ X p (cid:67) ห B ๐ธ p ๐ { qq Since ๐ ฤ ยจ ๐ธ p ๐, ๐ q , we have ฮฃ ฤ ๐ where ๐ is the hypersurface ๐ : โ ยจ ๐ธ p ๐, ๐ q X p (cid:67) ห B ๐ธ p ๐ { qq โ ๐ธ p ๐ { q ห B ๐ธ p ๐ { q Combining the above facts and applying the monotonicity of surface area under inclusion ofconvex domains, we ๏ฌnd thatarea p ฮฃ q โ area pB ๐ q ยด area p ฮฃ q ฤ area pB ๐ธ p ๐ { , ๐ { qq ยด area p ๐ q By Lemma 3.7 and direct calculation, we compute the areas of B ๐ธ p ๐ { , ๐ { q and ๐ to bearea pB ๐ธ p ๐ { , ๐ { qq ฤ ยด { ยจ ๐๐ { area p ๐ q โ ๐ ยจ p ๐ ๐ q { โ ยจ p ๐ { q { ยจ p ๐ { ๐ q { ยจ ๐๐ { Now let ๐ต ฤ ยด { and choose ๐ ฤ ๐ { ๐ ฤ ๐ then2 ยด { ยด ยจ p ๐ { q { ยจ p ๐ { ๐ q { ฤ ๐ต By applying this inequality and the upper bound for area in Lemma 3.7, we ๏ฌnd that for some ๐ถ ฤ ๐ , ๐ and ๐ and an ๐ ฤ p ฮฃ q ฤ p ยด { ยด ยจ p ๐ { q { ยจ p ๐ { ๐ q { q ยจ ๐๐ { ฤ ๐ถ ยจ ๐๐ { ฤ area p ๐ q This yields the desired ara bound and concludes the proof of the lemma. (cid:3)
By plugging the bounds for ๐ด ๐ and area p ฮฃ q from Lemma 3.16 into Lemma 3.13, we acquirethe following variation of Corollary 3.15. Corollary 3.17.
Let ๐ be a convex domain with smooth boundary ๐ , such that ๐ธ p ๐, ๐ q ฤ ๐ ฤ ยจ ๐ธ p ๐, ๐ q .Then there exists a ๐ถ ฤ and ๐ ฤ independent of ๐ , ๐ and ๐ such that ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ยจ dvol ๐ ฤ ๐ถ ยจ area p ๐ q ๐ ยจ ล ๐ ๐ป dvol ๐ if ๐ { ๐ ฤ ๐ Proof Of Main Bound.
We now combine the results of ยง3.1-3.3 to prove Proposition 3.1.
Proof. (Proposition 3.1) By Lemma 3.6, we may assume that ๐ is sandwiched between standardellipsoid ๐ธ p ๐, ๐ q with 0 ฤ ๐ ฤ ๐ and a scaling. ๐ธ p ๐, ๐ q ฤ ๐ ฤ ยจ ๐ธ p ๐, ๐ q We begin by proving the lower bound, under this assumption. By Lemma 3.11, we have(3.26) Ru p ๐ q ฤ ๐ ยจ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q dvol ๐ By applying the lower bound in Corollary 3.15 and using the estimates for diameter, area, totalcurvature, volume and systolic ratio in Lemma 3.7, we see that for some ๐ถ ฤ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q ยจ dvol ๐ ฤ area p ๐ q ๐ ยจ diam p ๐ q ยจ ล ๐ ๐ป dvol ๐ ฤ ๐ถ ยจ ๐ ฤ vol p ๐ q { ยจ sys p ๐ q {
2D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 19
On the other hand, suppose that ๐๐ !
1. Due to Lemma 3.7, this is equivalent to sys p ๐ q !
1. ByCorollary 3.17 and the estimates in Lemma 3.7, there are constants
๐ด, ๐ต, ๐ถ ฤ ลผ ๐ ๐ p ๐ผ ๐ , ๐ผ ๐ q dvol ๐ ฤ ๐ด ยจ area p ๐ q ๐ ยจ ล ๐ ๐ป dvol ๐ ฤ ๐ต ยจ ๐ ฤ ๐ถ ยจ vol p ๐ q { ยจ sys p ๐ q ยด { By assembling the estimate (3.26) with the two estimates (3.27) and (3.28), we deduce the followinglower bound for some ๐ถ ฤ p ๐ q ฤ ๐ถ ยจ vol p ๐ q { ยจ sys p ๐ q ยด { After some rearrangement, this is the desired lower bound.The second inequality is easier to show. By using the upper bound in Lemma 3.11 and theestimate for the mean curvature in Lemma 3.7, we see that for some
๐ด, ๐ถ ฤ p ๐ q ฤ ลผ ๐ ๐ป dvol ๐ ฤ ๐ด ยจ ๐ ฤ ๐ถ ยจ vol p ๐ q { ยจ sys p ๐ q ยด { This implies the desired upper bound, and concludes the proof. (cid:3)
4. Non-Convex, Dynamically Convex Contact FormsIn this section, we use the methods of [1] to construct a dynamically convex contact form withsystolic ratio and volume close to 1, and arbitrarily small Ruelle invariant.
Proposition 4.1.
For every ๐ ฤ , there exists a dynamically convex contact form ๐ผ on ๐ with vol ` ๐ , ๐ผ ห โ p ๐ , ๐ผ q ฤ ยด ๐ Ru p ๐ , ๐ผ q ฤ ๐ Hamiltonian Disk Maps.
We begin with some notation and preliminaries on Hamiltonianmaps of the disk that we will need for the rest of the section.Let (cid:68) ฤ (cid:82) denote the unit disk in the plane with ordinary coordinates p ๐ฅ, ๐ฆ q and radialcoordinates p ๐, ๐ q . We use ๐ and ๐ to denote the standard Liouville form and symplectic form. ๐ : โ ๐ ๐ ๐ โ p ๐ฅ๐๐ฆ ยด ๐ฆ๐๐ฅ q and ๐ : โ ๐๐๐ ^ ๐ ๐ โ ๐๐ฅ ^ ๐๐ฆ Let ๐ : r , s ห (cid:68) ร (cid:68) be a the Hamiltonian ๏ฌow (for ๐ก P r , s ) generated by a time-dependentHamiltonian on (cid:68) vanishing on the boundary, i.e. ๐ป : (cid:82) { (cid:90) ห (cid:68) ร (cid:82) with ๐ป | B (cid:68) โ ๐ ๐ป denote the Hamiltonian vector ๏ฌeld and adopt the convention that ๐ ๐ ๐ป ๐ โ ๐๐ป . Thedi๏ฌerential of ๐ de๏ฌnes a map ฮฆ : (cid:82) ห (cid:68) ร Sp p q with ฮฆ | ห (cid:68) โ Id, which lifts uniquely to a map(4.1) r ฮฆ : (cid:82) ห (cid:68) ร ฤ Sp p q satisfying r ฮฆ p ๐ ` ๐ , ๐ง q โ r ฮฆ p ๐ , ๐ ๐ p ๐ง qq r ฮฆ p ๐, ๐ง q There are two key functions on (cid:68) associated to the family of Hamiltonian di๏ฌeomorphisms ๐ .First, there is the action and the associated Calabi invariant. De๏ฌnition 4.2.
The action ๐ ๐ : (cid:68) ร (cid:82) and Calabi invariant
Cal p (cid:68) , ๐ q P (cid:82) of ๐ are de๏ฌned by(4.2) ๐ ๐ โ ลผ ๐ ห ๐ก p ๐ ๐ ๐ป ๐ ` ๐ป q ยจ ๐๐ก and Cal p (cid:68) , ๐ q โ ลผ (cid:68) ๐ ยจ ๐ The action measures the failure of ๐ to preserve ๐ , as captured by the following formula.(4.3) ๐ ห ๐ ยด ๐ โ ๐ ๐ ๐ Next, there is the rotation map and the associated Ruelle invariant. To discuss these quantities,we require the following lemma.
Lemma 4.3.
Let ๐ : r , s ห (cid:68) ร (cid:68) be the ๏ฌow of a Hamiltonian ๐ป : (cid:82) { (cid:90) ห (cid:68) ร (cid:68) with ๐ ๐ ฤ . Thenthe sequences ๐ ๐ : (cid:68) ร (cid:82) and ๐ ๐ : (cid:68) ร (cid:82) given by ๐ ๐ p ๐ง q : โ ๐ ๐ ห r ฮฆ p ๐, ๐ง q and ๐ ๐ p ๐ง q : โ ๐ ๐ ยด รฟ ๐ โ ๐ ๐ ห ๐ ๐ p ๐ง q converge in ๐ฟ p (cid:68) q to ๐ ๐ and ๐ ๐ , respectively. The map ๐ ยด ๐ also converges to ๐ ยด ๐ in ๐ฟ p (cid:68) q .Proof. We apply Kingmanโs sub-additive ergodic theorem [15] to the map ๐ ๐ โ ๐ ๐ ` ๐ถ forsu๏ฌciently large ๐ถ ฤ
0. Applying (4.1) and the quasimorphism property of ๐ , we ๏ฌnd that ๐ ๐ ` ๐ ฤ ๐ ๐ ` ๐ ๐ ห ๐ ๐ By Kingmanโs ergodic theorem, this implies that ๐ ๐ ๐ has a limit ๐ in ๐ฟ p (cid:68) q . Since } ๐ ๐ ยด ๐ ๐ } ๐ฟ isbounded, we acquire the same result for ๐ ๐ .By Birkho๏ฌโs ergodic theorem, ๐ ๐ converges to a limit ๐ P ๐ฟ p (cid:68) q . Note that for some ๐ ฤ ๐ ยด ฤ ๐ ๐ ฤ ๐ and therefore ๐ ยด ฤ ๐ ๐ ฤ ๐ Thus ๐ ฤ ๐ ยด is well-de๏ฌned almost everywhere. Since | ๐ ๐ | ยด ฤ ๐ , we can apply the dominated convergence theorem to conclude that ๐ ยด is integrableand ๐ ยด ๐ ร ๐ ยด in ๐ฟ . A similar argument applies to ๐ ๐ { ๐ ๐ , which converges to ๐ { ๐ . (cid:3) De๏ฌnition 4.4.
The rotation ๐ ๐ : (cid:68) ร (cid:82) and Ruelle invariant Ru p (cid:68) , ๐ q P (cid:82) of ๐ are de๏ฌned by(4.4) ๐ ๐ : โ lim ๐ ร8 ๐ ๐ and Ru p (cid:68) , ๐ q โ ลผ (cid:68) ๐ ๐ ยจ ๐ Remark 4.5.
Our Ruelle invariant Ru p (cid:68) , ๐ q of a symplectomorphism of the disk agrees with theone introduced by Ruelle in [18].The action, rotation, Calabi invariant and rotation invariant depend only on the homotopy classof ๐ relative to the endpoints, or equivalently the element in the universal cover of Ham p (cid:68) , ๐ q .We conclude this review with a discussion of periodic points and their invariants. De๏ฌnition 4.6. A periodic point ๐ of ๐ : (cid:68) ร (cid:68) is a point such that ๐ ๐ p ๐ q โ ๐ for some ๐ ฤ L p ๐ q , action A p ๐ q and rotation number ๐ p ๐ q of ๐ are given, respectively, by(4.5) L p ๐ q : โ min (cid:32) ๐ ฤ | ๐ ๐ p ๐ q โ ๐ ( A p ๐ q โ L p ๐ qยด รฟ ๐ โ ๐ ๐ ห ๐ ๐ p ๐ q ๐ p ๐ q : โ ๐ ห r ฮฆ p L p ๐ q , ๐ q Note that the rotation number can also be written as ๐ p ๐ q โ L p ๐ q ยจ ๐ ๐ p ๐ q .4.2. Open Books Of Disk Maps.
We next review the construction of contact forms on ๐ fromsymplectomorphisms of the disk, using open books. Construction 4.7.
Let ๐ป : (cid:82) { (cid:90) ห (cid:68) ร (cid:82) be a Hamiltonian with ๏ฌow ๐ : r , s ห (cid:68) ร (cid:82) such that(i) Near B (cid:68) , ๐ป is of the form ๐ป p ๐ก, ๐, ๐ q โ ๐ถ ยจ ๐ p ยด ๐ q for some ๐ถ ฤ ๐ ๐ of the Hamiltonian is positive everywhere.We now construct the open book contact form ๐ผ on ๐ associated to p (cid:68) , ๐ q . We proceed by pro-ducing two contact manifolds p ๐ , ๐ผ q and p ๐ , ๐ฝ q , then gluing them by a strict contactomorphism.To construct ๐ , we consider the contact form ๐๐ก ` ๐ on (cid:82) ห (cid:68) . Due to the identity ๐ ๐ ๐ โ ๐ ห ๐ ยด ๐ in (4.3), the map ๐ de๏ฌned by ๐ : (cid:82) ห (cid:68) ร (cid:82) ห (cid:68) ๐ p ๐ก, ๐ง q โ p ๐ก ยด ๐ ๐ p ๐ง q , ๐ p ๐ง qq is a strict contactomorphism. Thus, we can form the manifold ๐ as the following quotient space. ๐ โ (cid:82) ห (cid:68) { โ de๏ฌned by p ๐ก, ๐ง q โ ๐ p ๐ก, ๐ง q D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 21
The contact form ๐๐ก ` ๐ descends to a contact form ๐ผ on ๐ . Note that a fundamental domain ofthis quotient is given by ฮฉ โ tp ๐ก, ๐ง q| ฤ ๐ก ฤ ๐ ๐ p ๐ง qu To construct ๐ , we choose a small ๐ ฤ ๐ : โ (cid:82) { ๐ (cid:90) ห (cid:68) p ๐ q ๐ฝ : โ p ยด ๐ q ๐๐ก ` ๐ถ ๐ ๐ ๐ Here (cid:68) p ๐ q ฤ (cid:67) is the disk of radius ๐ , ๐ก is the (cid:82) { ๐ (cid:90) coordinate and p ๐, ๐ q are radial coordinateson (cid:68) p ๐ q . There is a strict contactomorphism ฮฆ identifying subsets of ๐ and ๐ , given by ฮจ : ๐ zp (cid:82) { ๐ (cid:90) ห q ร ๐ with ฮจ p ๐ก, ๐, ๐ q : โ p ๐ถ ยจ ๐ , a ยด ๐ , ๐ก ยด ๐ถ ๐ q We now de๏ฌne ๐ โ int p ๐ q Y ฮจ ๐ as the gluing of the interior of ๐ and ๐ via ฮฆ , and ๐ผ as theinherited contact form. Since ๐ is Hamiltonian isotopic to the identity, the resulting contact form p ๐, ๐ผ q is contactomorphic to standard contact ๐ .In order to relate various invariants associated to p ๐ , ๐ผ q and its Reeb orbits to correspondingstructures for p (cid:68) , ๐ q , we need to introduce a certain trivialization of ๐ over ๐ . Construction 4.8.
Let p ๐ , ๐ | ๐ q be as in Construction 4.7. We let ๐ denote the continuous trivial-ization of ๐ | ๐ de๏ฌned as follows. On the fundamental domain ฮฉ , we let(4.6) ๐ : ฮฉ ร Hom p ๐ | ๐ , (cid:82) q given by ๐ p ๐ก, ๐ง q : โ exp p ๐ ๐๐ก { ๐ ๐ p ๐ง qq ห ฮฆ p ๐ก { ๐ ๐ p ๐ง q , ๐ง q ห ฮ (cid:68) Here ฮฆ : r , s ห (cid:68) ร (cid:68) is the di๏ฌerential ๐ ๐ of the ๏ฌow ๐ : r , s ห (cid:68) ร (cid:68) and ฮ (cid:68) : ๐ ร ๐ (cid:68) denotes projection to the (canonically trivial) tangent bundle ๐ (cid:68) of (cid:68) . Note also that ห denotescomposition of bundle maps.To check that ๐ descends to a well-de๏ฌned trivialization on ๐ , we must check that it iscompatible with the quotient map ๐ : (cid:82) ห (cid:68) ร (cid:82) ห (cid:68) . Indeed, we have ๐ p ๐ ๐ p ๐ง q , ๐ง q โ ฮฆ p , ๐ง q ห ฮ (cid:68) โ ๐ p , ๐ p ๐ง qq ห ๐ ๐ ๐ p ๐ง q ,๐ง This precisely states that projection commutes with the isomorphism identifying tangent spacesin the quotient, so ๐ descends from ฮฉ to ๐ . Lemma 4.9.
Let ๐ : ๐ | ๐ ร (cid:82) be the trivialization in Construction 4.8. Then(a) The restriction ๐ | ๐พ of ๐ to any compact subset ๐พ ฤ int p ๐ q of the interior of ๐ is the restriction ofa global trivialization of ๐ on ๐ .(b) The local rotation number rot ๐ : ๐ ร (cid:82) of p ๐ , ๐ผ | ๐ q with respect to ๐ agres with the restrictionof the local rotation number rot : ๐ ร (cid:82) of p ๐ , ๐ผ q with respect to the global trivialization.Proof. Let ๐ โ (cid:82) { ๐ (cid:90) ห (cid:68) p ๐ q and ฮจ be as in Construction 4.7. For any ๐ฟ ฤ ๐ , we let ๐ p ๐ฟ q ฤ ๐ and ๐ p ๐ฟ q ฤ ๐ denote ๐ p ๐ฟ q : โ (cid:82) { ๐ (cid:90) ห ๐ท p ๐ฟ q ฤ ๐ and ๐ p ๐ฟ q : โ int p ๐ qz int p ฮจ p ๐ p ๐ฟ qqq The sets ๐ p ๐ฟ q are an exhaustion of int p ๐ q by compact, Reeb-invariant contact sub-manifolds.To show (a), we assume that ๐พ โ ๐ p ๐ฟ q . The homotopy classes of trivializations T of ๐ over ๐ p ๐ฟ q are in bฤณection with ๐ป p ๐ p ๐ฟ q ; (cid:90) q ยป (cid:90) . A map to (cid:90) classifying elements of T is given by T ร (cid:90) given by ๐ รร sl p ๐พ , ๐ q Here sl p ๐พ , ๐ q is the self-linking number (in the trivialization ๐ ) of the following transverse knot. ๐พ : (cid:82) { ๐ (cid:90) ร ๐ p ๐ฟ q ๐พ p ๐ q โ ฮจ p , ๐ , ๐ q โ p ๐ถ ๐ , a ยด ๐ , ยด ๐ถ ๐ q The knot ๐พ bounds a Seifert disk ฮฃ โ ห (cid:68) p ๐ q in ๐ ฤ ๐ . The foliation ๐ X ฮฃ has a singlepositive elliptic singularity, so the self-linking number of the boundary ๐พ with respect to theglobal trivialization is sl p ๐พ q โ ยด p ๐พ , ๐ q , we push ๐พ into ฮฃ along a collar neighborhood to acquire a nowhere zerosection ๐ : (cid:82) { ๐ (cid:90) ร ๐ and then compose with ๐ to acquire a map ๐ ห ๐ : (cid:82) { ๐ (cid:90) ร (cid:82) z
0. Up toisotopy through nowhere zero sections, we can compute that ๐ ห ๐ p ๐ q โ ๐ ๐ ๐ P (cid:67) โ (cid:82) On the other hand, the self-linking number can be computed as the negative of the windingnumber of this map. sl p ๐พ , ๐ q โ ยด wind p ๐ ห ๐ q โ ยด ๐ agrees with the restriction of the global trivialization.To show (b), note that since ๐ p ๐ฟ q is compact, we can choose a global trivialization of ๐ on ๐ ๐ : ๐ ยป (cid:82) such that ๐ | ๐ p ๐ฟ q โ ๐ | ๐ p ๐ฟ q By Proposition 2.11(c), rot ๐ โ rot on ๐ and so the local rotation numbers satisfyrot | ๐ p ๐ฟ q โ rot ๐ | ๐ p ๐ฟ q โ rot ๐ | ๐ p ๐ฟ q Since this holds for any ๐ฟ , this shows (b) on all of int p ๐ q . Note that we assiduously avoidedextending ๐ itself from int p ๐ q to ๐ in this argument. (cid:3) Proposition 4.10 (Open Book) . Let ๐ป and ๐ be as in Construction 4.7. Then there exists a contact form ๐ผ on ๐ with the following properties.(a) (Surface Of Section) There is an embedding ๐ : (cid:68) ร ๐ such that ๐ p (cid:68) q is a surface of section withreturn map ๐ and ๏ฌrst return time ๐ , and such that ๐ โ ๐ ห ๐ ๐ผ .(b) (Volume) The volume of p ๐ , ๐ผ q is given by the Calabi invariant of p (cid:68) , ๐ q , i.e. vol ` ๐ , ๐ผ ห โ Cal p (cid:68) , ๐ q (c) (Ruelle) The Ruelle invariant of p ๐ , ๐ผ q is given by a shift of the Ruelle invariant of p (cid:68) , ๐ q . Ru p ๐ , ๐ผ q โ Ru p (cid:68) , ๐ q ` ๐ (d) (Binding) The binding ๐ โ ๐ pB (cid:68) q is a Reeb orbit of action ๐ and rotation number ` { ๐ถ .(e) (Orbits) Every simple orbit ๐พ ฤ ๐ z ๐ corresponds to a periodic point ๐ of ๐ that satis๏ฌes lk p ๐พ , ๐ q โ L p ๐ q A p ๐พ q โ A p ๐ q ๐ p ๐พ q โ ๐ p ๐ q ` L p ๐ q Proof.
We prove each of these properties separately.
Surface Of Section.
De๏ฌne the inclusion ๐ : (cid:68) ร ๐ as the following composition. ๐ : (cid:68) โ ห (cid:68) ร (cid:82) ห (cid:68) ๐ รร ๐ ยป ๐ The surface 0 ห (cid:68) is transverse to the Reeb vector ๏ฌeld B ๐ก of (cid:82) ห (cid:68) and intersects every ๏ฌowline (cid:82) ห ๐ง . Also, p (cid:82) ห ๐ง qX ฮฉ has action ๐ ๐ p ๐ง q and ends on p ๐ ๐ p ๐ง q , ๐ง q โ p , ๐ p ๐ง qq . Thus ๐ p (cid:68) q โ ๐ p ห (cid:68) q is a surface of section with return time ๐ ๐ and monodromy ๐ . Finally, note that ๐ ห p ๐ ๐ผ q โ ๐ p ๐๐ก ` ๐ q| ห (cid:68) โ ๐ This veri๏ฌes all of the properties of ๐ : (cid:68) ร ๐ ยป ๐ listed in (a). Calabi Invariant.
This property follows from a simple calculation of the volume using thefundamental domain ฮฉ .vol p ๐, ๐ผ q โ ลผ ๐ ๐ผ ^ ๐ ๐ผ โ ลผ ฮฉ ๐๐ก ^ ๐ ๐ โ ลผ (cid:68) ๐ ๐ ยจ ๐ โ Cal p (cid:68) , ๐ q D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 23
Ruelle Invariant.
Let rot : ๐ ร (cid:82) be the local rotation number of p ๐ , ๐ผ q . By Lemma 4.9, therestriction of rot to the (open) fundamental domain ฮฉ ฤ ๐ coincides with rot ๐ . Since ๐ z ฮฉ ismeasure 0 in ๐ , we thus have(4.7) Ru p ๐ , ๐ผ q โ ลผ ๐ rot ยจ ๐ผ ^ ๐ ๐ผ โ ลผ ฮฉ rot ๐ ยจ ๐๐ก ^ ๐ โ ลผ (cid:68) ๐ ห rot ๐ ยจ ๐ ๐ ๐ Here ๐ ห rot ๐ denotes the pullback of rot ๐ via the map ๐ : (cid:68) ร ๐ from (a). We have used the Reebinvariance of rot ๐ , i.e. the fact that rot ๐ p ๐ก, ๐ง q โ ๐ ห rot ๐ p ๐ง q .To apply this alternative formula for Ru p ๐ , ๐ผ q , let ๐ ๐ denote the ๐ th positive time that the Reebtrajectory ๐พ : r ,
8q ร ๐ intersects the surface of section ๐ p (cid:68) q . Then ๐ ห rot ๐ โ lim ๐ ร8 ๐ ห r ฮฆ ๐ p ๐ ๐ , ยดq ๐ ๐ โ lim ๐ ร8 ๐ ห r ฮฆ p ๐, ยดq ` ๐ ล ๐ ยด ๐ โ ๐ ๐ ห ๐ ๐ โ ๐ ๐ ` ๐ ๐ Here the maps ๐ ๐ and ๐ ๐ are the averaged rotation and action maps constructed in Lemma 4.3.By construction, these maps are invariant under pullback by ๐ . Thus ลผ (cid:68) ๐ ๐ ` ๐ ๐ ยจ ๐ ๐ ๐ โ ๐ ๐ ยด รฟ ๐ โ ลผ (cid:68) r ๐ ๐ s ห p ๐ ๐ ` ๐ ๐ ยจ ๐ ๐ ๐ q โ ลผ (cid:68) ๐ ๐ ` ๐ ๐ ยจ ๐ ๐ ๐ where ๐ ๐ โ ๐ ๐ ยด รฟ ๐ โ ๐ ๐ ห ๐ ๐ By Lemma 4.3, we know that ๐ ๐ ร ๐ ๐ in ๐ฟ p (cid:68) q . Thus, by combining the above formula in the ๐ ร 8 limit with (4.7), we acquire the desired property.Ru p ๐ , ๐ผ q โ ลผ (cid:68) ๐ ๐ ` ๐ ๐ ยจ ๐ ๐ ยจ ๐ โ ลผ (cid:68) ๐ ๐ ` ๐ ๐ ยจ ๐ ๐ ยจ ๐ โ ลผ (cid:68) p ๐ ๐ ` q ยจ ๐ โ Ru p (cid:68) , ๐ q ` ๐ Binding.
Let ๐ โ ๐ pB (cid:68) q be the binding which coincides with (cid:82) { ๐ (cid:90) ห ๐ . First note thatthe Reeb vector ๏ฌeld is given on p ๐ , ๐ฝ q by the following formula.(4.8) ๐ ๐ฝ โ B ๐ก ` ๐ถ B ๐ Thus ๐ is a Reeb orbit. Since ๐ bounds a symplectic disk ๐ p (cid:68) q ฤ ๐ of area ๐ , the action is ๐ . Tocompute ๐ p ๐ q , note that there is a natural trivialization of ๐ | ๐ โ ker p ๐ฝ q given by ๐ : ๐ | ๐ ฤ ๐๐ ๐ รร ๐ (cid:68) p ๐ q โ (cid:82) The Reeb ๏ฌow ๐ : (cid:82) ห ๐ ร ๐ and the linearized Reeb ๏ฌow ฮฆ ๐ : (cid:82) ห ๐ ร Sp p q with respect to ๐ can be calculated from (4.8), as follows. ๐ ๐ก p ๐ , ๐ง q โ p ๐ ` ๐ก, ๐ ๐๐ก { ๐ถ ยจ ๐ง q ฮฆ ๐ p ๐ก, ๐ , ๐ง q โ ๐ ๐๐ก { ๐ถ Thus the rotation number ๐ p ๐, ๐ q of ๐ in the trivialization ๐ is 1 { ๐ถ . Finally, to compute the rotationnumber ๐ p ๐ q โ ๐ p ๐, ๐ q with respect to the global trivialization ๐ on ๐ , we note that ๐ p ๐, ๐ q ยด ๐ p ๐, ๐ q โ ๐ p ๐ ห ๐ ยด | ๐ q โ ๐ p ๐ | ๐ p (cid:68) q , ๐ q ยด ๐ p ๐ | ๐ p (cid:68) q , ๐ q โ ยด ๐ p ๐ | ๐ p (cid:68) q , ๐ q Here ๐ : ๐ p Sp p qq ร (cid:90) is the Maslov index and ๐ p ๐ | ๐ p (cid:68) q , ยดq is the relative Chern class of ๐ | ๐ p (cid:68) q with respect to a given trivialization over ๐ pB (cid:68) q , which vanishes for ๐ .On the other hand, the trivialization ๐ is speci๏ฌed by the section of ๐ | ๐ p (cid:68) q given by pushing ๐ pB (cid:68) q into ๐ p (cid:68) q along a collar neighborhood. Thus, ยด ๐ p ๐ | ๐ p (cid:68) q , ๐ q is precisely the self-linkingnumber sl p ๐ q of ๐ . This number can be calculated as a signed count of singularities of thefoliation ๐ X ๐ p (cid:68) q , which has 1 elliptic singularity. Thus sl p ๐ q โ ยด ๐ p ๐ q โ ` { ๐ถ . Orbits.
An embedded closed orbit ๐พ : (cid:82) { ๐ฟ (cid:90) ร ๐ of ๐ผ that is disjoint from the binding ๐ is equivalent to a closed orbit of p ๐ , ๐ผ | ๐ q . The orbit ๐พ intersects the surface of section ๐ p (cid:68) q transversely at ๐ ฤ ๐ โ , ๐ , . . . , ๐ ๐ โ ๐ฟ . Let ๐ ๐ P (cid:68) be such that ๐ p ๐ ๐ q โ ๐พ p ๐ ๐ q X ๐ p (cid:68) q Since ๐ p (cid:68) q is a surface of section, we have ๐ ๐ ` โ ๐ p ๐ ๐ q and since ๐พ is closed, ๐ ๐ โ ๐ . Thus ๐ โ ๐ is a periodic point of period L p ๐ q โ ๐ โ ๐ ห r (cid:68) s ยจ r ๐พ s โ lk p ๐พ , ๐ q Next, note that on the interval r ๐ ๐ , ๐ ๐ ` s , ๐พ restricts to a map r ๐ ๐ , ๐ ๐ ` s ร ฮฉ given by ๐พ p ๐ก q โp ๐ก, ๐ p ๐ ๐ qq , from which it follows that A p ๐พ q โ ๐ ยด รฟ ๐ โ ลผ ๐ ๐ ` ๐ ๐ ๐พ ห p ๐๐ก ` ๐ผ q โ ๐ ยด รฟ ๐ โ ลผ ๐ p ๐ ๐ q ๐๐ก โ ๐ ยด รฟ ๐ โ ๐ ห ๐ ๐ p ๐ q โ A p ๐ q Finally, due to Lemma 4.9 we may use the trivialization ๐ to compute the rotation number. Forthe purpose of abbreviation, we adopt the notation ๐ฆ ๐ โ ๐ p ๐ ๐ q โ ๐พ p ๐ ๐ q ๐ฟ ๐ โ ๐ ๐ ` ยด ๐ ๐ โ ๐ ๐ p ๐ ๐ q Note that the lifted linearized Reeb ๏ฌow with respect to ๐ at time ๐ฟ can be written as(4.9) r ฮฆ ๐ p ๐ฟ, ๐พ p qq โ r ฮฆ ๐ p ๐ฟ ๐ ยด , ๐ฆ ๐ ยด q r ฮฆ ๐ p ๐ฟ ๐ ยด , ๐ฆ ๐ ยด q . . . r ฮฆ ๐ p ๐ฟ , ๐ฆ q The linearized Reeb ๏ฌow r ฮฆ ๐ p ๐ฟ ๐ , ๐ฆ ๐ q takes place along a trajectory connecting p , ๐ ๐ q to p ๐ ๐ p ๐ ๐ q , ๐ ๐ q in the fundamental domain ฮฉ . We may be directly compute from (4.6) that(4.10) ฮฆ ๐ p ๐ก, ๐ฆ ๐ q โ exp p ๐ ๐๐ก { ๐ ๐ p ๐ ๐ qq ห ฮฆ p ๐ก { ๐ ๐ p ๐ง q , ๐ ๐ q and so r ฮฆ ๐ p ๐ฟ ๐ , ๐ฆ ๐ q โ r ฮ ยจ r ฮฆ p , ๐ ๐ q Here r ฮ is the unique lift of Id P Sp p q with ๐ p r ฮ q โ
1. This is a central element of ฤ Sp p q , socombining (4.9) and (4.10) we have r ฮฆ ๐ p ๐ฟ, ๐พ p qq โ r ฮ ๐ ยจ r ฮฆ p , ๐ ๐ ยด p ๐ qq ยจ r ฮฆ p , ๐ ๐ ยด p ๐ qq ยจ ยจ ยจ r ฮฆ p , ๐ q โ r ฮ ๐ ยจ r ฮฆ p ๐, ๐ q Since ๐ p r ฮ ยจ r ฮจ q โ ` ๐ p r ฮจ q for any r ฮจ P ฤ Sp p q , we can conclude that ๐ p ๐พ q โ ๐ ห r ฮฆ ๐ p ๐ฟ, ๐พ p qq โ ๐ ห r ฮฆ p ๐, ๐ q ` ๐ โ ๐ p ๐ q ` L p ๐ q This completes the proof of (e), and the entire proposition. (cid:3)
Radial Hamiltonians.
A Hamiltonian ๐ป : (cid:82) { (cid:90) ห (cid:68) ร (cid:82) that is rotationally invariant willbe called radial . In other words, ๐ป is radial if it can be written as ๐ป p ๐ก, ๐, ๐ q โ โ p ๐ก, ๐ q for a map โ : (cid:82) { (cid:90) ห r , s ร (cid:82) We will require a few lemmas regarding radial Hamiltonians.
Lemma 4.11.
Let ๐ป : (cid:68) ร (cid:82) be an autonomous, radial Hamiltonian with ๐ป โ โ ห ๐ . Then (4.11) ๐ ๐ p ๐, ๐ q โ โ p ๐ q ยด ๐ โ p ๐ q and ๐ ๐ p ๐, ๐ q โ ยด โ p ๐ q ๐ ๐ Proof.
We calculate the Hamiltonian vector ๏ฌeld ๐ ๐ป and the action function ๐ ๐ as follows. ๐ ๐ป โ ยด โ ๐ ยจ B ๐ and and ๐ ๐ p ๐, ๐ q โ ลผ ๐ ห ๐ก pยด ๐ โ p ๐ q ` โ p ๐ qq ยจ ๐๐ก โ โ p ๐ q ยด ๐ โ p ๐ q Here we use the fact that the Hamiltonian ๏ฌow ๐ preserves any function of ๐ . Next, we note thatthe di๏ฌerential ฮฆ : (cid:82) ห (cid:68) ร (cid:68) of the ๏ฌow ๐ is given by ฮฆ p ๐ก, ๐ง q ๐ฃ โ exp p ยด โ ๐ ยจ ๐๐ก q ๐ฃ ` ๐๐ก p ๐ โ ยด โ q ๐ ยจ exp p ยด โ ๐ ยจ ๐๐ก q ๐ง ยจ ๐๐ p ๐ฃ q Note that if we use ๐ โ ๐๐ง {| ๐ง | , then ๐๐ p ๐ฃ q โ
0. Thus, if r ฮฆ : (cid:82) ห (cid:68) ร ฤ Sp p q denotes the lift of ฮฆ ,and ๐ ๐ denotes the rotation number relative to ๐ (see De๏ฌnition 2.4) then(4.12) ฮฆ p ๐ก, ๐ง q ๐ โ exp p ยด โ p ๐ q ๐ ยจ ๐๐ก q ๐ and thus ๐ ๐ ห r ฮฆ p ๐ , ๐ง q โ ๐ ยจ ยด โ p ๐ q ๐ ๐ Since ๐ ๐ : ฤ Sp p q ร (cid:82) and ๐ : ฤ Sp p q ร (cid:82) are equivalent quasimorphisms (Lemma 2.5), we have D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 25 ๐ ๐ โ lim ๐ ร8 ๐ ห r ฮฆ p ๐ , ยดq ๐ โ lim ๐ ร8 ๐ ๐ ห r ฮฆ p ๐ , ยดq ๐ โ ยด โ ห ๐ ๐ ๐ in ๐ฟ p (cid:68) q This concludes the proof of the lemma. (cid:3)
More generally, a Hamiltonian ๐ป : (cid:82) { (cid:90) ห (cid:68) ร (cid:82) is called radial around ๐ P (cid:68) if ๐ป is invariantunder rotation around ๐ , i.e. if ๐ป can be written as ๐ป p ๐ก, ๐ฅ, ๐ฆ q โ โ p ๐ก, ๐ ๐ q for a map โ : (cid:82) { (cid:90) ห r , s ร (cid:82) Here ๐ ๐ : (cid:68) ร (cid:82) be the distance from ๐ , i.e. ๐ ๐ p ๐ง q โ | ๐ง ยด ๐ | . Lemma 4.12.
Let ๐ป : (cid:68) ร (cid:82) be an autonomous Hamiltonian that is radial around ๐ โ p ๐, ๐ q P (cid:68) , with ๐ป โ โ ห ๐ ๐ , in a neighborhood ๐ of ๐ . Then on ๐ , we have (4.13) ๐ ๐ โ โ p ๐ ๐ q ยด ๐ ๐ โ p ๐ ๐ q ` ๐ข ๐ ยด ๐ ห ๐ข ๐ and ๐ ๐ โ ยด โ p ๐ ๐ q ๐ ๐ ๐ Here the map ๐ข ๐ : (cid:68) ร (cid:82) is given by ๐ข ๐ p ๐ฅ, ๐ฆ q โ p ๐๐ฅ ยด ๐ ๐ฆ q{ .Proof. Let ๐ ๐ be the radial Liouville form on p (cid:68) , ๐ q centered at ๐ . That is, ๐ ๐ is given by ๐ ๐ โ pp ๐ฅ ยด ๐ q ๐๐ฆ ยด p ๐ฆ ยด ๐ q ๐๐ฅ q โ ๐ ` ๐๐ข ๐ Let ๐ : (cid:68) ร (cid:82) be the function decribed in (4.13). Then by Lemma 4.11, we know that on ๐ wehave ๐ ๐ โ p ๐ ห ๐ ๐ ยด ๐ ๐ q ` p ๐ ห ๐๐ข ๐ ยด ๐ข ๐ q โ ๐ ห ๐ ยด ๐ โ ๐ ๐ ๐ Thus it su๏ฌces to check that ๐ ๐ p ๐ q โ ๐ p ๐ q . Since ๐ โ p ๐ q โ ๐ข ๐ p ๐ q โ ๐ข ๐ p ๐ p ๐ qq โ
0, we seethat ๐ p ๐ q โ โ p q โ ๐ป p ๐ q . On the other hand, ๐ ๐ป p ๐ q โ
0, we see that ๐ ๐ p ๐ q โ ลผ ๐ ห ๐ก p ๐ p ๐ ๐ป q ` ๐ป q ๐๐ก โ ลผ โ p q ๐๐ก โ ๐ p ๐ q Thus ๐ ๐ p ๐ q โ ๐ p ๐ q . The formula for ๐ ๐ follows from identical arguments to Lemma 4.11. (cid:3) A Special Hamiltonian Map.
We next construct a special Hamiltonian ๏ฌow ๐ : r , s ห (cid:68) ร (cid:68) whose corresponding contact form will provide our counterexample. We de๏ฌne ๐ as a product ๐ โ ๐ ๐ป โ ๐ ๐บ Here ๐ ๐บ : r , s ห (cid:68) ร (cid:68) and ๐ ๐ป : r , s ห (cid:68) ร (cid:68) are autonomous ๏ฌows generated by ๐บ and ๐ป , and the product occurs in the universal cover of the group Ham p (cid:68) , ๐ q of Hamiltoniandi๏ฌeomorphisms of p (cid:68) , ๐ q . We denote the Hamiltonian generating ๐ by ๐ป ๐บ : (cid:82) { (cid:90) ห (cid:68) ร (cid:82) To construct ๐บ and ๐ป , we must ๏ฌx the following setup (which will be used for the rest of ยง4.4). Setup 4.13.
Fix an integer ๐ ฤ
10 and let (cid:83) p ๐, ๐ q ฤ (cid:68) for 0 ฤ ๐ ฤ ๐ ยด ๐ ๐ { ๐ ฤ ๐ ฤ ๐ p ๐ ` q{ ๐ .Let ๐ ฤ (cid:68) be a ๏ฌnite union of disjoint disks in (cid:68) such that each of the component disks ๐ท ฤ ๐ is contained in one of the sectors (cid:83) p ๐, ๐ q and such that for every ๐ท ฤ ๐ the disk ๐ ๐ ๐ { ๐ ยจ ๐ท is acomponent disk of ๐ as well. Finally, let ๐ฟ ฤ ๐ท , smaller than the distance between any two of the disks ๐ท and ๐ท , and smaller thanthe distance between ๐ท and the boundary of any of the sectors (cid:83) p ๐, ๐ q .For any subset ๐ ฤ (cid:68) , we use the notation ๐ p ๐ q : โ t ๐ง P (cid:68) || ๐ง ยด ๐ | ฤ ๐ฟ for some ๐ P ๐ u The neighborhoods ๐ pB ๐ท q , ๐ p ๐ท q , ๐ p ๐ q and ๐ pB ๐ q will be of particular importance. We now introduce the two Hamiltonians ๐ป and ๐บ in some detail. Construction 4.14.
We let ๐ป : (cid:68) ร (cid:82) denote the radial Hamiltonian given by the formula(4.14) ๐ป p ๐, ๐ q : โ ๐ p ๐ ` q ๐ ยจ p ยด ๐ q The Hamiltonian vector ๏ฌeld ๐ ๐ป โ ๐ p ๐ ` q ๐ ยจ B ๐ and so the Hamiltonian ๏ฌow is given by(4.15) ๐ ๐ป : (cid:82) ห (cid:68) ร (cid:68) with ๐ ๐ป p ๐ก, ๐ง q โ exp p ๐ p ๐ ` q ๐ ยจ ๐๐ก q ยจ ๐ง In particular, the time 1 ๏ฌow is rotation by ๐ ๐ and preserves the collection ๐ . Construction 4.15.
We let ๐บ : (cid:68) ร (cid:82) denote a Hamiltonian that is invariant under rotation byangle 2 ๐ { ๐ and that vanishes away from ๐ p ๐ q . That is(4.16) ๐บ p ๐ง q โ ๐บ p ๐ ๐ ๐ { ๐ ยจ ๐ง q and ๐บ | (cid:68) z ๐ p ๐ q โ ๐ท be a component disk of ๐ that is centered at ๐ P (cid:68) and with radius ๐ . Thenwe also assume that ๐บ is radial about ๐ in the neighborhood ๐ p ๐ท q of ๐ท , i.e.(4.17) ๐บ | ๐ p ๐ท q โ ๐ ห ๐ ๐ for a function ๐ : r , ๐ ` ๐ฟ s ร (cid:82) Finally, we assume that the function ๐ satis๏ฌes the following conditions.(4.18) ๐ p ๐ q โ ยด ๐ ยจ p ยด ๐ฟ q ยจ p ๐ ยด ๐ q if ๐ ฤ ๐ ยด ๐ฟ (4.19) ๐ ฤ ฤ ๐ ฤ ๐ ยจ p ยด ๐ฟ q ยจ p ๐ ยด ๐ฟ q if ๐ ยด ๐ฟ ฤ ๐ ฤ ๐ ` ๐ฟ Note that (4.18) speci๏ฌes ๐บ on the region ๐ท z ๐ pB ๐ท q and (4.19) speci๏ฌes ๐บ on the region ๐ pB ๐ท q .A crucial fact that we will use later without comment is that ๐ ๐บ and ๐ ๐ป commute as elementsof the universal cover of Ham p (cid:68) , ๐ q . That is ๐ ๐บ โ ๐ ๐ป โ ๐ ๐ป โ ๐ ๐บ and ๐บ ๐ป โ ๐ป ๐บ up to isotopy in ๐ก relative to 0 , ๐ . Lemma 4.16 (Action of ๐ ) . The action map ๐ ๐ : (cid:68) ร (cid:82) and Calabi invariant Cal p (cid:68) , ๐ q satisfy (4.20) ๐ ๐ โ ๐ p ` ๐ q ยด รฟ ๐ท ฤ ๐ area p ๐ท q ยจ ๐ ๐ท ` ๐ p ๐ฟ q on (cid:68) z ๐ pB ๐ q (4.21) ๐ { ฤ ๐ ๐ ฤ ๐ on all of (cid:68) (4.22) Cal p (cid:68) , ๐ q โ ๐ p ` ๐ q ยด รฟ ๐ท ฤ ๐ area p ๐ท q ` ๐ p ๐ฟ q Proof.
Since ๐ ๐บ and ๐ ๐ป commute, we have ๐ ๐บ ห ๐ ๐ป โ ๐ ๐บ and therefore ๐ ๐ โ ๐ ๐บ ห ๐ ๐ป ` ๐ ๐ป โ ๐ ๐บ ` ๐ ๐ป Thus we must compute the action map of ๐บ and ๐ป . First, we note that ๐ป is radial by (4.14). Thuswe apply Lemma 4.11 to see(4.23) ๐ ๐ป โ ๐ p ` ๐ q on all of (cid:68) Next we compute the action map of ๐บ . Let ๐ท be a component disk of ๐ centered at ๐ and ofradius ๐ . We can apply Lemma 4.12 to see that ๐ ๐บ โ ยด ๐ ๐ ` ๐ฟ ยจ pยด ๐ ๐ q ` p ๐ข ๐ ยด r ๐ ๐บ s ห ๐ข ๐ q โ ยด p ๐ท q ` ๐ p ๐ฟ q on ๐ท z ๐ pB ๐ท q D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 27
Here the ๐ข ๐ ยด r ๐ ๐บ s ห ๐ข ๐ is an ๐ p ๐ฟ q term because ๐ ๐บ is a rotation of angle ๐๐ฟ on ๐ท z ๐ pB ๐ท q . Since ๐ ๐บ โ ๐ p ๐ท q , we thus acquire the formula(4.24) ๐ ๐บ โ ยด รฟ ๐ท ฤ ๐ area p ๐ท q ยจ ๐ ๐ท ` ๐ p ๐ฟ q on (cid:68) z ๐ pB ๐ q Adding (4.23) and (4.24) yields the desired formula (4.20) and implies (4.21) away from ๐ pB ๐ q .On the neighborhood ๐ pB ๐ q , we have the formula | ๐ ๐บ | ฤ | ๐ p ๐ ๐ q ยด ๐ p ๐ ๐ q| ` ๐ p ๐ฟ q ฤ ๐ ๐ ` ๐ p ๐ฟ q ฤ ๐ ๐ pB ๐ q By adding this to the formula (4.23) for ๐ ๐ป , we immediately acquire (4.21) on ๐ pB ๐ q . Finally,since ๐ pB ๐ q has area ๐ p ๐ฟ q , the Calabi invariant agrees with the integral of (4.20) over (cid:68) z ๐ pB ๐ q up to an ๐ p ๐ฟ q term. This proves (4.22). (cid:3) Lemma 4.17 (Rotation of ๐ ) . The rotation map ๐ ๐ : (cid:68) ร (cid:82) and the Ruelle invariant Ru p (cid:68) , ๐ q satisfy (4.25) ๐ ๐ โ p ` ๐ q ยด รฟ ๐ท ฤ ๐ ๐ ๐ท ` ๐ p ๐ฟ q on (cid:68) z ๐ pB ๐ q (4.26) ยด ` ๐ ` ๐ฟ ฤ ๐ ๐ ฤ ` ๐ on all of (cid:68) (4.27) Ru p (cid:68) , ๐ q โ ๐ p ` ๐ q ยด รฟ ๐ท ฤ ๐ area p ๐ท q ` ๐ p ๐ฟ q Proof.
In the universal cover of Ham p (cid:68) , ๐ q , the time ๐ ๏ฌow ๐ ๐ of ๐บ ๐ป can be factored in terms ofthe time 1 ๏ฌow ๐ ๐บ : r , s ห (cid:68) ร (cid:68) of ๐บ and the time 1 ๏ฌow ๐ ๐ป : r , s ห (cid:68) ร (cid:68) of ๐ป , as follows. ๐ ๐ โ p ๐ ๐ป โ ๐ ๐บ q ๐ โ ๐ ๐ป โ ๐ ๐บ โ ๐ ๐ป โ ยจ ยจ ยจ โ ๐ ๐ป โ ๐ ๐บ This factorization is inherited by the lifted di๏ฌerential ห ฮฆ : (cid:82) ห (cid:68) ร ฤ Sp p q of ๐ : (cid:82) ห (cid:68) ร (cid:68) dueto the cocycle property of r ฮฆ .(4.28) r ฮฆ p ๐, ๐ง q โ r ฮฆ ๐ป p , ๐ ๐บ ห ๐ ๐ ยด p ๐ง qq โ r ฮฆ ๐บ p , ๐ ๐ ยด p ๐ง qq โ r ฮฆ ๐ป p , ๐ ๐บ ห ๐ ๐ ยด p ๐ง qq โ ยจ ยจ ยจ โ r ฮฆ ๐บ p , ๐ง q To apply this, we note that the di๏ฌerential ฮฆ ๐ป : r , s ห (cid:68) ร Sp p q of the ๏ฌow of ๐ป is given by(4.29) ฮฆ ๐ป p ๐ก, ๐ง q โ exp p ๐ p ` { ๐ q ยจ ๐๐ก q for any ๐ง P (cid:68) Likewise, the di๏ฌerential ฮฆ ๐บ : r , s ห (cid:68) ร Sp p q of the ๏ฌow of ๐บ is given by the formula(4.30) ฮฆ ๐บ p ๐ก, ๐ง q โ exp pยด p ยด ๐ฟ q ๐ ยจ ๐๐ก q if ๐ง P ๐ z ๐ pB ๐ q and ฮฆ ๐บ p ๐ก, ๐ง q โ Id if ๐ง P (cid:68) z ๐ p ๐ท q By combining (4.29) and (4.30) with the decomposition (4.28), we acquire the following formula.(4.31) ๐ ห r ฮฆ p ๐, ๐ง q โ ๐ ยจ ` ` ๐ ยด รฟ ๐ท ฤ ๐ ๐ ๐ท p ๐ง q ` ๐ p ๐ฟ q ห if ๐ง P (cid:68) z ๐ pB ๐ q By dividing (4.31) by ๐ and taking the limit as ๐ ร 8 , we acquire the ๏ฌrst formula (4.25).Next, we examine the rotation number in the region ๐ pB ๐ท q . Fix a component disk ๐ท ฤ ๐ centered at ๐ and a point ๐ง P ๐ pB ๐ท q . Let ๐ ฤ ๐ pB ๐ท q be a circle centered at ๐ with ๐ง P ๐ , and let ๐ข P ๐ ๐ง ๐ be a unit tangent vector to ๐ at ๐ง . Finally, let ๐ ๐ โ ๐ ๐ p ๐ q ๐ง ๐ โ ๐ ๐ p ๐ง q ๐ค ๐ โ ๐ ๐บ ห ๐ ๐ p ๐ง q ๐ข ๐ โ ฮฆ p ๐, ๐ง q ๐ข ๐ฃ ๐ โ ฮฆ ๐บ p , ๐ ๐ p ๐ง qq ฮฆ p ๐, ๐ง q ๐ข Note that these points and vectors satisfy ๐ง ๐ P ๐ ๐ , ๐ค ๐ P ๐ ๐ , ๐ข ๐ P ๐ ๐ง ๐ ๐ ๐ and ๐ฃ ๐ P ๐ ๐ค ๐ ๐ ๐ for each ๐ . Byapplying the decomposition (4.28) and the additivity property (2.7) of ๐ ๐ , we see that(4.32) ๐ ๐ข p r ฮฆ p ๐, ๐ง qq โ ๐ ยด รฟ ๐ โ ๐ ๐ข ๐ p r ฮฆ ๐บ p , ๐ง ๐ qqq ` ๐ ยด รฟ ๐ โ ๐ ๐ฃ ๐ p r ฮฆ ๐ป p , ๐ค ๐ qq Since ๐ ๐ป is just an orthogonal rotation, we can use (4.29) to immediately conclude that(4.33) ๐ ๐ข ๐ p r ฮฆ ๐บ p , ๐ง ๐ qqq โ ` ๐ On the other hand, since ๐ฃ ๐ is tangent to the circle ๐ ๐ , we may use the formula (4.12) to see that(4.34) ๐ ๐ฃ ๐ p r ฮฆ ๐ป p , ๐ง ๐ qqq โ ยด ๐ p ๐ ๐ p ๐ง qq ๐ ๐ ๐ p ๐ง q Here ๐ is the function such that ๐บ | ๐ p ๐ท q โ ๐ ห ๐ ๐ . By our hypotheses, we know that ยด ` ๐ฟ ฤ ยด p ยด ๐ฟ qp ๐ ยด ๐ฟ q ๐ ` ๐ฟ ฤ ยด ๐ p ๐ ๐ p ๐ง qq ๐ ๐ ๐ p ๐ง q ฤ ๐ ๐ข ห r ฮฆ p ๐, ๐ง q as follows. ๐ ยจ pยด ` ๐ ` ๐ฟ q ฤ ๐ ๐ข ห r ฮฆ p ๐, ๐ง q ฤ ๐ ยจ p ` ๐ q We can therefore estimate ๐ ๐ . Since ๐ ๐ข and ๐ are equivalent (Lemma 2.5) we ๏ฌnd that ๐ ๐ p ๐ง q โ lim ๐ ร8 ๐ ๐ข ห r ฮฆ p ๐, ๐ง q ๐ and thus ยด ` ๐ ` ๐ฟ ฤ ๐ ๐ p ๐ง q ฤ ` ๐ Finally, since ๐ pB ๐ q has area ๐ p ๐ฟ q , the Ruelle invariant agrees with the integral of (4.25) over (cid:68) z ๐ pB ๐ q up to an ๐ p ๐ฟ q term. This proves (4.27). (cid:3) Lemma 4.18 (Periodic Points of ๐ ) . The periodic points of ๐ : (cid:68) ร (cid:68) satisfy (4.35) A p ๐ q ฤ ๐ and ๐ p ๐ q ` L p ๐ q ฤ Proof.
First, consider the center ๐ โ P (cid:68) , where ๐ โ ๐ ๐ป . This periodic point has period L p ๐ q โ
1. Thus, due to Lemmas 4.16 and 4.17, the action and rotation number are given by A p ๐ q โ ๐ ๐ p ๐ q โ ๐ p ` ๐ q ๐ p ๐ q โ ๐ ๐ p ๐ q โ ` ๐ Any other periodic point ๐ of ๐ป has period L p ๐ q ฤ ๐ , since ๐ rotates the sector (cid:83) p ๐, ๐ q to thesection (cid:83) p ๐, ๐ ` q . Since ๐ ฤ ๐ ๐ ฤ ๐ { ๐ is lower bounded,as follows. A p ๐ q โ L p ๐ qยด รฟ ๐ โ ๐ ๐ p ๐ ๐ p ๐ qq ฤ ๐ ยจ L p ๐ q ฤ ๐ Likewise, we apply Lemma 4.17 to see that the rotation number of ๐ is lower bounded as follows. ๐ p ๐ q โ L p ๐ q ยจ ๐ ๐ p ๐ q ฤ L p ๐ q ยจ pยด ` ๐ ` ๐ฟ q ฤ ยด L p ๐ q ` ` ๐ฟ In particular, the rotation number satis๏ฌes ๐ p ๐ q ` L p ๐ q ฤ (cid:3) Main Construction.
We conclude this construction by proving Proposition 4.1. The resultwill be an easy consequence of Proposition 4.10 and the properties of the special ๏ฌow ๐ of ยง4.4. Proof. (Proposition 4.1) Let ๐ ฤ
0. Choose an integer ๐ , a union of disks ๐ ฤ (cid:68) and a number ๐ฟ ฤ
0, satisfying the properties of Setup 4.13. Additionally, choose a ๐ ฤ ๐ท ฤ ๐ satisfy(4.36) ๐ ยด ๐ ฤ รฟ ๐ท ฤ ๐ area p ๐ท q ฤ ๐ and area p ๐ท q ฤ ๐๐ Let ๐ : r , s ห (cid:68) ร (cid:68) be the associated family of Hamiltonian di๏ฌeomorphisms from ยง4.4. Bydirect calculation and Lemma 4.16, we know that ๐บ ๐ป โ ๐ p ` ๐ q ยจ p ยด ๐ q near B (cid:68) and ๐ ๐ ฤ D CONVEX CONTACT FORMS AND THE RUELLE INVARIANT 29
Therefore we can associate a contact form ๐ผ on ๐ to ๐ via Construction 4.7. We now show that(a scaling of) this contact form has all of the desired properties.First, by Proposition 4.10(b) and Lemma 4.16, the volume of p ๐ , ๐ผ q is given by the formulavol ` ๐ , ๐ผ ห โ Cal p (cid:68) , ๐ q โ ๐ p ` ๐ q ยด รฟ ๐ท ฤ ๐ area p ๐ท q ` ๐ p ๐ฟ q Thus, by applying the inequalities in (4.36), we acquire the following estimates for the volume. ๐ p ` ๐ q ` ๐ p ๐ฟ q ฤ vol ` ๐ , ๐ผ ห ฤ ๐ p ยด ๐ q ` ๐ p ๐ฟ q Next, by Proposition 4.10(c) and Lemma 4.17, the Ruelle invariant of p ๐ , ๐ผ q satis๏ฌesRu p ๐ , ๐ผ q โ Ru p (cid:68) , ๐ q ` ๐ โ ๐ p ` ๐ q ยด รฟ ๐ท ฤ ๐ area p ๐ท q ` ๐ p ๐ฟ q Again, we can then use the inequalities in (4.36) to acquire estimates for the Ruelle invariant. ๐ ๐ ` ๐ ` ๐ p ๐ฟ q ฤ Ru p ๐ , ๐ผ q ฤ ๐ ๐ ` ๐ p ๐ฟ q Last, by Proposition 4.10(d) the binding ๐ โ ๐ pB (cid:68) q in ๐ has action and rotation number given by A p ๐ q โ ๐ ๐ p ๐ q โ ` ` { ๐ ฤ p ๐ , ๐ผ q other than ๐ satis๏ฌes A p ๐พ q ฤ ๐ ๐ p ๐พ q ฤ ๐ผ is a dynamically convex contact form. To conclude the proof, we now note thatby choosing ๐ฟ and ๐ su๏ฌciently small, and choosing ๐ su๏ฌciently large, we can guarantee thatRu p ๐ , ๐ผ q vol p ๐ , ๐ผ q { ฤ ๐ { ๐ ` ๐ ` ๐ p ๐ฟ q ๐ p ยด ๐ ` ๐ p ๐ฟ qq { ฤ ๐ andsys p ๐, ๐ผ q โ min t A p ๐พ q| ๐พ is an orbit of ๐ผ u vol p ๐ , ๐ผ q ฤ ๐ ๐ p ` { ๐ ` ๐ p ๐ฟ qq ฤ ยด ๐ By scaling ๐ผ so that vol p ๐, ๐ผ q โ
1, we arrive at a contact form satisfying all of the properties ofProposition 4.1. This ๏ฌnishes the proof and the main construction of this section. (cid:3)
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Email address : [email protected] Department of Mathematics, University of California at Berkeley, Berkeley, CA, 94720, USA
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