aa r X i v : . [ m a t h . S G ] M a y A DYNAMICAL CONSTRUCTION OF LIOUVILLE DOMAINS
YANG HUANGA
BSTRACT . We first present a general construction of Liouville domains as partial mapping tori. Then westudy two examples where the (partial) monodromies exhibit certain hyperbolic behavior in the sense of Dy-namical Systems. The first example is based on Smale’s attractor, a.k.a., solenoid; and the second example isbased on certain hyperbolic toral automorphisms. A Liouville domain ( W n , ω, X ) is a triple where W is a compact manifold with (nonempty) boundary, ω is a symplectic form and X is a vector field, called the Liouville vector field , such that L X ω = ω and X isoutward pointing along ∂W . Define the Liouville form λ := i X ω . Then by Cartan’s formula, the symplecticform ω = dλ is exact. A Liouville domain ( W, ω, X ) is called Weinstein if, in addition, X is gradient-likewith respect to a Morse function on W .It turns out that the Weinstein condition is rather restrictive on the topology of W . Indeed, any Weinsteindomain admits a handle decomposition which contains only handles of index at most n . On the otherhand, since the first example of McDuff [McD91], it has been known that general Liouville manifolds arenot subject to such topological constraints. See [Gei95, Mit95, Gei94, MNW13] for more constructions ofLiouville, but not Weinstein, domains. Unfortunately, no good methods are currently available to distinguishbetween Liouville and Weinstein structures besides the obvious topological distinction.The goal of this note is produce a few (exotic) examples of Liouville domains where the dynamics of theLiouville vector field X can be explicitly described. It turns out that the dynamics of X is only interestingwhen restricted to the skeleton of ( W, ω, X ) which we now introduce.Given a Liouville domain ( W, ω, X ) , the skeleton Sk(
W, ω, X ) is defined by Sk(
W, ω, X ) := \ t> φ X − t ( W ) , where φ Xt denotes the time- t flow of X . Clearly Sk(
W, ω, X ) contains all the information of the symplecticstructure, but it is, in general, not invariant under Liouville homotopies. It is a very interesting problem tounderstand how Sk(
W, ω, X ) and Sk(
W, ω, X ′ ) are related to each other if X, X ′ are two different Liouvillevector fields on the same symplectic manifold. In the case of Weinstein domains, an on-going project ofAlvarez-Gavela, Eliashberg and Nadler [AGEN] aims at simplifying Sk(
W, ω, X ) , up to Weinstein homo-topy, such that it contains only arboreal singularities introduced by Nadler [Nad17]. In the following wesometimes simply write Sk( W ) for the skeleton if there is no risk of confusion.Now let’s present the main construction of this note: Liouville domains as partial mapping tori . Let M n − be a compact manifold with boundary and α be a contact form on M . For the rest of this note, everycontact manifold comes with a chosen contact form.
Definition 0.1.
A compact contact manifold ( M, α ) admits a contraction if there exists a map φ : M → M ,which satisfies the following properties:(D1) φ ( M ) ⊂ int( M ) ;(D2) φ is a diffeomorphism onto its image;(D3) φ ∗ ( α ) = e − g α , where g : M → R > is a positive function. It is not enough, however, to merely know, say, the topological type of
Sk(
W, ω, X ) whose embedding in W can be very wild. Note that (D3) implies ∂M = ∅ for volume considerations. We start with a not so interesting example. Example . Let Y be a closed manifold and consider the 1-jet space J Y equipped with the standardcontact form α = dz − pdq . Let M ⊂ J Y be a closed tubular neighborhood of the 0-section Y ⊂ J Y .Then the map φ : M → M defined by φ ( z, q, p ) = ( z/ , q, p/ is clearly a contraction. In this example, φ ( M ) is a deformation retract of M , but this is not necessarily the case in general.We construct a Liouville domain W ( M,φ ) , as a partial mapping torus, in three steps as follows. Firstly,let R × M be the symplectization of ( M, α ) with Liouville form λ = e s α , where s ∈ R ; secondly, let G : M → R > be a smooth extension of the function g ◦ φ − : φ ( M ) → R > ; finally, define the partialmapping torus W ( M,φ ) := { ( s, x ) ∈ R × M | ≤ s ≤ G ( x ) } (cid:14) (0 , x ) ∼ ( G ( x ) , φ ( x )) . We claim that λ descends to a 1-form on W ( M,φ ) . Indeed, define Φ : R × M → R × M by Φ( s, x ) :=( s + G ( x ) , φ ( x )) , then Φ ∗ ( λ ) = e s + G ◦ φ φ ∗ α = e s + g e − g α = λ as desired. Abusing notations, we also write λ for the descendent 1-form on W ( M,φ ) .Define the vertical boundary of W ( M,φ ) by ∂ v W ( M,φ ) := { ( s, x ) ∈ R × ∂M | ≤ s ≤ G ( x ) } , and the horizontal boundary ∂ h W ( M,φ ) as the closure of ∂W M,φ \ ∂ v W M,φ . It follows from the constructionthat the Liouville vector field X is outward-pointing along ∂ h W ( M,φ ) and is tangent to ∂ v W ( M,φ ) . By slightlytilting ∂ v W ( M,φ ) , we can assume that X is everywhere outward-pointing along ∂W ( M,φ ) . More precisely,let ∂M × [ − ǫ, ⊂ M \ φ ( M ) be a collar neighborhood of the boundary such that ∂M is identified with ∂M × { } , where ǫ > is small. Suppose, without loss of generality, that G is constant on ∂M × [ − ǫ, .Define P ⊂ [0 , G ] × ( ∂M × [ − ǫ, by P := { ( s, x ) ∈ [0 , G ] × ( ∂M × [ − ǫ, | − ǫs/G < τ ( x ) ≤ } , where τ : ∂M × [ − ǫ, → [ − ǫ, denotes the projection. Then by construction W ( M,φ ) \ P has a piecewisesmooth boundary which is everywhere transverse to the Liouville vector field X = ∂ s . Finally, one caneasily round the corners on the boundary of W ( M,φ ) \ P as shown in Figure 1 to result in a smooth Liouvilledomain. By abuse of notation, we denote the resulting Liouville domain, again, by W ( M,φ ) . Of course as asymplectic manifold W ( M,φ ) depends on various parameters involved in the construction above, but W ( M,φ ) is a well-defined Liouville domain up to deformation, i.e., Liouville homotopy. In what follows we will notdistinguish between Liouville domains which are deformation equivalent.F IGURE
1. Round the corners on W ( M,φ ) \ P .We conclude our general construction with an obvious lemma which describes the skeleton of W ( M,φ ) . DYNAMICAL CONSTRUCTION OF LIOUVILLE DOMAINS 3
Lemma 0.3.
The skeleton of W ( M,φ ) is given by the mapping torus Sk( W ( M,φ ) ) = { ( s, x ) ∈ R × K | ≤ s ≤ G ( x ) } / (0 , x ) ∼ ( G ( x ) , φ ( x )) , where K := T i ≥ φ i ( M ) ⊂ M . It is clear that if the input ( M, φ ) is as in Example 0.2, then the resulting W ( M,φ ) is Weinstein and theskeleton is a smooth Lagrangian submanifold S × Y . In the following, we give two explicit examples of ( M, φ ) of very different nature, such that the resulting Liouville domains W ( M,φ ) are “more interesting”. Example . Consider M = S × D equipped with thecontact form α = dx + ydθ , where θ ∈ S and ( x, y ) ∈ D ⊂ R . Define φ : M → M by φ ( θ, x, y ) = (cid:0) θ, x + cos θ, (cid:0) y + sin θ (cid:1)(cid:1) . Clearly φ satisfies (D1)–(D2). But φ ( M ) is not a deformation retract of M . Instead, it winds around M twice along the S -factor. It is straightforward to compute that φ ∗ α = α , and therefore φ satisfies (D3).Hence φ is a contraction and we have a well-defined Liouville domain W SA := W ( M,φ ) .Let’s examine the skeleton Sk( W SA ) . First observe that K = T i ≥ φ i ( M ) is itself a mapping torus of aCantor set. Namely, for each fixed θ ∈ S , the intersection K ∩ ( { θ } × D ) ⊂ D is a Cantor set. Such K is known as a solenoid in Dynamical Systems, and is a hyperbolic attractor . In particular, it is stable under C ∞ -small perturbations. We refer the interested readers to the comprehensive monograph [KH95] for moredetails. In light of Lemma 0.3, Sk( W SA ) is a mapping torus of the solenoid K .It follows from the construction that, as a smooth -manifold, W SA can be built by handles of index atmost . On the other hand, the Hausdorff dimension of Sk( W SA ) is strictly greater than . Hence it is a veryintriguing question to ask whether W SA is Liouville homotopic to a Weinstein domain. More generally, onecan ask whether W SA × T ∗ Y is Liouville homotopic to Weinstein for any smooth manifold Y . Example . The updates presented here were established soon after the firstappearance of this note on arXiv in October 2019.First, let’s generalize Example 0.4 as follows. Let Y be a compact manifold without boundary and M ⊂ J Y be a closed tubular neighborhood of the -section. Then M inherits the standard contact form α = dz − pdq on J Y . Denote ξ = ker α .Suppose Λ ⊂ ( M, ξ ) is a Legendrian submanifold which is diffeomorphic to Y . Weinstein’s neighbor-hood theorem implies that a sufficiently small closed tubular neighborhood N ǫ (Λ) of Λ is contactomorphicto ( M, ξ ) . Indeed, it is easy to see that if ǫ is small, then there exists φ : M ∼ −→ N ǫ (Λ) ⊂ int( M ) such that φ ∗ α = e − g α for some g : M → R > . In other words, such φ is a contraction in the sense ofDefinition 0.1. Hence we have the associated Liouville domain W ( Y, Λ) := W ( M,φ ) . Clearly Example 0.4 isa special case of this construction where both Y and Λ are circles.Next, we make the following observation: all such W ( Y, Λ) are Weinstein up to homotopy . This answersmy own question above in a somewhat disappointing way. Indeed, according to the Creation Lemma dis-cussed in [HH, § box-fold based on, say, φ ( M ) to make W ( Y, Λ) Weinstein. Forexample, the Liouville domain W SA constructed in Example 0.4 is homotopic to a Weinstein domain whichcan be built by one -handle, two -handles and one -handle. In fact, it seems most likely that there existsno nontrivial upper bound on the dimension of Sk( W ) , i.e., other than dim Sk( W ) < dim W , in generalunder the assumption that W is homotopic to a Weinstein domain. Strictly speaking, one needs to slightly tilt the vertical sides of the box-fold since Liouville homotopies correspond to graphicalperturbations in the contactization.
YANG HUANG
However, since no details of the argument are given here, we formulate our observation as a conjectureas follows.
Conjecture . The Liouville structure on W ( Y, Λ) , in particular W SA from Example 0.4, is homotopic to aWeinstein structure. Finally, we present a variation of Example 0.4. Namely, instead of considering the neighborhood of aLegendrian knot as in Example 0.4, let M = S × D be a neighborhood of a transverse knot, equippedwith a contact form α = dθ − ydx , where θ ∈ S and ( x, y ) ∈ D . Up to rescaling, we can identify S = R / Z so that the length of the transverse knot S × { } is one.Let Λ ⊂ int( M ) be a Legendrian knot. Then there exist coordinates ¯ θ, ¯ x, ¯ y in an ǫ -neighborhood N ǫ (Λ) ∼ = R / Z × D ( ǫ ) of Λ ∼ = R / Z × { } such that α | N ǫ (Λ) = d ¯ x + ¯ yd ¯ θ . Fix < δ ≪ c ≪ ǫ tobe specified later. Define K := R / Z × { (0 , δ ) } to be a transverse push-off of Λ in N ǫ (Λ) . Consider thefollowing diffeomorphism ψ : R / Z × D → R / Z × D ( ǫ )( θ, x, y ) (cid:16) θ − cxδ , cx, cδc − δy (cid:17) onto its image, which sends S × { } to K . We compute ψ ∗ α = cdx + cδc − δy ( dθ − cδ dx ) = cδc − δy α. It follows that ψ is a contraction in the sense of Definition 0.1 if c ≫ δ .The resulting Liouville domain W ( M,ψ ) is also homotopic to a Weinstein domain since K is sufficientlyclose to a Legendrian. This can be considered as a reminiscent of a general principle that transverse knottheory is a stabilized version of Legendrian knot theory. The details of the argument, again, will be omitted. Example . Let T n = R n / Z n be the n -dimensional torus and M = D n − × T n where D n − ⊂ R n − denotes the unit ball. Suppose A ∈ SL ( n, Z ) has real eigenvalues λ , · · · , λ n such that < λ n < | λ i | for all ≤ i ≤ n − . In particular < λ n < . The existence of such A will be establishedin the Appendix. View A as an automorphism of T n . Then there exists linear 1-forms β i , ≤ i ≤ n , on T n such that A ∗ ( β i ) = λ i β i . In particular β i , ≤ i ≤ n , are linearly independent. Define the contact form α := β n + X ≤ i ≤ n − y i β i on M , where ( y , · · · , y n − ) ∈ D n − . The map φ A : M → M defined by φ A ( y , · · · , y n − , x ) = (cid:16) λ n λ y , · · · , λ n λ n − y n − , Ax (cid:17) is clearly a contraction. Indeed, we have φ ∗ A ( α ) = λ n α . We denote the resulting Liouville domain by W A .Then the skeleton Sk( W A ) is a smooth ( n + 1) -manifold given by the mapping torus of A : T n → T n . For n = 2 , we recover the examples of Mitsumatsu [Mit95].An answer to the following question is desirable, but unfortunately is unknown to the author. Question . Is W A stably homotopic to a Weinstein structure, i.e., is W A × R m Weinstein for m ≫ ? It was pointed out to the author by G. Dimitroglou-Rizell that W A × R , being diffeomorphic to thecotangent bundle of Sk( W A ) , cannot be symplectomorphic to the cotangent bundle since everything in W A × R is displaceable due to the R -factor. Remark . In a recent preprint of Eliashberg, Ogawa and Yoshiyasu [EOY], the authors proved that everysufficiently stabilized Liouville manifold is symplectomorphic to a (flexible) Weinstein manifold. How-ever, the symplectomorphism constructed therein is in general not compactly supported. In particular, the
DYNAMICAL CONSTRUCTION OF LIOUVILLE DOMAINS 5 contact boundary at infinity is not preserved under such construction. It is still unknown, to the best ofmy knowledge, that whether a sufficiently stabilized Liouville manifold is necessarily compactly supporteddeformation equivalent to a Weinstein manifold.A
PPENDIX
A. T
ORAL AUTOMORPHISM WITH REAL SPECTRUM
The goal of this appendix is to construct hyperbolic toral automorphisms with real spectrum. The materialpresented here came from a joyful discussion with Luis Diogo, who deserves every bit of this wonderful (butmaybe trivial) result, in the summer of 2019 in Uppsala.
Proposition A.1.
Fix n ≥ . For any ǫ > and any tuple ( µ , . . . , µ n − ) ∈ R n − , there exists a matrix A ∈ SL ( n, Z ) which is diagonalizable in SL ( n, R ) such that the eigenvalues λ i , ≤ i ≤ n , satisfy(1) | λ i − µ i | < ǫ for ≤ i ≤ n − .(2) | λ n − | > /ǫ and | λ n | < ǫ .Proof. Using the Frobenius companion matrix, it suffices to find infinitely many tuples k := ( k , . . . , k n − ) ∈ Z n − such that the polynomial P k := x n − k n − x n − + · · · + ( − n − k x + ( − n has n roots λ . . . λ n which satisfy the conditions in the Proposition. Indeed P k is the characteristic polyno-mial of the matrix A k := · · · − n +1 · · · − n k · · · − n − k ... ... . . . ... ... · · · k n − ∈ SL ( n, Z ) , which is exactly what we look for. Let σ ( n ) i = σ ( n ) i ( λ , · · · , λ n ) , ≤ i ≤ n , be the i -th elementarysymmetric polynomial in n variables, i.e., σ ( n ) i := X ≤ j < ···
Here σ ( n − n − ≡ and σ ( n − ≡ by convention. Plugging Eq. (2) and Eq. (4) into Eq. (3), we have(5) σ ( n − j + ( k − σ ( n − ) σ ( n − j − + σ ( n − j − (cid:14) σ ( n − n − = k j , ≤ j ≤ n − , which is a system of equations without λ n − and λ n . Clearly λ n − , λ n can be solved easily from Eq. (2)and Eq. (4) once we determine the values of λ i , ≤ i ≤ n − .S TEP Passing to a discrete dynamical system.
The idea is that for any fixed λ , · · · , λ n − , we can view the left-hand side of Eq. (5) as a discretedynamical system as k runs through the integers, while the right-hand side ( k , · · · , k n − ) ∈ Z n − forms alattice in R n − . Then the existence of approximate solutions to Eq. (5) is, roughly speaking, a consequenceof the ergodicity of such dynamical system.The technical heart of this argument is a theorem due to Weyl and von Neumann which we now recall.See [Sin76, Lect. 3] for an excellent exposition on this topic. Let T m = R m / Z m be an m -dimensionaltorus. Fix a vector r = ( r , · · · , r m ) ∈ R m . Define the translation τ r : T m → T m by τ ([ x ]) = [ x + r ] . Theorem A.2 (Weyl-von Neumann) . The translation τ r is ergodic if and only if r is irrational , i.e., thecomponents of r are linearly independent over Z . In order to apply Theorem A.2 to our case, choose λ i , ≤ i ≤ n − , such that the vector r := ( σ ( n − , · · · , σ ( n − n − ) ∈ R n − is irrational. For example, it suffices to choose λ i , ≤ i ≤ n − , to be algebraically independent. Inparticular λ i = λ j whenever i = j . Let’s rewrite Eq. (5) as(6) x ( r ) + k r = k ′ , where k ′ := ( k , · · · , k n − ) ∈ Z n − and x ( r ) := ( x , · · · , x n − ) ∈ R n − with x j := σ ( n − j − σ ( n − σ ( n − j − + σ ( n − j − (cid:14) σ ( n − n − , for ≤ j ≤ n − . It follows from Theorem A.2 that for any ǫ > and K > , there exists k > K and k ′ ∈ Z n − such that | x + k r − k ′ | < ǫ . Switching point of view, one can think of the prescribed tuple ( λ , · · · , λ n − ) as an approximate solution to Eq. (5) with suitable choices of k i , ≤ i ≤ n − .S TEP From approximate solutions to exact solutions.
Observe that the tuple ( λ , · · · , λ n − ) uniquely determines the vector r which approximately solvesEq. (6) with k > K . By choosing K sufficiently large, there exists an exact solution r ′ of Eq. (6) which isclose to r . It remains to argue that r ′ corresponds to a tuple ( λ ′ , · · · , λ ′ n − ) which is close to ( λ , · · · , λ n − ) .Indeed, consider the map Π : R n − → R n − defined by Π( λ , · · · , λ n − ) = ( σ ( n − , · · · , σ ( n − n − ) . The Jacobian Jac (Π) = 0 if λ i , ≤ i ≤ n − , are algebraically independent. By the Inverse Func-tion Theorem, Π − ( r ′ ) exists and is close to ( λ , · · · , λ n − ) . Abusing notations, let us write Π − ( r ′ ) =( λ , · · · , λ n − ) .To wrap up the proof, let us rewrite Eq. (2) and Eq. (4) as follows:(7) λ n − + λ n = k − σ ( n − , λ n − λ n = 1 (cid:14) σ ( n − n − . Since both σ ( n − and σ ( n − n − = 0 are finite numbers, Eq. (7) admits a solution ( λ n − , λ n ) with | λ n − | > /ǫ and | λ n | < ǫ as long as k > K is sufficiently large. (cid:3) DYNAMICAL CONSTRUCTION OF LIOUVILLE DOMAINS 7
Acknowledgements.
The author is grateful to Ko Honda for many years of collaboration which essentiallyshaped his understanding of contact and symplectic structures. He also wants to thank his friends andcolleagues at Uppsala and Nantes for their interest and curiosity in this work. Finally, the author thanks ananonymous referee for his/her comments on the first draft.R
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