aa r X i v : . [ m a t h . S G ] F e b Higher P -symmetric Ekeland-Hofer capacities Kun Shi and Guangcun Lu ∗ February 1, 2021
Abstract
This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplecticcapacities for P -symmetric subsets in the standard symplectic space ( R n , ω ), which is moti-vated by Long and Dong’s study P -symmetric closed characteristics on P -symmetric convexbodies. We study the relationship between these capacities and other capacities, and givesome computation examples. Moreover, we also define higher real symmetric Ekeland-Hofercapacities as a complement of Jin and the second named author’s recent study of the realsymmetric analogue about the first Ekeland-Hofer capacity. Motivated by studies of closed characteristics, Ekeland and Hofer [6, 7] introduced the conceptof the symplectic capacities and constructed a sequence of symplectic capacities c jEH , nowdayscalled the Ekeland-Hofer (symplectic) capacities. They are symplectic invariants for subsetsin the standard symplectic space ( R n , ω ), and play important actions in symplectic topologyand Hamiltonian dynamics. Recently, in [11, 12, 13] Rongrong Jin and the second namedauthor gave several generalizations of the first Ekeland-Hofer capacity c EH and studied theirproperties and applications.Our purpose of this paper is to define analogues of higher Ekeland-Hofer symplectic ca-pacities for subsets with some kind of symmetric in ( R n , ω ). Without special statements,we always use J to denote standard complex structure on R n , and h· , ·i R n standard in-ner product on R n . Define P = diag( − I n − κ , I κ , − I n − κ , I κ ) for some integer κ ∈ [0 , n ). Asubset A ⊂ R n is said to P - symmetric if P A = A , that is, x ∈ A implies P x ∈ A . Let B ( R n ) = { B ⊂ R n | P B = B & B ∩ Fix( P ) = ∅} .Recall that the distinguished line bundle of a compact smooth connected hypersurface S inthe standard symplectic space ( R n , ω ) is defined by L S = { ( x, ξ ) ∈ T S | ω x ( ξ, η ) = 0 , ∀ η ∈ T x S} . A C embedding z : R /T Z → S is called a closed characteristic on S if ˙ z ( t ) ∈ ( L S ) z ( t ) for all t ∈ R /T Z . When the hypersurface S is P -symmetric, such a closed characteristic z on S is saidto be P - symmetric if it also satisfies z ( t + T ) = P z ( t ) for all t ∈ R /T Z . Similarly, if S is onlythe boundary of convex body D in ( R n , ω ), we call a nonconstant absolutely continuous curve x : [0 , T ] → ∂D (for some T >
0) to be a generalized closed characteristic on S if x (0) = x ( T )and ˙ x ( t ) = J N S ( x ( t )) , a.e., where N S ( x ) = { y ∈ R n | h u − x, y i R n , ∀ u ∈ D } ; when D is also P -symmetric, the generalized closed characteristic x on S is said to be P -symmetric if ∗ Corresponding author2020
Mathematics Subject Classification. ( t + T ) = P x ( t ) for all x ∈ [0 , T ]. As usual the action of a (generalized) closed characteristic x is defined by A ( x ) = 12 Z T h− J ˙ x, x i dt. Let us writeΣ S = { k A ( x ) > | x is a closed characteristic on S , k ∈ N } , Σ P S = { k A ( x ) > | x is a P -symmetric closed characteristic on S , k ∈ N } . Hereafter N always denotes the set of all positive integers. They are called the action spectrum and the P -symmetric action spectrum of S or D , respectively.Dong and Long [5] first studied the existence of P -symmetric closed characteristics onboundaries of P -symmetric convex bodies in R n . By considering such closed characteristicswe define analogues of higher Ekeland-Hofer symplectic capacities, c jP : B ( R n ) → [0 , + ∞ ] , j = 1 , , · · · , which are called P - symmetric higher Ekeland-Hofer symplectic capacities . As in [6, 7] the fol-lowing proposition can be proved easily. Proposition 1.1. ∀ j ∈ N , c jP has the following property: (i)(Monotonicity) If B , B ∈ B ( R n ) and B ⊂ B , then c jP ( B ) c jP ( B ) ; (ii)(Conformality) c jP ( λB ) = λ c jP ( B ) , ∀ λ ∈ R , ∀ B ∈ B ( R n ) ; (iii) c jP ( h ( B )) = c jP ( B ) for each P -equivariant h ∈ Symp( R n ) . (iv) c jP : B ( R n ) → R is continuous with respected to Hausdroff distance on B ( R n ) . For the first P -Ekeland-Hofer symplectic capacity c P , we have: Theorem 1.2.
For a P -symmetric convex bounded domain D ⊂ R n with C , boundary S = ∂D , if it contains a fixed point of P , then c P ( D ) = c P ( S ) = min Σ P S . As arguments below [12, Remark 1.12] the condition “ C , ” for S can be removed if wereplace min Σ P S by min {A ( x ) > | x is a P -symmetric generalized characteristic on S} .A compact smooth connected hypersurface S in R n is said to be of restricted contact type if there exists a Liouville vector field η on R n such that L η ω = ω and η points transverselyoutward at S . Denote B S by the bounded component of R n \ S , and by L ( R n ) = { B S ⊂ B ( R n ) | S is a hypersurface of restricted contact type } . Proposition 1.1(i) implies that each c jP ( B S ) is always finite. Theorem 1.3. c jP ( S ) = c jP ( B S ) ∈ Σ P S . The following theorem gives relationships between the Ekeland-Hofer symplectic capacities,the generalized Ekeland-Hofer-Zehnder symplectic capacities and the P -symmetric Ekeland-Hofer symplectic capacities with P = diag( − I n − κ , I κ , − I n − κ , I κ ) for some integer κ ∈ [0 , n ]. Theorem 1.4. If D ∈ B ( R n ) is a bounded convex domain, then c P ( D ) = c EH ( D ) for κ = 0 , (1.1) c P ( D ) = 2 c P EHZ ( D ) for ≤ κ < n, (1.2) where c P EHZ is the generalized Ekeland-Hofer-Zehnder symplectic capacities defined in [11]. f κ >
0, it is possible that (1.1) fails. See an counterexample in Remark 1.6(ii) below.
Proof of Theorem 1.4.
By an approximation as in [12, Section 4.3] we can assume that D isstrictly convex and has a C ∞ boundary S . Then c EH ( D ) = min {A ( x ) > | x is a closed characteristic on S} . This and Theorem 1.2 lead to c EH ( D ) ≤ c P ( D ).If κ = 0, then D = − D . By [1, Corollary 2.2] any closed characteristic of minimal actionon the boundary S is itself centrally symmetric. Hence c EH ( D ) = min {A ( x ) > | x is a centrally symmetric closed characteristic on S} . Moreover, in the present case a closed characteristics on S is P -symmetric if and only if it iscentral symmetric. It follows from these and Theorem 1.2 that c P ( D ) = c EH ( D ).Let 1 κ < n and let x : [0 , T ] → S be a P -symmetric closed characteristics on S . Since x ( t + T ) = P x ( t ), y := x | [0 , T ] is a P -characteristic on S in the sense of [11, Definition 1.1]and A ( x ) = 2 A ( y ). Conversely, if y : [0 , T ] → S is a P -characteristic on S , then x ( t ) = (cid:26) y ( t ) , if t ∈ [0 , T ] ,y ( t − T ) , if t ∈ [ T, T ]is a P -symmetric closed characteristics on S , and A ( x ) = 2 A ( y ). Thus by these two claims,and [11, Theorem 1.9] and Theorem 1.2, we deduce that c P ( D ) ≤ c P EHZ ( D ) and c P ( D ) ≥ c P EHZ ( D ), respectively.In the following we consider computations of the P -Ekeland-Hofer capacities of ellipsoidsand polydiscs. Let r = ( r , · · · , r n ) with r i > i = 1 , · · · , n . We call sets E ( r ) := { z ∈ R n | n X i =1 x i + y i r i < } and D ( r ) := B ( r ) × · · · × B ( r n )the ellipsoid and the polydisc of radius r , respectively. Define a set σ P ( r ) = { mπr j | m ∈ N + 1 , j = 1 , · · · , n − κ } [ { mπr j | m ∈ N , j = n − κ + 1 , · · · , n } and a map φ : N × { , · · · , n } → σ P ( r ) , ( k, j ) kr j . The multiplicity of d ∈ σ P ( r ) is defined by m ( d ) := ♯φ − ( d ) . From σ P ( r ) we construct a nondecreasing sequence of numbers { d i } i , where d i = d i ( σ P ( r )) , i = 1 , , · · · , such that each d ∈ σ P ( r ) is repeated m ( d ) times. Theorem 1.5. c jP ( E ( r )) = d j ( σ P ( r )) for each j ∈ N . For r = (1 , · · · , c jP ( B n (1)) = π if j = 1 , · · · , n − κ, π if j = n − κ + 1 , · · · , n, π if j = n + 1 , · · · , n − κ, π if j = 2 n − κ + 1 , · · · , n, · · · emark 1.6. For r = ( r , · · · , r n ), let ¯ r = ( r n − κ +1 , r , · · · , r n − κ , r , r n − κ +2 , · · · , r n ). (i) If r = r n − κ +1 , then the corresponding sequence { ¯ d i } i to σ P (¯ r ) is different from thesequence { d i } i of σ P ( r ) though it is possible that sets σ P (¯ r ) and σ P ( r ) coincide. HenceTheorem 1.5 implies that there is no P -equivariant symplectic diffeomorphisms from E ( r ) to E (¯ r ). (ii) If κ > { r , · · · , r n − κ } > { r n − κ +1 , · · · , r n } , we can directly compute c P ( E ( r )) = 2 min { r n − κ +1 , · · · , r n } and c EH ( E ( r )) = min { r n − κ +1 , · · · , r n } by Theorem 1.5 and [7, Proposition 4], respectively, and so c P ( E ( r )) = c EH ( E ( r )).For r = ( r , · · · , r n ), defineˆ r = min { r i | i = 1 , · · · , n − κ } and r ′ = min { r i | i = n − κ + 1 , · · · , n } . Let σ ′ P ( r ) = { (2 m − π ˆ r , mπr ′ | m ∈ N } . By a similar construction to { d i ( σ P ( r )) } i , wecan get a sequence of numbers { d i ( σ ′ P ( r )) } i . Theorem 1.7. c jP ( D ( r )) = d j ( σ ′ P ( r )) . Since E ( r ) ⊂ D ( r ), c jP ( D ( r )) > c jP ( E ( r )) > j = 1, noting thatmin σ P ( r ) = π min { ˆ r , r ′ } and min σ ′ P ( r ) = π min { ˆ r , r ′ } by the definitions of σ P ( r ) and σ ′ P ( r ), we have c P ( D ( r )) = c P ( E ( r )).Consider the Lagrangian bidisk in ( R ( x , x , y , y ) , ω ) with ω = P i =1 dx i ∧ dy i ([14]), D × L D = { ( x , x , y , y ) ∈ R | x + x < , y + y < } By [3, Proposition 2.2] the action sprectrum of ∂ ( D × L D ) isΣ ∂ ( D × L D ) = { n cos( θ k,n ) | k, n ∈ N , θ k,n ∈ J n } ∪ { nπ | n ∈ N } where J n = { (2 k − π/ n | ≤ k ≤ ( n − / } if n is odd, and J n = { kπ/n | ≤ k ≤ n/ − } if n is even. In general, it is difficult to determinate Σ P∂ ( D × L D ) . It was shown that c EH ( D × L D ) = 4 ([2, 3]) and c EH ( D × L D ) = 3 √ c EH ( D × L D ) = 8 ([14, 3]).By (1.1) c P ( D × L D ) = c EH ( D × L D ) = 4 if P = diag( − , − , − , − P we have: Theorem 1.8. If P = diag( − , , − , , then c P ( D × L D ) ∈ { π, } and thus there existsa P -symmetric generalized closed characteristic x with action A ( x ) ∈ { π, } on the ∂ ( D × L D ) . But for c P ( D × L D ) we can only get the estimate: c P ( D × L D ) ∈ [4 , π √ −
45 ) ∩ Σ ∂ ( D × L D ) . A real symplectic manifold is a triple (
M, ω, τ ) consisting of a symplectic manifold (
M, ω )and an anti-symplectic involution τ on ( M, ω ), i.e. τ ∗ ω = − ω and τ = id M . The standardsymplectic space ( R n , ω ) is real with respect to the canonical involution τ : R n → R n given by τ ( x, y ) = ( x, − y ). For τ -invariant subsets in ( R n , ω ) Jin and the second namedauthor defined a symmetrical version of the first Ekeland-Hofer capacity of such subsets in [12].In Section 7, we will also give symmetrical versions of the higher Ekeland-Hofer capacities ofsuch subsets.The paper is organized in the following way. In Section 2 we first present our variationalframe and then give variational definitions of the higher P -symmetric Ekeland-Hofer capacities.Section 5 proves Theorem 1.5 and Theorem 1.7. In Sections 3, 4 and 6 we give proofs ofTheorems 1.2, 1.3 and 1.8, respectively. Variational definitions of higher P -symmetric Ekeland-Hofer capacities Our method follows [7] basically. The variational frame is slight modifications of that of [7].For clearness and completeness we state necessary definitions and results.
Let S = R /Z and let X P = { x ∈ L ( S ; R n ) | x ( t + ) = P x ( t ) a.e.t ∈ R } . Then x ∈ L ( S ; R n ) sits in X P if and only if coefficients of its Fourier series x = P j ∈ Z e πjtJ x j satisfyconditions: P x j = − x j for all j ∈ Z + 1, and P x j = x j for j ∈ Z . For s ≥
0, consider theHilbert space E s = x ∈ L ( S ; R n ) (cid:12)(cid:12)(cid:12) x = X j ∈ Z e πjtJ x j , x j ∈ R n , X j ∈ Z | j | s | x j | < ∞ with inner product and associated norm given by h x, y i s = h x , y i R n + 2 π X j ∈ Z | j | s | x j | , k x k s = h x, x i s (2.1)([7]). Then E sP := E s ∩ X P is a closed subspace of E s . Throughout, we write E := E / and E P := E ∩ X P . (2.2)There exists an orthogonal splitting E P = E − P ⊕ E P ⊕ E + P , where E P = Fix( P ) ≡ R k and E − P = { x ∈ E | x = X j< x j e πjtJ } , E + P = { x ∈ E | x = X j> x j e πjtJ } . Denote P + , P − , P by the orthogonal projections onto E + P , E − P , E P , respectively. Then every x ∈ E P has the unique decomposition x = x − + x + x + , where x + = P + x, x = P x and x − = P − x . Define the functional A : E P → R by A ( x ) = 12 k x + k − k x − k if x = x + + x + x − . It is easy to prove that ∇A ( x ) = x + − x − ∈ E P and A ( x ) = 12 Z h− J ˙ x, x i R n dt, ∀ x ∈ C ( S , R n ) ∩ E P . By Propositions 3,4 on pages 84-85 of [10] we immediately obtain:
Proposition 2.1. (i) If t > s > , the inclusion map I t,s : E tP → E sP is compact. (ii) Assume s > . Then each x ∈ E sP is continuous and satisfies x ( t + ) = P x ( t ) for all t ∈ R . Moreover, there exists a constant c > such that sup t ∈ S | x ( t ) | c k x k s , ∀ x ∈ E sP . Let j : E P → L ( S ; R n ) be the inclusion map, and j ∗ : L ( S ; R n ) → E P be the adjointoperator of j defined by h j ( x ) , y i L = h x, j ∗ ( y ) i , ∀ x ∈ E P , y ∈ L ( S ; R n ) . As in the proof of Proposition 5 on page 86 of [10] we have roposition 2.2. For y ∈ L , j ∗ ( y ) ∈ E P and satisfies k j ∗ ( y ) k k y k L . Consequently, j ∗ is a compact operator. Let ˆ H ( R n ) be the set of nonnegative smooth function H ∈ C ∞ ( R n ) satisfying the fol-lowing condition: (H1) H ( z ) = H ( P z ) , ∀ z ∈ R n ; (H2) There is an open set U ⊂ R n such that H | U ≡ (H3) When | z | is large enough, H ( z ) = a | z | , where a ∈ ( π, ∞ ) \ N π .For H ∈ ˆ H ( R n ), we have a number M > | H ( z ) | M | z | and | d H ( z ) | M for all z ∈ R n . Define functionals b H , A H : E P → R by b H ( x ) = Z H ( x ( t )) dt, x ∈ L and A H ( x ) := A ( x ) − b H ( x ) . These two functionals are also well-defined on E . Note that the gradient ∇ b H ( x ) ∈ E P isequal to the orthogonal projection onto E P of the gradient of b H at x as a functional on E .By Lemma 4 on page 87 of [10] we have: Proposition 2.3.
The map b H : E P → R is continuously differentiable. The gradient ∇ b H : E P → E P is continuous and compact. Moreover, for all x, y ∈ E P there hold k∇ b H ( x ) − ∇ b H ( y ) k M k x − y k and | b H ( x ) | M k x k L . Proposition 2.4.
Each critical point x of A H in E P belongs to C ∞ ( S , R n ) and satisfies ˙ x ( t ) = J ∇ H ( x ( t )) and x ( t + 12 ) = P x ( t ) , ∀ t ∈ S . Proof.
Since ∇ b H ( x ) = j ∗ ( ∇ H ( x )), we have x + − x − = j ∗ ( ∇ H ( x )) and so h x + − x − , v i = h j ∗ ∇ H ( x ) , v i = ( ∇ H ( x ) , j ( v )) L (2.3)For v = P j ∈ Z e πjtJ v j ∈ E P . Substitute the Fourier series of x and ∇ H , x = X j ∈ Z e πjtJ x k and ∇ H ( x ) = X j ∈ Z e πjtJ a k , into (2.3), we obtain a j = 2 πjx j for all j ∈ Z . It follows that a = 0 = Z ∇ H ( x ( t )) dt and X | j | | x j | X | a j | = k∇ H ( x ) k L < ∞ . This implies that x ∈ E P , and therefore x is continuous by Proposition 2.1(ii). Others arerepeating of the proof of Lemma 5 on page 88 of [10].In the proof of Lemma 6 on page 89 of [10] replacing E by E P leads to: Proposition 2.5. If H ∈ ˆ H ( R n ) , then every sequence { x j } j ⊂ E P with ∇A H ( x j ) → hasa convergent subsequence. In particular, A H satisfies (PS) condition. Similarly, the proof of Lemma 7 on page 90 of [10] yields
Proposition 2.6.
The flow φ t ( x ) of the gradient equation ˙ x = −∇A H ( x ) on E P has repre-sentation φ t ( x ) = e t x − + x + e − t x + + K ( t, x ) for all t ∈ R and x = x − + x + x + ∈ E P , where K : R × E P → E P is continuous andmaps bounded sets to precompact sets. In addition, K ( t, · ) : E P → E P is P -equivariant, forall t ∈ R . .2 Defining higher P -symmetric Ekeland-Hofer capacities Consider the natural S -action on E P : θ ⋆ x ( t ) = x ( θ + t ) , ∀ θ, t ∈ S . Let E be the collectionof S -invariant subsets of E P . As in [8] we can assign a Fadell-Rabinowitz index i S ,α ( X ) = sup { m ∈ N | f ∗ ( α m − ) = 0 } to each nonempty X ∈ E , where α ∈ H ( C P ∞ ; Q ) is the Euler class of the classifying vectorbundle ES = S ∞ → BS = C P ∞ , and f ∗ : H ∗ ( C P ∞ ; Q ) → H ∗ S ( X ; Q ) is the homomor-phism induced by the classifying map f : ( X × S ∞ ) /S → C P ∞ given by f ([ x, s ]) = [ s ]. Wealso define i S ,α ( ∅ ) = 0 for convenience. Then i S ,α satisfies the properties in [8, Theorem 5.1].Since Fix( S ) = E P = Fix( P ) ≡ R k , [8, Corollary 7.6] showed that i S ,α has the 2-dimensionproperty, that is, i S ,α ( V m ∩ S P ) = m for each S -invariant 2 m -dimension subspace V m of E P such that V m ∩ Fix( S ) = { } , where S P = { x ∈ E P | k x k / = 1 } .Consider the group Γ of homeomorphisms h : E P → E P of form h ( x ) = e γ + ( x ) P + ( x ) + P ( x ) + e γ − ( x ) P − ( x ) + K ( x ) , where i) K : E P → E P is a S -equivariant continuous map, and maps bounded set to precom-pact set; ii) γ + , γ − : E P → R + is S -invariant continuous function, and maps bounded set tobounded set; iii) there exists a constant c > γ + ( x ) = γ − ( x ) = 0 and K ( x ) = 0 foreach x ∈ E P satisfying A ( x ) k x k / > c .Following [4, Definition1.2] we define the pseudoindex i ∗ S ,α of i S ,α relative to Γ by i ∗ S ,α ( ξ ) := inf { i ( h ( ξ ) ∩ S + P ) | h ∈ Γ } ∀ ξ ∈ E , (2.4)where S + P := E + P ∩ S P . Repeating the proof of [7, Proposition 1] we can obtain: Proposition 2.7.
Suppose that a S -invariant subspace X of E + P has even dimension dim( X ) =2 p . Then i ∗ S ,α ( E − P ⊕ E P ⊕ X ) = p . Let B ⊂ R n be bounded, P -symmetric and disjoint with Fix( P ). For each H ∈ ˆ H ( R n )we define c iP ( H ) := inf { sup A H ( ξ ) | ξ ∈ Σ , i ∗ S ,α ( ξ ) > i } , i = 1 , , · · · . (2.5)Clearly, for H, K ∈ ˆ H ( R n ) we have (i) ( Monotonicity ) c iP ( H ) ≥ c iP ( K ) if H ≤ K . (ii) ( Continuity ) | c iP ( H ) − c iP ( K ) | ≤ sup {| H ( z ) − K ( z ) | (cid:12)(cid:12) z ∈ R n } . (iii) ( Homogeneity ) c iP ( λ H ( · /λ )) = λ c iP ( H ) for λ = 0. Proposition 2.8.
For a given H ∈ ˆ H ( R n ) , if the associated a ∈ ( π, ∞ ) \ N π is as in (H3),and j ∈ N satisfies a ∈ ( jπ, ( j + 1) π ) , then for some β > there hold < β c P ( H ) c P ( H ) · · · c mP ( H ) < + ∞ , where m is half of the dimension of X j = { x ∈ E + P | x k = 0 for k > j } , which is equal to nj if j is even and n ( j −
1) + 2( n − κ ) if j is odd (by a direct computation).Proof. By the definition of H there exists a P -symmetric open set U ⊂ R n such that H = 0in U . Taking x ∈ U ∩ Fix( P ) and arguing as in [7] or [12], we can find ǫ > A H | x + ǫS + P > β > β >
0. Pick h ∈ Γ such that h ( S + P ) = x + ǫS + P . Let ξ ∈ E satisfy i ∗ S ,α ( ξ ) >
1. SinceΓ is a group, h − ∈ Γ. By the definition of i ∗ S ,α ( ξ ) we have i S ,α ( h − ( ξ ) ∩ S + P ) ≥ i ∗ S ,α ( ξ ) ≥ nd hence h − ( ξ ) ∩ S + P = ∅ or equivalently ∅ 6 = ξ ∩ h ( S + P ) = ξ ∩ ( x + ǫS + P ) . It follows fromthis and (2.6) that c P ( H ) > β >
0. By the monotonicity c kP ( H ) > c P ( H ) > β > k ≥ A H ( E − P ⊕ E P ⊕ X j ) < + ∞ . (2.7)In fact, for any x ∈ E − P ⊕ E P ⊕ X j , since π P ji =1 | i || x i | πj P ji =1 | x i | πj R | x ( t ) | dt , A H ( x ) π j X i =1 | i || x i | − Z H ( x ( t )) dt ≤ Z πj | x ( t ) | − H ( x ( t )) dt. (2.8)By the definition of H there exists a R > H ( z ) = a | z | and so πj | z | − H ( z ) z / ∈ B n ( R ). Moreover, by the compactness of B n ( R ) we have a constant C > πj | z | − H ( z ) C for all z ∈ B n ( R ). Thus πj | z | − H ( z ) max { C, } for all z ∈ R n .From these and (2.8) we derive that A H ( x ) R | x ( t ) | − H ( x ( t )) dt max { C, } and so (2.7).Since dim X j = nj if j is even, and dim X j = ( n ( j −
1) + 2( n − κ )) if j is odd, we have i ∗ S ,α ( E − P ⊕ E P ⊕ X j ) = nj j is even ,i ∗ S ,α ( E − P ⊕ E P ⊕ X j ) = 12 n ( j −
1) + ( n − κ ) if j is oddby Proposition 2.7. The desired inequalities follows immediately.As in [7, Lemma 1], using Proposition 2.5 we can get: Proposition 2.9.
For H ∈ ˆ H ( R n ) and a S -invariant open neighborhood N of K c ( A H ) = { x ∈ E P | A H ( x ) = c, A ′ H ( x ) = 0 } , there exist ε > and h ∈ Γ such that h ( A c + εH \ N ) ⊂ A c − εH , where A dH = { x ∈ E P |A H d } . As usual, for each H ∈ ˆ H ( R n ), if c jP ( H ) < ∞ , then c jP ( H ) is critical value of A H . Let H ∈ H ( B ) := { H ∈ ˆ H ( R n ) | H = 0 in a neighborhood of B } For a P -symmetric bounded subset B ⊂ R n with B ∩ Fix( P ) = ∅ , we define the j -th P - symmetric Ekeland-Hofer capacity by c jP ( B ) := inf H ∈H ( B ) c jP ( H ) , j = 1 , , · · · . (2.9)If B ⊂ R n is a P -symmetric unbounded set with B ∩ Fix( P ) = ∅ , we define the j -th P - symmetric Ekeland-Hofer capacity of it by c jP ( B ) := sup { c jP ( B ′ ) | B ′ ⊂ B is bounded, P -symmetric, and B ′ ∩ Fix( P ) = ∅} . (2.10) Proof of Theorem 1.2.
Let x ∈ D be a fixed point of P . Since the translation R n → R n , x x − x is a P -equivariant symplectomorphism, by Proposition 1.1(iii) we assume0 ∈ int( D ). Let j D : R n → R be the Minkowski functional of D , and let H ( z ) = j D ( z ).For ǫ >
0, let F ǫ ( D ) consist of f ◦ H , where f : [0 , + ∞ ) → [0 , + ∞ ) satisfies f ( s ) = 0 if s , f ′ ( s ) > s > , f ′ ( s ) = α ∈ R \ Σ PS if f ( s ) > ǫ. rguing as in the proof of Lemma 6 on page 89 of [10], for f ◦ H ∈ F ǫ ( D ) we can deduce that A f ◦ H satisfies (PS) condition and thus that c P ( f ◦ H ) is a positive critical value of A f ◦ H . Let x ∈ E P be a critical of A f ◦ H with A f ◦ H ( x ) >
0. Then it is nonconstant and satisfies (cid:26) ˙ x = f ′ ( H ( x )) J ∇ H ( x ) ,x ( t + ) = P x ( t ) . It follows that y ( t ) := √ s x ( t/T ) is a solution of (cid:26) ˙ y = J ∇ H ( y ) ,y ( t + T ) = P y ( t ) , where T = f ′ ( s ) and s = H ( x (0)). A straightforward computation yields H ( y ( t )) ≡ A f ◦ H ( x ) = 12 Z h− J ˙ x ( t ) , x ( t ) i dt − Z f ◦ H ( x ( t )) dt = 12 Z h f ′ ( H ( x )) ∇ H ( x ) , x ( t ) i dt − Z f ◦ H ( x ( t )) dt = f ′ ( s ) s − f ( s ) , which lead to f ′ ( s ) = A ( y ) ∈ Σ PS . By the definition of f we deduce that f ( s ) < ǫ and so A f ◦ H ( x ) = f ′ ( s ) s − f ( s ) > f ′ ( s ) − ǫ > min Σ PS − ǫ because A f ◦ H ( x ) > s = H ( x (0)) > ǫ > G ∈ H ( D ) there exists f ◦ H ∈ F ǫ ( D ) such that f ◦ H > G .We deduce that c P ( G ) > c P ( f ◦ H ) > min Σ PS − ǫ . Hence c P ( D ) > min Σ PS .Next, we prove c P ( D ) min Σ PS . Let γ := min Σ PS . It suffices to prove that for any ǫ > H ∈ H ( D ) such that c P ( H ) < γ + ǫ, (3.1)which is reduced to prove thatΩ h := { x ∈ h ( S + P ) | A H ( x ) < γ + ǫ } 6 = ∅ , ∀ h ∈ Γ . (3.2)In fact, this implies that ξ h := { θ ⋆ x | θ ∈ S , x ∈ Ω h } is nonempty. Define ξ = ∪ h ∈ Γ ξ h .For h ∈ Γ, take x ∈ Ω h − . Then x ∈ h − ( S + P ) ∩ ξ . This implies h ( x ) ∈ h ( ξ ) ∩ S + P , and thus i ∗ S ,α ( ξ ) >
1. Note that sup x ∈ ξ A H ( x ) < γ + ǫ . (3.1) holds.It remains to prove (3.2). For τ >
0, by the definition of H ( D ) there exists H τ ∈ H ( D )such that H τ > τ ( H − (1 + ǫ γ )) . (3.3)As the proof of [11, Theorem 1.9], for h ∈ Γ, we can choose x ∈ h ( S + P ) such that A ( x ) γ Z H ( x ( t )) dt. We claim that for τ > H := H τ satisfies (3.2). Case 1. If R H ( x ( t )) dt (1 + ǫγ ), then by H τ >
0, we have A H τ ( x ) A ( x ) γ Z H ( x ( t )) dt γ (1 + ǫγ ) < γ + ǫ. ase 2. If R H ( x ( t )) dt > (1 + ǫγ ), then Z H τ ( x ( t )) dt > τ ( Z H ( x ( t )) dt − (1 + ǫ γ )) > τ ǫ γ (1 + ǫγ ) − Z H ( x ( t )) dt because(1 + ǫ γ ) = (1 + ǫ γ )(1 + ǫγ ) − (1 + ǫγ ) < (1 + ǫ γ )(1 + ǫγ ) − Z H ( x ( t )) dt and 1 − (1 + ǫ γ )(1 + ǫγ ) − = (1 + ǫγ ) − [(1 + ǫγ ) − (1 + ǫ γ )] = ǫ γ (1 + ǫγ ) − . Pick τ > τ ǫ γ (1 + ǫγ ) − > γ. Then Z H τ ( x ( t )) dt > γ Z H ( x ( t )) dt and hence A H τ ( x ) = A ( x ) − Z H τ ( x ( t )) dt A ( x ) − γ Z Z H ( x ( t )) dt . In summary, we have A H τ ( x ) < γ + ǫ . (3.2) is proved.Recall that we have assumed 0 ∈ D . Let F ǫ ( ∂D ) consist of f ◦ H , where f : R → R satisfies f ( s ) = 0 if s near 1 , f ′ ( s ) s , f ′ ( s ) > s > ,f ′ ( s ) = α ∈ R \ Σ PS if f ( s ) > ǫ, and α > π . Repeating the above proof, we can get c P ( f ◦ H ) > min Σ PS − ǫ, ∀ f ◦ H ∈ F ǫ ( ∂D ) . It follows that c P ( ∂D ) ≥ γ . By the monotonicity of c P , we can get c P ( ∂D ) = γ . Lemma 4.1 ([15, Proposition 7.4]) . Let S be a hypersurface of restricted contact type in ( R n , ω ) . Then the interior of Σ( S ) := { A ( x ) = Z x λ > | x is a closed characteristic on S} in (0 , ∞ ) is empty. Moreover, if Σ( S ) is nonempty then it contains a smallest element.Proof of Theorem 1.3. The ideas are following the proof of [6, Proposition 6]. Since S is a P -symmetric smooth hypersurface of restricted contact type, we can pick a Liouville vectorfield η on R n such that L η ω = ω and η points transversely outward at S . Moreover, η can be required to be P -equivariant in a small neighbor of S and to have linear growth. Let λ = ι η ω , then dλ = ω . Thus the flow Ψ ǫ generated by η is P -equivariant in a small neighborof S . Define S ǫ := Ψ ǫ ( S ). For ǫ > = 1 so that ∪ ǫ ∈ ( − , S ǫ is contained in the small neighborhood. It is easy to compute thatΨ ∗ ǫ ω = e ǫ ω .Define r = diam( ∪ ǫ ∈ [ − , ] S ǫ ) and for m ∈ N fix a number b > ( m + ) πr . Pick a smoothmap g : R → R such that g ( s ) = b if s r , g ( s ) = ( m + ) πs if s large ,g ( s ) > ( m + ) πs if s > r , < g ′ ( s ) (2 m + 1) πs if s > r . (cid:27) (4.1)In addition, let φ : R → R be a smooth map such that for suitable 0 < β < β < , φ ( s ) = 0 if s < β , φ ′ ( s ) > β < s < β , φ ( s ) = b if s > β . (4.2)Define a Hamiltonian H ∈ H ( B S ) by H ( z ) = z ∈ B S ,φ ( ǫ ) if z ∈ S ǫ ,b if z / ∈ B S β & | z | r ,g ( | z | ) if | z | > r . (4.3)Note that a critical point x of A H with positive action must satisfy x ([0 , ⊂ S ǫ for some ǫ ∈ ( β , β ), and that Σ P S = e ǫ Σ P S ǫ .By the definition of H , we can see that the positive critical levels only depend on the choiceof φ but not on g . In particular c jP ( H ) does not depend on the choice of g . Let H ( B S ) b,m = { H ∈ H ( B S ) | H ( z ) b for | z | r , H ( z ) = ( m + ) π | z | for | z | large. } Define c jP ( B S ) b,m = inf { c jP ( H ) | H ∈ H ( B S ) b,m } . Let ˆ H ( B S ) b,m consist of H ∈ H ( B S ) b,m as in (4.3), where g is as in (4.1) and φ as in (4.2)with 0 < β < β < small enough such that φ ′ ( s ) / ∈ Σ P S s if φ ( s ) ∈ [ β , b − β ] . (4.4)Noting that for any given H ∈ H ( B S ) b,m we can get an ˜ H ∈ ˆ H ( B S ) b,m such that ˜ H > H (bymodifying g if necessary), there exists a sequence { H l } l ⊂ ˆ H ( B S ) b,m such that c jP ( H l ) → c jP ( B S ) b,m . Taking critical points x l of A H l with action A H l ( x l ) = c jP ( H l ) we have x l ([0 , ⊂ S ǫ l forsome ǫ l ∈ ( β l , β l ), and therefore φ l ( ǫ l ) ∈ [0 , β l ) or φ l ( ǫ l ) ∈ ( b − β l , b ]because of (4.4). Note that A ( x l ) = φ ′ l ( ǫ l ). We get | c jP ( H l ) − A ( x l ) | β l or | c jP ( H l ) − A ( x l ) + b | β l . Since the restricted contact condition implies that λ does not vanish outside of the zero sectionof L S , we have c =: min (cid:8) | λ ( x, ξ ) || ξ | (cid:12)(cid:12) ( x, ξ ) ∈ L S , | ξ | = 1 (cid:9) > , and thus | λ ( x, ξ ) | ≥ c | ξ | for all ( x, ξ ) ∈ L S . Moreover, the condition of the restricted contacttype is C -open, we can get for ǫ > | λ ( x, ξ ) | ≥ c/ | ξ | , ∀ ( x, ξ ) ∈ L S ǫ , ∀ ǫ ∈ [0 , ǫ ] . t follows that length( x l ) c A ( x l ) for l large enough. As in [9, page 3] or [10, page 109], wecan use Ascoli-Arzela Theorem to find a T b ∈ Σ P S such that c jP ( B S ) b,m = T b or c jP ( B S ) b,m = T b − b. Note that the map b → c jP ( B S ) b,m is nonincreasing. We claim thatthe map b → T b must be nonincreasing in both case. (4.5)In fact, the first case is obvious. Assume that (4.5) does not hold in the second case. Thenthere exists b < b such that T b < T b . For each T ∈ ( T b , T b ), define∆ T = { b ∈ ( b , b ) | T b > T } . Since T b > T and c jP ( B S ) b ,m ≤ c jP ( B S ) b,m ≤ c jP ( B S ) b ,m for any b ∈ ( b , b ), we obtain if b ∈ ( b , b ) sufficiently close to b , c jP ( B S ) b,m + b = T b > T . Hence ∆ T = ∅ . Set b = inf ∆ T ,then b ∈ [ b , b ).Let { b ′ i } i ⊂ ∆ T satisfy b i ↓ b . Since c jP ( B S ) b ′ i ,m ≤ c jP ( B S ) b ,m , we have T < T b ′ i ≤ c jP ( B S ) b ,m + b ′ i , for all i ∈ N , and thus T ≤ T b by picking i → ∞ .Suppose that T < T b . Since T > T b , thus b = b and so b > b . Thus b ∈ ∆ T .For b ′ ∈ ( b , b ), c jP ( B S ) b ′ ,m ≥ c jP ( B S ) b ,m implies that T b ′ > T if b ′ is close to b . Thiscontradicts to the definition of b . Therefore, T = T b . This implies ( T b , T b ) ⊂ Σ P S . ButΣ P S = { k A ( x ) | x is a P -symmetric closed characteristic on S , k ∈ N } . It easily follows fromLemma 4.1 that Σ P S has the empty interior in (0 , ∞ ). Hence we get a contradiction again.(4.5) is proved.From (4.5) and c jP ( B S ) b,m ≥ c jP ( B S ) we deduce that the second case is impossible forlarge b . Hence c jP ( B S ) b,m = T b . Moreover, since c jP ( B S ) b,m → c jP ( B S ) and Σ P S has the emptyinterior as claimed above, we can get a T ∈ Σ P S such that c jP ( B S ) = T .The monotonicity leads to c jP ( B S ) > c jP ( S ).In order to prove the converse inequality, for a >
1, let τ a : R n → R such that τ a ( s ) = a if s − a , τ ′ a ( s ) < − a < s < − a , τ a ( s ) = 0 if s > − a . Define γ a : R n → R by γ a ( z ) = a if z ∈ B S − a ,τ a ( ǫ ) if z ∈ S ǫ , − a < ǫ , z / ∈ B S For any H ∈ H ( B S ), define H a ( z ) = H ( z ) + γ a ( z ). Then when x is a nonconstant 1-periodicsolution of ˙ x = X H a ( x ) , x ( t + 12 ) = P x ( t ) , x (0) ∈ B S , we may deduce that x (0) ∈ S ǫ for some ǫ ∈ ( − /a,
0) and A H a ( x ) = τ ′ a ( ǫ ) − τ a ( ǫ ) <
0. Hencethe positive critical levels of A H a and A H are the same. Since c jP ( H + sγ a ) = A H + sγ a ( x ) = A H ( x ) for some x and the map s → c jP ( H + sγ a ) has to be continuous, it follows that themap s → c jP ( H + sγ a ) is constant. Moreover for every e H ∈ H ( S ) there exist a H ∈ H ( B S )and a γ a such that e H H + γ a . Thus, c jP ( e H ) > c jP ( H + γ a ) = c jP ( H ) > c jP ( B S ). This implies c jP ( S ) > c jP ( B S ). Proof of Theorem 1.5 and Theorem 1.7
Proof of Theorem 1.5.
We first assume that r i /r j ( i = j ) are irrational. Then the sequence d j ( r ) is strictly monotonic. Define q ( z ) := n X i =1 x i + y i r i . It is the gauge function of E ( r ). For given ǫ > l ∈ N , we pick a smooth increase function f l,ǫ : [0 , + ∞ ) → [0 , + ∞ ) such that f l,ǫ ( s ) = 0 if s ≤ , f l,ǫ ( s ) = ( d l ( σ P ( r )) + ǫ ) s if s large enough ,f ′ l,ǫ ( s ) < d l ( σ P ( r )) + 2 ǫ ∀ s and f ′ l,ǫ ( s ) = d m ( σ P ( r )) holds only at s m , for m = 1 , · · · , l .Since for every f l,ǫ ◦ q , there exists H ∈ H ( B ) such that f l,ǫ ◦ q H , and for any H ∈ H ( B ),there also exists f l,ǫ ◦ q such that H f l,ǫ ◦ q , we obtain c jP ( B ) = inf { c jP ( f ◦ q ) | f = f l,ǫ is as above } . Fix f = f l,ǫ as above. The critical points of A f ◦ q are the solutions of the problem (cid:26) ˙ w = f ′ ( q ( w )) J ∇ q ( w ) ,w ( t + ) = P w ( t ) . (5.1)For each solution w of (5.1), z ( t ) := w ( t/T ) is a solution of (cid:26) ˙ z = J ∇ q ( z ) ,z ( t + T ) = P z ( t ) , (5.2)where T = f ′ ( q ( w (0))). Let us identify R n ( x , · · · , x n ; y , · · · , y n ) with R ( x , y ) ⊕ · · · ⊕ R ( x n , y n ), and write z ( t ) = ( z ( t ) , · · · , z n ( t )) with z j ( t ) = ( x j ( t ) , y j ( t )), j = 1 , · · · , n . Thenit is easy to compute that z ( t ) satisfies ˙ z = J ∇ q ( z ) if and only if z j ( t ) = e t/r j J (2)0 z j (0) , j n, where J (2)0 is the standard complex structure on R . When z is also required to satisfy thecondition z ( t + T ) = P z ( t ), by the assumption that r i /r j ( i = j ) are irrational, we get thatthe family of solutions for (5.2) has the form: z ( j ) ( t ) = (0 , · · · , , z j ( t ) , , · · · , , j = 1 , · · · , n, where z j ( t ) = e t/r j J (2)0 z j (0) with z j (0) ∈ R has period T = (2 m + 1) πr j ( j = 1 , · · · , n − κ )and T = 2 mπr j ( j = n − κ + 1 , · · · , n ) for m ∈ N . It follows that T = f ′ ( q ( w (0))) ∈ σ P ( r ).By the construction of f = f l,ǫ , there exists a m ∈ { , · · · , l } such that q ( w ) ≡ s m . Then(5.1) is translated into (cid:26) ˙ w = d m ( σ P ( r )) J ∇ q ( w ) ,w ( t + ) = P w ( t ) . Since h∇ q ( z ) , z i = 2 q ( z ) and q ( w ( t )) ≡ q ( w (0)), we get A f ◦ q ( w ) = 12 Z h− J ˙ w ( t ) , w ( t ) i dt − Z f ◦ q ( w ( t )) dt = 12 Z h f ′ ( q ( w )) ∇ q ( w ) , w ( t ) i dt − Z f ◦ q ( w ( t )) dt = f ′ ( q ( w (0))) q ( w (0)) − f ( q ( w (0)))= f ′ ( s m ) s m − f ( s m ) herefore, the critical value of A f l,ǫ ◦ q has the form f ′ l,ǫ ( s m ) s m − f l,ǫ ( s m ).Picking l large enough and fixing an integer j ≤ l , for m = 1 , · · · , j let X m denote thespace spanned by solutions of (cid:26) ˙ w = d m ( σ P ( r )) J ∇ q ( w ) ,w ( t + ) = P w ( t ) , and put ξ j := E − P ⊕ E P ⊕ jm =1 X m . Then by Theorem 2.7, we have i ∗ P ( ξ j ) = j . By choosing f l,ǫ such that s m is close enough to 1for m = 1 , · · · , l , and ǫ > A f l,ǫ ◦ q ( ξ j ) = f ′ l,ǫ ( s j ) s j − f l,ǫ ( s j )and thus c jP ( f l,ǫ ◦ q ) f ′ l,ǫ ( s j ) s j − f l,ǫ ( s j ). Note that s j can be chosen to be sufficiently closeto 1. We get that c jP ( f l,ǫ ◦ q ) d j ( σ P ( r )) and therefore c jP ( E ( r )) d j ( σ P ( r )) . Next we prove the converse inequality. By a similar proof to that of [7, Proposition 4] or[4, Formula (4.2)], we can show that c jP ( f l,ǫ ◦ q ) < ∞ and that j c jP ( f l,ǫ ◦ q ) is strictlyincreasing. The choice of s j implies that j f ′ l,ǫ ( s j ) s j − f l,ǫ ( s j ) is also strictly increasing.Now both { f ′ l,ǫ ( s j ) s j − f l,ǫ ( s j ) | ≤ j ≤ l } and { c jP ( f l,ǫ ◦ q ) | ≤ j ≤ l } are subsets of the critical value set of A f l,ǫ ◦ q and the latter is contained in the former. Hence c jP ( f l,ǫ ◦ q ) > f ′ l,ǫ ( s j ) s j − f l,ǫ ( s j ) , j = 1 , · · · , l. On the other hand, since c jP ( f l,ǫ ◦ q ) > f ′ l,ǫ ( s j ) s j − f l,ǫ ( s j ) > d j ( σ P ( r )) s j − ( d j σ P ( r ) + 2 ǫ )( s j − > d j ( σ P ( r )) + 2 ǫ (1 − s j ) , we can take ǫ so small that c jP ( f l,ǫ ◦ q ) > d j ( σ P ( r )) and hence c jP ( E ( r )) > d j ( σ P ( r )).Finally, the general case may follow from the continuity of c jP and the above special case. Proof of Theorem 1.7.
Firstly, we consider the case ˆ r /r ′ is irrational. Without loss of gen-erality we can assume ˆ r = r and r ′ = r n − k +1 , and complete the proof in two steps. Step 1 . Prove the inequality c jP ( D ( r )) d j ( σ ′ P ( r )). For given ǫ > l ∈ N , we choosea smooth increase function f l,ǫ : [0 , + ∞ ) → [0 , + ∞ ) such that f l,ǫ ( s ) = 0 if s ǫ, f l,ǫ ( s ) = ( d l ( σ ′ P ( r )) + 12 π ) s if s large enough f ′ l,ǫ ( s ) = 2 d l ( σ ′ P ( r ) s and s > ǫ = ⇒ f l,ǫ ( s ) ǫ and s ǫ,f ′′ l,ǫ ( s ) > s > ǫ. Define ϕ : R n ( x , · · · , x n ; y , · · · , y n ) ≡ R ( x , y ) ⊕ · · · ⊕ R ( x n , y n ) → R by w = ( x , · · · , x n ; y , · · · , y n ) → ϕ ( w ) = p x + y r + q x n − κ +1 + y n − κ +1 r n + k − . hen w ∈ E P is a critical point of A f l,ǫ ◦ ϕ if and only if w is a solution of (cid:26) ˙ w = f ′ l,ǫ ( ϕ ( w )) J ∇ ϕ ( w ) ,w ( t + ) = P w ( t ) . (5.3)For each solution w of (5.3), z ( t ) := w ( t/T ) with T = f ′ l,ǫ ( ϕ (0)) is a solution of (cid:26) ˙ z = J ∇ ϕ ( z ) ,z ( t + T ) = P z ( t ) . (5.4)As before we write z ( t ) = ( z ( t ) , · · · , z n ( t )) with z j ( t ) = ( x j ( t ) , y j ( t )), j = 1 , · · · , n . Then itis easy to compute that z ( t ) satisfies ˙ z = J ∇ ϕ ( z ) if and only if z j ( t ) = e t/r j J (2)0 z j (0) , j = 1 , n − κ + 1 , z j ( t ) ≡ const for other j, where J (2)0 is the standard complex structure on R . If we also require that z satisfies z ( t + T ) = P z ( t ), since ˆ r /r ′ is irrational, we get that the family of solutions for (5.4) has the form: z ( j ) ( t ) = (0 , · · · , , z j ( t ) , , · · · , , j n, where z ( t ) = e t/r J (2)0 z (0) with z (0) ∈ R has period T = 2(2 m − πr with m ∈ N , z n − κ +1 ( t ) = e t/r n − κ +1 J (2)0 z n − κ +1 (0) with z n − κ +1 (0) ∈ R has period T = 4 mπr n − κ +1 with m ∈ N , and z j ( t ) ≡ const for other j .For each integr l large enough and a fixed an integer j ≤ l , and m = 1 , · · · , j let X m denotethe space spanned by solutions of (cid:26) ˙ x = 2 d m ( σ ′ P ( r )) J ∇ ϕ ( x ) ,x ( t + ) = P x ( t ) , and put ξ j = E − P ⊕ E P ⊕ jm =1 X m . By Theorem 2.7, we have i ∗ P ( ξ j ) = j . As in the proof ofTheorem 1.5, we deduce thatsup A f j,ǫ ◦ ϕ ( ξ j ) = 12 f ′ j,ǫ ( s ) s − f j,ǫ ( s ) d j ( σ ′ P ( r )) s − f j,ǫ ( s ) d j ( σ ′ P ( r ))(1 + 2 ǫ ) . Here s satisfies f ′ j,ǫ ( s ) = 2 d j ( σ ′ P ( r )) s . Since ǫ > f l,ǫ can be chosen toarbitrarily small, we obtain the desired inequality. Step 2 . Prove c jP ( D ( r )) ≥ d j ( σ ′ P ( r )). By monotonicity it suffices to prove c jP ( D ( r )) > d j ( σ ′ P ( r )) (5.5)for r i = ˆ r if i = 1 , · · · , n − κ , and r i = r ′ if i = n − κ + 1 , · · · , n .Obviously, ∂D has closed characteristics R / Z ∋ t γ z s ( t ) = (0 , · · · , , e πtJ (2)0 z s , , · · · , z s = ( x s , y s ) has norm ˆ r (1 ≤ s ≤ n − κ ) or r ′ ( n − κ + 1 ≤ s ≤ n ). The m -th iterationof it, ( γ z s ) m , is defined by ( γ z s ) m ( t ) = γ z s ( mt ). For l ∈ N define Σ (2 l +1) consisting ofΥ j , ··· ,j p z s , ··· ,z sp := ( γ z s ) j + · · · + ( γ z sp ) j p (5.6)where 1 ≤ s < s < · · · < s p ≤ n − κ and j ν ∈ N − ν = 1 , · · · , p , satisfy j + · · · + j p = 2 l +1;and also define Σ (2 l ) consisting ofΥ j , ··· ,j p z s , ··· ,z sp := ( γ z s ) j + · · · + ( γ z sp ) j p (5.7) here n − κ + 1 ≤ s < s < · · · < s p ≤ n and j ν ∈ N , ν = 1 , · · · , p , satisfy j + · · · + j p = 2 l .Then for Υ j , ··· ,j p z s , ··· ,z sp in (5.6) (resp. (5.7)) we have A (Υ j , ··· ,j p z s , ··· ,z sp ) = (2 l + 1) π ˆ r (resp. A (Υ j , ··· ,j p z s , ··· ,z sp ) = 2 lπr ′ ) . (5.8)Let us pick a sequence of smooth functions, f m : R → R , m = 1 , , · · · , such that f m ( s ) = 0 if s m , f m = ( d m ( σ ′ P ( r )) + 12 ) | s | if s > m , (5.9) f ′′ m ( s ) > s > m , (5.10) s > m and f ′ m ( s ) = 2 d m ( σ ′ P ( r )) s = ⇒ f m ( s ) m . (5.11)Define H m : R n ( x , · · · , x n ; y , · · · , y n ) ≡ R ( x , y ) ⊕ · · · ⊕ R ( x n , y n ) → R by w = ( x , · · · , x n ; y , · · · , y n ) → H m ( w ) = n − κ X i =1 f m ( p x i + y i ˆ r ) + n X i = n − κ +1 f m ( p x i + y i r ′ ) . As in the proof of Theorem 1.5, we can get c jP ( H m ) → c jP ( D ( r )) , as m → ∞ . (5.12)Moreover, it follows from Theorem 1.3 that c jP ( D ( r )) ∈ σ ′ P ( r ). Claim 5.1.
For each m ∈ N , let z ( m ) = ( x ( m )1 , y ( m )1 , · · · , x ( m ) n , y ( m ) n ) be a critical point of A H m with A H m = c jP ( H m ) . Then the sequence { z ( m ) } m has a subsequence to converge to an orbiton ∂D ( r ) of form (5.6) or (5.7). In fact, by (5.9) and (5.11) we respectively deduce(1 + 1 m ) r i ≤ q ( x ( m ) i ( t )) + ( y ( m ) i ( t )) ≤ (1 + 2 m ) r i ∀ t, i = 1 , · · · , n and0 ≤ H m ( z ( m ) ( t )) ≤ nm ∀ t. Since c jP ( H m ) = A H m ( z ( m ) ) = A ( z ( m ) ) − R H m ( z ( m ) ( t )) dt , we obtain | c jP ( H m ) − A ( z ( m ) ) | nm . By a similar argument to the paragraph below (4.4) in the proof of Theorem 1.3, there existsa constant c such that 12 h J ddt z ( m ) , z ( m ) i R n ≥ c | ddt z ( m ) | . This leads to length( z ( m ) ) ≤ c − A ( z ( m ) ). As in [9, page 3] or [10, page 109] using Ascoli-ArzelaTheorem we can find a subsequence converge to an orbit on ∂D ( r ) of form (5.6) or (5.7).If j c jP ( D ( r )) is strictly increasing we can prove (5.5) by the same arguments as inthe proof of Theorem 1.5. Otherwise, let ¯ j be the first j ∈ { , , · · · } such that c jP ( D ( r )) = c j +1 P ( D ( r )). Then c iP ( D ( r )) = d i ( σ ′ P ( r )) ∀ i ≤ ¯ j, and c ¯ jP ( D ( r )) = d ¯ j ( σ ′ P ( r )) = c ¯ j +1 P ( D ( r )) (5.13)and these numbers take values in { lπr ′ , (2 l ′ + 1) π ˆ r | l, l ′ ∈ N } . Since we have assumed thatˆ r /r ′ is irrational, sets { lπr ′ | l ∈ N } and { (2 l + 1) π ˆ r | l ∈ N } are disjoint, and thus onlyone of the following two cases occurs: ase 1. c ¯ jP ( D ( r )) = (2 l + 1) π ˆ r for some l ∈ N ; Case 2. c ¯ jP ( D ( r )) = 2 lπr ′ for some l ∈ N .Because proofs for two case are similar we only consider Case 1. By (5.12) and (5.13) ̺ := d ¯ j ( σ ′ P ( r )) − d ¯ j − ( σ ′ P ( r )) = lim m →∞ ( c ¯ jP ( H m ) − c ¯ j − P ( H m )) > . Note that Σ (2 l +1) consists of finitely many S -orbits. We have i S ,α (Σ (2 l +1) ) = 1. Bythe continuity of i S ,α there exists a S -invariant open neighborhood U of Σ (2 l +1) such that i S ,α ( U ) = i S ,α (Σ (2 l +1) ) = 1. By (5.12), when m is large enough, both c ¯ jP ( H m ) and c ¯ j +1 P ( H m )are close enough to c jP ( D ( r )) = c j +1 P ( D ( r )). It follows from Claim 5.1 that the critical sets of A H m on levels c ¯ jP ( H m ) and c ¯ j +1 P ( H m ) are contained in U for m sufficiently large. Fix such a m . For ǫ > δ ∈ (0 , ̺ ), we can find a h ∈ Γ, h ( A c ¯ j +1 P ( H m )+ ǫH m \ U ) ⊂ A c ¯ j +1 P ( H m )+ ǫ − δH m . By the definition c j +1 P ( H m ), i ∗ S ,α ( A c ¯ j +1 P ( H m )+ ǫH m ) ≥ ¯ j + 1 . It follows from this, the subadditivity and the supervariance of i ∗ S ,α that i ∗ S ,α ( A c ¯ j +1 P ( H m )+ ǫ − δH m ) > i ∗ S ,α ( h ( A c ¯ j +1 P ( H m )+ ǫH m \ U )) > i ∗ S ,α ( A c ¯ j +1 P ( H m )+ ǫH m \ U ) > i ∗ S ,α ( A c ¯ j +1 P ( H m )+ ǫH m ) − i S ,α ( U ) > ¯ j. This leads to d ¯ j ( σ ′ P ( r )) = c ¯ jP ( D ( r )) c ¯ j +1 P ( H m ) + ǫ − δ . But we can ǫ > δ ∈ (0 , ̺ ) arbitrarily close to ̺ . Combing these with the definition of ̺ , we obtain d ¯ j ( σ ′ P ( r )) d ¯ j ( σ ′ P ( r )) − ̺ d ¯ j − ( σ ′ P ( r )) , which is impossible. So c ¯ jP ( D ( r ) is strictly increasing. The proof is complete in this case.The case that ˆ r /r ′ is rational can be derived from the monotonicity and continuity of c ¯ jP . We use the method in [3] to compute c P ( D × L D ). Slightly modifying the proof of [3,Proposition 4.1] can lead to: Lemma 6.1.
Let D ∈ B ( R n ) be a bounded convex domain, and let j D : R n → R be theMinkowski functional of D . For any c > , define A c : E P → R by A c ( x ) := A ( x ) − c Z j D ( x ( t )) dt. If W ⊂ E P is a S -invariant subset with i ∗ S ,α ( W ) > k such that A c | W , then c kP ( D ) c .Proof of Theorem 1.7. Firstly, by the monotonicity we have c jP ( D × L D ) ≥ c jP ( B (1)) ∀ j ∈ N , where B (1) is the unit ball in R . Specially, c P ( D × L D ) ≥ c P ( B (1)) = 2 π . ext, let W := E − P ⊕ E P ⊕ span { ( e πit , , (0 , e πit ) } . Then Theorem 2.7 yields i ∗ S ,α ( W ) = 2. For x ( t ) = ( x ( t ) , x ( t )) = ( αe πit , βe πit ) + w − + w ∈ W with α, β ∈ C , w − ∈ E − P , w ∈ E P , since the gauge function r of D × L D is r (( z , z )) = | z | + | z | z + z )2 , a straightforward computation leads to A c ( x ) = A ( x ) − c Z r ( x ( t )) dt = π ( | α | + 2 | β | ) − k w − k − c | α | + | β | ) − c k w + w − k L − c Z | Re( x ( t ) + x ( t ) ) | dt. We want to find c such that A c ( x ) , ∀ x ∈ W . For this purpose, as in the proof of [3,Theorem 4.6] we can estimate A c ( x ) to get A c ( x ) | α | ( π − c − c c + 80 π ) + | β | (2 π − c − c c c + 96 π ) − C, where C > α, β . Thus when π − c − c c + 80 π < π − c − c c c + 96 π < , (6.1)we can deduce that A c ( x ) <
0. Solving the system of inequalities in (6.1), we obtain c > π √ − . Approximating domain of D × L D as in [3] we can use Lemma 6.1 to derive c P ( D × L D ) < π √ − . Moreover, by [3, Proposition 2.2] we know that the intersection of the action spectrum and[2 π, π √ − ) is equal to { π, } . Hence c P ( D × L D ) ∈ { π, } , which implies the conclu-sion.To compute c P ( D × D ), let us consider W := E − P ⊕ E P ⊕ span { ( e πit , } . By Theorem 2.7, we get i ∗ S ,α ( W ) = 1. For x ∈ W , repeating the above arguments as in case β = 0, we obtain A c ( x ) | α | ( π − c − c c + 80 π ) − C. Thus when c satisfies π − c − c c + 80 π < , (6.2)we have A c ( x ) <
0. Solving the inequality in (6.2), we get c > π √ − . By approximatingdomain D × L D as in [3] we may use Lemma 6.1 to get c P ( D × L D ) < π √ − . Hence c P ( D × L D ) ∈ (cid:2) π, π √ − (cid:1) ∩ Σ ∂ ( D × D ) . Moreover, c P ( D × L D ) ≥ c EHZ ( D × L D ) = 4 by Theorem 1.2 and [2, Theorems 1.3, 1.7].The conclusion follows from these. Higher real symmetric Ekeland-Hofer capacities
As a complementary for the symmetrical version of the first Ekeland-Hofer capacity studiedby Jin and the second named author in [12], we outline constructions of higher real symmetricEkeland-Hofer capacities.The real part in the standard real symplectic space ( R n , ω , τ ) is L := Fix( τ ) = { ( x, y ) ∈ R n | y = 0 } . Following [12] consider the Hilbert subspace of the Hilbert space E in (2.2), E τ = { x ∈ L ( S ; R n ) | x = X j ∈ Z e πjtJ x j , x j ∈ L , X j ∈ Z | j || x j | < ∞} . Denote S = { x ∈ E τ | k x k / = 1 } . Define a Z -action T = { T , T } on E τ as follows: T : E τ → E τ , x ( t ) → x ( t ) , T : E τ → E τ , x ( t ) → x ( t + 12 ) . Let H be the set of T -equivariant maps from E τ to itself, and let F denote the fixed point setof this Z action, i.e., F = { x ∈ E τ | x = X j ∈ Z e πjtJ x j , x j ∈ L , x j = 0 , if j is odd } . There exists an orthogonal splitting of E τ , E τ = F ⊕ G , where G = { x ∈ E τ | x = X j ∈ Z e πjtJ x j , x j ∈ L , x j = 0 , if j is even } . Denote by P and P the orthogonal projections to F and G , respectively. Let E T = { A ⊂ E τ | A is closed and T -invariant } . There exists a natural index i τ : E T → N ∪ { , ∞} (cf. [16] for example) given by i τ ( A ) = min { k ∈ N | ∃ f ∈ C ( A, R k \ { } ) satisfying f ( T x ) = − f ( x ) ∀ x ∈ A } for each nonempty A ∈ E T , where i τ ( A ) is defined to be ∞ if there is no such k ∈ N . Of course, i τ ( A ) = 0 if A = ∅ . Notice that A ∩ F = ∅ implies i τ ( A ) = ∞ (since f ( x ) = f ( T x ) = − f ( x )for x ∈ A ∩ F ). i τ possess the properties of the index stated in [4, Definition 1.1] (see [16] forproofs). We have also Proposition 7.1. i τ satisfies -dimension property.Proof. For a T -invariant subspace V k with dim( V k ) = k , if V k ∩ F = { } , then V k ∩ S is thesphere of V k , dim( V k ) = k . It is obvious that i τ ( V k ∩ S ) k .Note that P ∈ H . Denote V = P ( V k ∩ S ). For l < k , and f : V → R l ⊂ R k satisfying f ( T x ) = − f ( x ). Since V ⊂ G , f ( T x ) = f ( − x ) = − f ( x ) ∀ x ∈ V , that is, f is odd. By Borsuktheorem, there must be some x ∈ V such that f ( x ) = 0. So i τ ( V k ∩ S ) i τ ( P ( V k ∩ S )) k .This implies i τ ( V k ∩ S ) = k .Consider the orthogonal composition of E τ E τ = E − τ ⊕ E τ ⊕ E + τ , where E τ = L =Fix( τ ) = { ( x, y ) ∈ R n | y = 0 } , and E − τ = { x ∈ E τ | x = X j< x j e πjtJ } and E + τ = { x ∈ E τ | x = X j> x j e πjtJ } . Denote P + , P − , P by the orthogonal projections to E + τ , E − τ , E τ , respectively. Consider thesubgroup of H ,Γ T := { h : E τ → E τ | h ( x ) = e γ + ( x ) P + ( x ) + P ( x ) + e γ − ( x ) P − ( x ) + K ( x ) } , here i) K : E τ → E τ is a T -equivariant continuous map, and maps bounded set to precom-pact set, ii) γ + , γ − : E τ → R + is T -invariant continuous function, and maps bounded set tobounded set, and iii) there exists a constant c > x ∈ E τ satisfies A ( x ) k x k / > c , then γ + ( x ) = γ − ( x ) = 0 and K ( x ) = 0.For ξ ∈ E T , we define the pseudoindex of i τ relative to Γ T : i ∗ τ ( ξ ) := inf { i ( h ( ξ ) ∩ S ∩ E + τ ) | h ∈ Γ T } . Let H ( R n ) be the set of nonnegative function H ∈ C ∞ ( R n ) satisfying the following condition: (H1) H ( z ) = H ( τ z ) , ∀ z ∈ R n ; (H2) There is an open set U ⊂ R n such that H | U ≡ (H3) When | z | is large enough, H ( z ) = a | z | , where a > π and a / ∈ N π .Let B ⊂ R n be bounded, τ -invariant and B ∩ L = ∅ , and let F ( R n , B ) consist of H ∈ H ( R n ) such that H = 0 in a neighborhood of B . For H ∈ H ( R n ), define A H : E τ → R , x Z h− J ˙ x, x i dt − Z H ( x ( t )) dt and c jτ ( H ) := inf { sup A H ( ξ ) | ξ ∈ E T & i ∗ τ ( ξ ) > j } , ∀ j ∈ N . As in the proof of Proposition 2.8 we have
Proposition 7.2.
There is β > such that < β c τ ( H ) c τ ( H ) · · · c kτ ( H ) . As before, if c jτ ( H ) < ∞ , then c jτ ( H ) is critical value of A H , and c kτ ( H ) > c kτ ( H ) forany two H , H ∈ H ( R n ) with H > H , k = 1 , , · · · . Let B τ ( R n ) be the set of τ -invariantsubset B ⊂ R n with B ∩ L = ∅ . For each j = 1 , , · · · , if B ∈ B τ ( R n ) is bounded we call c jτ ( B ) := inf H ∈F ( R n ,B ) c jτ ( H )the j -th real symmetric Ekeland-Hofer capacity , and if B ∈ B τ ( R n ) is unbounded we define c jτ ( B ) := sup { c jτ ( B ′ ) | B ′ ⊂ B is bounded and B ′ ∈ B τ ( R n ) } . It is easy to verify that c jτ also satisfies the corresponding properties in Proposition 1.1. Bya similar argument to the proof of [12, Theorem 1.11], we have the following representationformula for c τ . Theorem 7.3.
Let D ⊂ R n be a τ -invariant convex bounded domain with C , boundary S = ∂D and contain a fixed point of τ . Then c τ ( D ) = min {A ( x ) > | x is a τ -brake closed characteristic on S} , where by the definition of [12] a τ brake closed characteristic on S is a closed characteristic z : R /T Z → S on S satisfying z ( T − t ) = τ z ( t ) . Moreover, if both ∂D and D contain fixedpoints of τ , then c τ ( D ) = c τ ( ∂D ) . It follows from this result and [12, Theorem 1.3] that c τ and the real symmetric Ekeland-Hofer capacity c EHZ ,τ defined in [12] coincide on any τ -invariant convex domain D ⊂ R n . eferences [1] A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closedcurves on the unit spheres, arXiv:1801.00242.[2] S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convexbodies, Int. Math. Res. Not. , , no.1, 165-193.[3] L. Baracco, M. Fassina, and S. Pinton, On the Ekeland-Hofer symplectic capacities ofthe real bidisc, Pacific J. Math. , (2020), no. 2, 423-446.[4] V. Benci, On the critical point theory for indefinite functionals in the presence of sym-metries, Trans. Amer. Math. Soc. , (1982), 533-572.[5] Y. Dong, Y. Long, Closed characteristics on partially symmetric compact convex hyper-surfaces in R n , J. Diff. Eq. , (2004), 226-248.[6] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z. , (1989), 355-378.[7] I. Ekeland and H. Hofer, Symplectic topology and Haniltonian dynamics II, Math.Z. , (1990), 553-567.[8] E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie groupactions with an application to bifurcation questions for Hamiltonian systems, Invent.Math. , (1978), no. 2, 139-174.[9] H. Hofer, E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math. , (1987), 1-9.[10] H, Hofer, E. Zehnder, Symplectic invariants and Hamiltonian dynamics . Birkh´userAdvanced Texts: Basler Lehrb´ucher. [Birkh¨uuser Advanced Texts: Basel Textbooks]Birkh¨user Verlag, Basel, 1994.[11] Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehndersymplectic capacities and applications, arXiv:1903.01116v2.[12] Rongrong Jin and Guangcun Lu, Representation formula for symmetrical symplecticcapacity and applications,
Discrete Contin. Dyn. Syst. , (2020), no. 8, 4705-4765.[13] Rongrong Jin and Guangcun Lu, Coisotropic Ekeland-Hofer capacities, arXiv:1910.14474.[14] V. G. B. Ramos, Symplectic embeddings and the Lagrangian bidisk, Duke Math. J. (2017), no.9, 1703-1738.[15] J.-C. Sikorav, Syst´emes Hamiltoniens et topologie symplectique. Dipartimento di Matem-atica dell’Universit´a di Pisa, 1990. ETS, EDITRICE PISA.[16] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamil-tonian systems, Mathematische Annalen , (1989), no. 2, 241-255.Laboratory of Mathematics and Complex Systems (Ministry of Education),School of Mathematical Sciences, Beijing Normal University,Beijing 100875, People’s Republic of ChinaE-mail address: [email protected], [email protected](1989), no. 2, 241-255.Laboratory of Mathematics and Complex Systems (Ministry of Education),School of Mathematical Sciences, Beijing Normal University,Beijing 100875, People’s Republic of ChinaE-mail address: [email protected], [email protected]