Hutchings' inequality for the Calabi invariant revisited with an application to pseudo-rotations
aa r X i v : . [ m a t h . S G ] F e b MEAN ACTION AND THE CALABI INVARIANT FOR AREA PRESERVING DISCMAPS: HUTCHING’S INEQUALITY WITH WEAKER BOUNDARY CONDITIONS
ABROR PIRNAPASOVA
BSTRACT . In [Hut16], Hutchings uses embedded contact homology to show the following forarea-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: If the as-ymptotic mean action on the boundary is bigger than the Calabi invariant, then the infimum of themean action of the periodic points is less than or equal to the Calabi invariant. In this note, weextend this to area-preserving disc diffeomorphisms, which are only a rotation on the boundary ofthe disc. Our strategy is to extend the diffeomorphism to a larger disc with nice properties and applyHutchings’ theorem. As a corollary, we observe that Hutchings’ inequality holds for disc diffeomor-phisms with a Diophantine rotation number on the boundary. Finally, as an application we show thatin almost all cases the Calabi invariant of a smooth pseudo-rotation is equal to its rotation number.
1. I
NTRODUCTION
Let φ be an area-preserving diffeomorphism of the closed 2 dimensional radius disc D with thearea form ω = rdr ∧ dθ. Let β = r dθ be a primitive of ω. A C ∞ function f : D → R is called anaction of φ if f satisfies the following property: φ ∗ β − β = df. (1.1)Such f : D → R exists, because the 1-form φ ∗ β − β is closed, i.e., d ( φ ∗ β − β ) = φ ∗ dβ − dβ = 0 and any closed 1-form on D is exact. Obviously f is not unique, we address this in a moment. Thequantity A φ ( x ) is called the asymptotic mean action with respect to f, if the following limit exists A ∞ φ ( x ) := lim n →∞ P n − i =0 f ( φ i ( x )) n . By Birkhoffs’ ergodic theorem, A ∞ φ ( x ) is well-defined for Lebesgue a.e. x ∈ D . If we choose x ∈ ∂ D , then A ∞ φ is well-defined and is independent of x. If f is the action of φ with respect to the primitive β, then for any c ∈ R , f + c is also and thecorresponding asymptotic mean action simply changes by adding c. For this reason we can definefor each a ∈ R f ( φ,a ) : D → R to be the unique action function for φ, for which A ∞ φ coincides with a on ∂ D . We define thecorresponding Calabi invariant by V ( φ, a ) := 1 R D ω Z D f ( φ,a ) ω . We say x is a periodic point of φ with period d, if φ d ( x ) = x for some d ∈ Z + and φ i ( x ) = x forevery ≤ i ≤ d − . We will use the symbol P ( φ ) to denote the set of periodic points of φ. If x is periodic point with periodic d , we define mean action A ( φ,a ) ( x ) := 1 d d − X i =0 f ( φ,a ) ( φ i ( x )) . Remark 1.1. • We denote the ”usual” asymptotic mean action and the Calabi invariant for maps φ with compactsupport by A ∞ ( φ ) ( x ) := A ∞ ( φ, ( x ) and V ( φ ) := V ( φ, . For the case where φ coincides with rotation by πα on the boundary of D , Hutchings in[Hut16] defines the asymptotic mean action and the Calabi invariant by A ∞ ( φ ) ( x ) := A ∞ ( φ,α ) ( x ) and V ( φ ) := V ( φ, α ) . • Clearly A ∞ φ ( x ) is well-defined at every x ∈ P ( φ ) and A ∞ ( φ,a ) ( x ) = A ( φ,a ) ( x ) . • From the definitions of the Calabi invariant and the asymptotic mean action, we have V ( φ, a ) = V ( φ,
0) + a and A ∞ ( φ,a ) ( x ) = A ∞ ( φ, ( x ) + a. • The Calabi invariant and the asymptotic mean action are independent of the primitive of ω anddo not change under area-preserving conjugacy. We refer for instance to [MS17] or [Sen20].In [Hut16], Hutchings used embedded contact homology to prove the following inequality: Theorem 1.2 ([Hut16]) . Let θ ∈ R , and let φ be an area-preserving diffeomorphism of D whichagrees with a rotation by angle πθ near the boundary. Suppose that V ( φ, θ ) < θ . (1.2) Then inf (cid:26) A ( φ,θ ) ( x ) | x ∈ P ( φ ) (cid:27) ≤ V ( φ, θ ) . (1.3)One of the conditions of Theorem 1.2 is that φ is a rigid rotation near the boundary i.e. on a neigh-borhood. In his blog post [Hut], Hutchings raises the question whether Theorem 1.2 holds when φ is a rigid rotation only on the boundary. The following result answers this question positively.T HEOREM Let φ be an area-preserving diffeomorphism of D which is a rotation on the bound-ary. Suppose that V ( φ, < . (1.4) Then inf (cid:26) A ( φ, ( x ) | x ∈ P ( φ ) (cid:27) ≤ V ( φ, . (1.5)By Remark 1.1, V ( φ, θ ) = V ( φ,
0) + θ nd inf (cid:26) A ( φ,θ ) ( x ) | x ∈ P ( φ ) (cid:27) = inf (cid:26) A ( φ, ( x ) | x ∈ P ( φ ) (cid:27) + θ , so that (1.4) and (1.5) are nothing else than reformulations of (1.2) and (1.3) respectively. Wefound this reformulation is helpful in the proof.Recall that an irrational number x is called Diophatine if there exist σ ≥ and constant γ > forwhich | x − pq | ≥ γq σ for all integers p and q = 0 . These form a set of full measure.The following corollary in some sense covers most non-rigid boundary conditions.
Corollary 1.3.
Let φ be an area-preserving diffeomorphism of D smoothly conjugate to rigidrotation on the boundary. Suppose that V ( φ, < . (1.6) Then inf (cid:26) A ( φ, ( x ) | x ∈ P ( φ ) (cid:27) ≤ V ( φ, . (1.7) Proof.
There are various ways one can extend this diffeomorphism to a smooth Hamiltonian dif-feomorphism of the whole disc. For example one can use [Ban74].Therefore, one can choose a smooth area-preserving diffeomorphism ψ such that ψ − ◦ φ ◦ ψ is arigid rotation on the boundary of D . By Remark 1.1, all assumptions of Theorem 1 are fulfiled. (cid:3) Remark 1.4.
By a deep result of Hermann and Yoccoz [Her79, Yoc84], we know any smoothcircle diffeomorphism with Diophantine rotation number is smoothly conjugate to a rigid rotation.In particular, Corollary 1.3 applies whenever the boundary rotation number of the disc map isDiophantine.
Applications to Pseudo-rotations.
In this section we apply our results to compute the Calabiinvariant of a large class of (smooth and area preserving) pseudo-rotations. In this article by apseudo-rotation we always mean a C ∞ -smooth and area preserving diffeomorphism of the closeddisc D having a unique periodic point. As is well known by results of Franks [Fra88] the uniqueperiodic point is an interior fixed point and to each pseudo-rotation φ there is a unique irrationalnumber we denote by ρ ( φ ) ∈ [0 , called the rotation number of φ which is characterised by thefollowing dynamical property: Every trajectory besides the unique fixed point has a well definedasymptotic mean winding number about the fixed point, and the value of this winding number is ρ ( φ ) . In particular ρ ( φ ) coincides with the rotation number of φ on the boundary of the disc. Alarge class of pseudo-rotations that are not conjugate to a rotation were discovered by Anosov andKatok in their seminal paper [KA72].Applying the inequality (1.7) above to pseudo-rotations yields the following identity: heorem 1.5 (Calabi identity for pseudo-rotations) . If φ : D → D is a pseudo-rotation thatis conjugate to a rotation on the boundary then its Calabi invariant coincides with its rotationnumber. In our notation (1.8) V ( φ ) = ρ ( φ ) or equivalently V ( φ, = 0 . In other words, the Calabi invariant of a pseudo-rotation coincides with the Calabi invariant of therigid rotation with the same rotation number.In the proof we will also need the following:
Proposition 1.6. [Bra15] If φ : D → D is a pseudo-rotation, then the action of its unique fixedpoint x is ρ ( φ ) . In other words A ( φ ) ( x ) = A ( φ,ρ ( φ )) ( x ) = ρ ( φ ) . It is also possible to prove Proposition 1.6 with more elementary methods using a generalisedversion of Theorem 1.1 in [Sen20].
Proof of Theorem 1.5.
The proof goes by contradiction. If the claim is false then either V ( φ, < or V ( φ, > . Note that the conclusion of Proposition 1.6 is equivalent to A ( φ, ( x ) = 0 in our notation. If V ( φ, < , then by Corollary 1.3, we have inf (cid:26) A ( φ, ( x ) | x ∈ P ( φ ) (cid:27) ≤ V ( φ, < A ( φ, ( x ) . Which implies the existence of another periodic point, and that is contradiction to the uniquenessof periodic points.If V ( φ, > Then V ( φ − , < and we apply Corollary 1.3 to the map φ − . (cid:3) By Remark 1.4, the Calabi identity Theorem 1.5 is true for all pseudo-rotations with Diophantinerotation number. This raises the question whether the identity holds also for pseudo-rotationswith Liouville rotation number (the irrational numbers that are not Diophantine). A subset ofthe Liouville numbers is the so called non-Brjuno numbers. For this the following result wasestablished by Avila, Fayad, Le Calvez, Xu, and Zhang.
Theorem 1.7 ([AFLC + . If φ : D → D is a C k k ≥ , pseudo-rotation with rotation numberthat is non-Brjuno, then φ is C k − − rigid: There exist a monotone increasing sequence n j suchthat φ n j converges to I D in C k − . This allows the identity (1.8) to be proven in the following non-Diophantine cases, complementingTheorem 1.5:
Theorem 1.8. If φ : D → D is a pseudo-rotation having non-Brjuno type rotation number, then itsCalabi invariant equals its rotation number. roof. In our notation we need to show V ( φ, = 0 . We use the following properties of the Calabi invariant:(1) V ( φ n ,
0) = n V ( φ, for every area preserving φ : D → D and for every n ∈ N , (2) if φ n : D → D converges to φ in C , then lim n →∞ V ( φ n ,
0) = V ( φ, . By Theorem 1.7, there exist a monotone increasing sequence n j ∈ N such that φ n j converges to I D in C . Then by property (1) and (2) of the Calabi invariant V ( I D ,
0) = lim j →∞ V ( φ n j ,
0) =lim j →∞ n j V ( φ,
0) = V ( φ,
0) lim j →∞ n j . From the last equality, we have V ( φ,
0) lim j →∞ n j = 0 , and if V ( φ, = 0 , then the left-hand sideof equality goes to + ∞ or −∞ . This contradiction to the right-hand side. (cid:3)
NoteRemark 1.9.
In [SHS20], the authors proved a version of Theorem 1 for Anosov Reeb flows usingthe Action-Linking Lemma. In [CGHHL21], authors recently show the same results as Theorem1.5 for all Reeb flow with only two periodic orbits on closed three-manifolds. Independently in[Jol21], the author proved Theorem 1.8 in more general case.
Question 1.10.
Continuing Hutchings’ question mentioned above: Does Theorem 1.2 (equiva-lently Theorem 1) hold without any assumptions on the boundary except the inequality (1.4)? ByCorollary 1.3, it suffices to understand disc maps with Liouvillean rotation number on the bound-ary.1.1.
Acknowledgement.
This work is part of the author’s Ph.D. thesis, written under the super-vision of Barney Bramham and Gerhard Knieper at the Ruhr University of Bochum. The authoris grateful to Alberto Abbondandolo for his suggestion of using generating functions and to DavidBechara Senior for the many helpful suggestions. The author was supported by the DFG SFB/TRR191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, Projektnummer 281071066-TRR 191. 2. P
ROOF OF T HEOREM φ continuously as a rotation to a larger disc, then approximate it by smooth mapsin such a way that we can apply Hutchings’ theorem. For the approximations, we use generatingfunctions. Throughout, smooth mean C ∞ -smooth, unless otherwise stated.We first state detailed facts about the generating functions given in [ABHS18].Let T : [ a, b ] × R → [ a, b ] × R denote the map T ( r, θ ) = ( r, θ + 2 π ) and set Ω = rdr ∧ dθ where ( r, θ ) ∈ [ a, b ] × R . Assume that Φ( r, θ ) = ( R ( r, θ ) , Θ( r, θ )) : [ a, b ] × R → [ a, b ] × R is a C map with the followingproperties: Φ ◦ T = T ◦ Φ; (2) Φ maps each connected component of ∂ ([ a, b ] × R ) into itself;(3) Φ ∗ Ω = Ω; (4) D R ( r, θ ) > . Property (4) implies that r R ( r, θ ) is an orientation preserving diffeomorphism of [ a, b ] ontoitself, for every θ. This defines a smooth diffeomorphism
Ψ : [ a, b ] × R → [ a, b ] × R with Ψ( r, θ ) = ( R ( r, θ ) , θ ) . From the fact above, one can work with ( R, θ ) coordinates on [ a, b ] × R . By property (2),(2.1) R ( a, θ ) = a for all θ ∈ R . Let us consider the 1-form: λ ( R, θ ) = R − r ( R, θ )2 dθ + R ( θ − Θ) dR (2.2)One can show easily that T ∗ λ = λ with (2.2).By property (3) of Φ , we compute the differential of λ : dλ = RdR ∧ dθ − rdr ∧ dθ − RdR ∧ dθ + RdR ∧ d Θ =Φ ∗ Ω − Ω = 0 . The last equality shows that the 1-form λ is closed and therefore exact on [ a, b ] × R . Hence, wecan choose W such that dW = λ. From the equality T ∗ λ = λ, one can prove that W ◦ T = W + c for some constant c ∈ R . By (2.1), λ ( a,θ ) [ ∂ θ ] = 0 for all θ ∈ R and so W is constant on the boundary of [ a, b ] × R . Therefore c = 0 , i.e., W ◦ T = W. We will call W a generating function for Ψ with respect to the area form Ω . It is defined up to theaddition of a constant and from the components of dW = λ we see that W satisfies: D W ( R, θ ) = R ( θ − Θ) , (2.3)and D W ( R, θ ) = R − r . (2.4) Remark 2.1.
We emphasize that if the property (4) is not true for the diffeomorphism Φ on thewhole of [ a, b ] × R but only true near { b } × R , then one can also define the generating function W near to { b } × R . In section 2.6 of [ABHS18], the authors computed the action of a diffeomorphism with respect tothe generating function of this diffeomorphism. heorem 2.2. The function
Σ : [ a, b ] × R → R given by (2.5) Σ := W ( R, θ ) + R Θ − R θ is an action of Φ( r, θ ) with respect to the one form β = r dθ. That is, Σ satisfies the equation (1.1).Moreover, Σ ◦ T = Σ . Note that Σ and W are unique up addition of a constant.We now prove a technical result that plays an important role in proving Theorem 1. Lemma 2.3.
Let W : [0 , × R → R and V : [1 , × R → R be smooth functions with thefollowing properties: • W ◦ T = W and V ◦ T = V, • ˆ W ( r, θ ) = ( V ( r, θ ) , if ≤ r ≤ W ( r, θ ) , if r ≤ is a C function.Then there exists a sequence { W n } n ∈ N approximating ˆ W , with the following properties:(1) W n ◦ T = W n , (2) W n is in C ∞ ([0 , × R , R ) , (3) W n (cid:12)(cid:12) r ≤ = W, (4) W n converges uniformly to ˆ W in C , as n → ∞ , (5) W n (cid:12)(cid:12) n +1 ≤ r ≤ = V. The proof of Lemma 2.3 is based on the following classical approximation theorem [Whi34] dueto Whitney.
Theorem 2.4.
Assume that M is a smooth manifold, possibly with boundary, and F : M → R is a C k function for some k ∈ Z ≥ . Given any δ > , there exists a C ∞ smooth function ˆ F : M → R such that || ˆ F − F | (cid:12)(cid:12) C k ≤ δ. If F is C ∞ smooth on a closed subset A ⊂ M, then ˆ F can be chosento be equal to F on A. Remark 2.5.
We say F : M → R is smooth on A ⊂ M if it has a smooth extension in aneighborhood of each point of A .For the continuous case of Theorem 2.4 ( k = 0) see Theorem 6.21 in [Lee13].Let us denote by D ( r ) ⊂ R the closed disc of radius r > which shares center with D . Define asmooth map ρ by ρ ( r, θ ) : [0 , × R → D (2) ,ρ ( r, θ ) = re iθ which restricts to a smooth covering map from (0 , × R to the punctured disc D (2) \ { } . roof of Lemma 2.3. The function W ρ = W ◦ ρ is a C function on D (2) and C ∞ on A n = { ( r, θ ) ∈ D (2) : r ≤ and n +1 ≤ r ≤ } . For each n ∈ N , we apply Theorem 2.4 to: thefunction W ρ , and as closed set A n as above and δ = n . Then we get the sequence of functions ˆ W n converges uniformly to W ρ in C , as n → ∞ . We now choose W n : [0 , × R → R such that ˆ W n = W n ◦ ρ and W n (cid:12)(cid:12) r ≤ = W. By constructions, W n fulfills all the properties we need. (cid:3) We know from the assumptions of Theorem 1 that φ is a rotation on the boundary of D . So we maywrite φ (cid:12)(cid:12) ∂ D = (1 , θ + 2 πθ ) for some θ ∈ R . We now start to apply the properties given above to our situation. It is clear that one can find asmooth ω -preserving diffeomorphim ψ : D → D which coincides with φ near the boundary of D and so that every x ∈ D ( δ ) is a fixed point of ψ for some sufficiently small δ > . For example if φ generated by a Hamiltonian we can multiply this by a cut off function vanishing on D ( δ ) . One can therefore find an Ω -preserving smooth diffeomorphism Υ( r, θ ) = ( R ( r, θ ) , Θ( r, θ )) : [0 , × R → [0 , × R with the property Υ ◦ ρ = ψ. Under the hypotheses of Theorem 1, D R (1 , θ ) = 1 for all θ ∈ R , since Θ is area preserving and D R (1 , θ ) = 0 , and therefore D R ( r, θ ) > (2.6)near { } × R by continuity.By Remark 2.1 and property (2.6), we can find a generating function W : (1 − ǫ, × R → R of Υ near { } × R and satisfying the following equalities:(2.7) D W (cid:12)(cid:12) r =1 = R ( θ − Θ) (cid:12)(cid:12) r =1 = − πθ and(2.8) D W (cid:12)(cid:12) r =1 = R − r (cid:12)(cid:12) r =1 = 0 on { } × R . By equation (2.8), we know W is constant on { } × R . We assumed also that the action f ( φ, on { } × R is 0. By equation (2.5) of Theorem 2.2, we can deduce that(2.9) W (cid:12)(cid:12) { }× R = − πθ . Now consider the following extension: ˆ W ( R, θ ) : (1 − ǫ, × R → R given by ˆ W ( R, θ ) = ( − θ πR , if ≤ R ≤ W ( R, θ ) , if R ≤ . sing equality (2.9) and equations (2.7) and (2.8), one can easily show that ˆ W ( R, θ ) is a C function. We now apply Lemma 2.3 to the ˆ W in order to get a sequence of smooth functions W n , for n ∈ N , which has following properties:(1) W n ◦ T = W n , (2) W n are in C ∞ ((1 − ǫ, × R , R ) , (3) W n (cid:12)(cid:12) r ≤ = W, (4) W n converges uniformly to ˆ W in C , as n → ∞ , (5) W n (cid:12)(cid:12) n +1 ≤ r ≤ = − θ πR . By solving equations (2.3) and (2.4) with respect to W n , one can get smooth diffeomorphisms ˆΦ n = ( R n , Θ n ) : [1 , × R → [1 , × R . Set ˆ φ n := ( R n ◦ ρ, Θ n ◦ ρ ) : [1 , × S → [1 , × S . The properties of W n imply ˆΦ n ◦ ρ (cid:12)(cid:12) n +1 ≤ r ≤ = ˆ φ n (cid:12)(cid:12) n +1 ≤ r ≤ = ( r, θ + 2 πθ ) (2.10)and ψ n = ( ˆ φ n , if ≤ r ≤ ,ψ, if r ≤ . is smooth. By construction of ψ , the map φ n = ( ˆ φ n , if ≤ r ≤ ,φ, if r ≤ . is also smooth.One can show the action of the diffeomorphism φ n : D (2) → D (2) is(2.11) f ( φ n , = ( Σ n , if ≤ r ≤ ,f ( φ, , if r ≤ , where by Theorem 2.2, Σ n := ˆΣ n ◦ ρ = ( W n ( R n , θ ) + R n Θ n − R n θ ◦ ρ. The following lemma is a straightforward consequence of the conditions above.
Lemma 2.6.
Let W n be given as above. Then for every k ∈ N there exists n k such that for every n ≥ n k and ( r, θ ) ∈ [1 , × R there holds • | R n − R | ≤ k , • | Θ n − Θ | ≤ k , • | W n ( R n , θ ) − W ( R, θ ) | ≤ k . We now prove the main theorem using the previous lemmas. roof of Theorem 1. We first give lower and upper bounds for the action and the Calabi invariantof the diffeomorphisms φ n .Claim 1: For every k ∈ N there exists n k , such that for every n ≥ n k and r ≥ the inequality | ˆΣ n ( r, θ ) | ≤ k holds. Moreover, (cid:12)(cid:12)(cid:12)(cid:12) Z [1 , × [0 , π ] ˆΣ n rdr ∧ θ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z [1 , × [0 , π ] | ˆΣ n r | dr ∧ θ ≤ πk . Proof: Suppose r ≥ . We know that ˆΣ n ( r, θ ) = W n ( R n , θ ) − R n θ − Θ n ) Lemma . ≤ k + ˆ W ( R, θ ) + R θ − Θ) = 2 k for sufficiently large n ∈ N . Similarly, we prove that: ˆΣ n ( r, θ ) ≥ − k (2.12)and (cid:12)(cid:12)(cid:12)(cid:12) Z [1 , × [0 , π ] ˆΣ n rdr ∧ θ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z [1 , × [0 , π ] | ˆΣ n r | dr ∧ θ ≤ k Z [1 , × [0 , π ] rdr ∧ θ = 6 πk . (2.13) (cid:3) Applying estimate (2.13), we get V ( φ n (cid:12)(cid:12) D (1+ n ) ,
0) = 1(1 + n ) π Z D (1+ n ) f ( φ n , ( r, θ ) rdr ∧ dθ . n ) π Z D f ( φ, ( r, θ ) rdr ∧ dθ + 1(1 + n ) π Z [1 , n ] × [0 , π ] Σ n ( r, θ ) rdr ∧ dθ =1(1 + n ) π Z D f ( φ, ( r, θ ) rdr ∧ dθ + 1(1 + n ) π Z [1 , n ] × [0 , π ] ˆΣ n ( r, θ ) rdr ∧ dθ ≥ n ) V ( φ, − n ) π k Z [1 , n ] × [0 , π ] rdr ∧ dθ ≥ n ) V ( φ, − k an estimate for the Calabi invariant of the diffeomorphisms φ n restricted to the disc D (1 + n ) . Similarly V ( φ n (cid:12)(cid:12) D (1+ n ) , ≤ n ) V ( φ,
0) + 32 k . (2.14)By the assumption V ( φ, < , we can choose k sufficiently large, so that for every k ≥ k n ) V ( φ, ≤ V ( φ, < − k . (2.15) he last two inequalities together with (2.12) imply the following estimate: V ( φ n (cid:12)(cid:12) D (1+ n ) , < Σ n ( r, θ ) − k (2.16)for every r ≥ . By property (2.10) the diffeomorphism φ n (cid:12)(cid:12) D (1+ n ) is a rotation near the boundaryof the disc D (1 + n ) and by (2.16), it satisfies V ( φ n (cid:12)(cid:12) D (1+ n ) , < . Hence, by Theorem 1.2 inf (cid:26) A ( φ n (cid:12)(cid:12) D (1+ 1 n ) , ( x ) | x ∈ P ( φ n (cid:12)(cid:12) D (1+ n ) ) (cid:27) ≤ V ( φ n (cid:12)(cid:12) D (1+ n ) , . (2.17)Since A ∞ ( φ n (cid:12)(cid:12) D (1+ 1 n ) , ( x ) = lim n →∞ P k − i =0 f ( φ n , ( φ in ( x )) k = lim n →∞ P k − i =0 Σ n ( φ in ( x )) k for every r ≥ , we have that A ∞ ( φ n (cid:12)(cid:12) D (1+ 1 n ) , ( x ) ≥ V (( φ n (cid:12)(cid:12) D (1+ n ) ) ,
0) + k for every ( r, θ ) ∈ [1 , × [0 , π ] . Therefore inf (cid:26) A ( φ n (cid:12)(cid:12) D (1+ 1 n ) , ( x ) | x ∈ P ( φ n (cid:12)(cid:12) D (1+ n ) ) ⊂ [1 , × S (cid:27) Remark . (cid:26) A ∞ ( φ n (cid:12)(cid:12) D (1+ 1 n ) , ( x ) | x ∈ P ( φ n (cid:12)(cid:12) D (1+ n ) ) ⊂ [1 , × S (cid:27) ≥ V ( φ n (cid:12)(cid:12) D (1+ n ) ,
0) + 12 k . (2.18)Now by inequalities (2.17) and (2.18): inf (cid:26) A ( φ, ( x ) | x ∈ P ( φ ) (cid:27) =inf (cid:26) A ( φ n (cid:12)(cid:12) D (1+ 1 n ) , ( x ) | x ∈ P ( φ n (cid:12)(cid:12) D (1+ n ) ) (cid:27) ≤V ( φ n (cid:12)(cid:12) D (1+ n ) , ≤ n ) V ( φ,
0) + 32 k .
Hence inf (cid:26) A ( φ, ( x ) | x ∈ P ( φ ) (cid:27) ≤ k ) V ( φ,
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