Geodesics of norms on the contactomorphisms group of R 2 n × S 1

. We prove that some paths of contactomorphisms of R 2 n × S 1 endowed with its standard contact structure are geodesics for diﬀerent norms deﬁned on the identity component of the group of compactly supported contactomorphisms and its universal cover. We characterize these geodesics by giving conditions on the Hamiltonian functions that generate them. For every norm considered we show that the norm of a contactomorphism that is the time-one of such a geodesic can be expressed in terms of the maximum of the absolute value of the corresponding Hamiltonian function. In particular we recover the fact that these norms are unbounded.


Introduction
Hofer [15] introduced in the 90's a conjugation invariant norm that comes from a Finsler structure on the group of compactly supported Hamiltonian symplectomorphisms of the standard symplectic Euclidean space pR 2n , ω st q.This norm has then been generalized to any symplectic manifold by Lalonde and McDuff [17].The Hofer norm has been intensively studied (see for example [3], [4], [16], [17], [19], [24], [27], [28]) since the non-degeneracy and the geodesics of this norm are notions that are intimately linked to symplectic rigidity phenomena such as nonsqueezing or Lagrangian intersection properties.By contrast, in the contact setting Fraser, Polterovich and Rosen [14] showed that there does not exist any conjugation invariant norm coming from a Finsler structure on the identity component of the group of compactly supported contactomorphisms of any contact manifold.More precisely they showed that any conjugation invariant norm on this group should be discrete, which means that there exists a positive constant such that any element that is not the identity has norm greater than this constant.In some sense this group is too big to carry a non-discrete conjugation invariant norm.Indeed, one important ingredient in their proof is an argument of contact flexibility: any Darboux ball can be contactly squeezed into an arbitrarily small one.Nevertheless, Sandon in [21] constructed an integer-valued unbounded conjugation invariant norm on the identity component of the group of compactly supported contactomorphisms of R 2n ˆS1 with its standard contact structure.Contact rigidity, in particular the existence of translated and discriminant points (see section 4 for a definition), plays a crucial role for existence of such a norm.Indeed, if we forget the contact structure, Burago, Ivanov and Polterovich [5] proved that there is no unbounded conjugation invariant norm on the identity component of the group of compactly supported diffeomorphism of R 2n ˆS1 .Since then several authors constructed different norms (conjugation invariant or not) on the identity component of the group of compactly supported contactomorphisms and on its universal cover [11], [14], [25], [30].
The idea of this paper is to study the geodesics of some of these norms in this context.We focus our study on the discriminant norm [11], the oscillation norm [11], the Shelukhin norm [25] and the Fraser-Polterovich-Rosen norm (FPR norm) [14] on Cont c 0 pR 2n ˆS1 , ξ st q, the identity component of the group of compactly supported contactomorphisms of R 2n ˆS1 endowed with its standard contact structure, and on its universal cover Č Cont c 0 pR 2n ˆS1 , ξ st q.As we recall below, for any co-oriented contact manifold, once a global contact form is fixed for the contact distribution, there is an explicit bijection between the space of smooth compactly supported time dependent functions on the contact manifold and the space of smooth paths of compactly supported contactomorphisms starting at the identity.While Shelukhin and Fraser, Polterovich and Rosen use this correspondence to define their norms in terms of the corresponding functions' size, the discriminant norm and the oscillation norm count in some sense the minimal number of discriminant points we should encounter while going from the identity to the considered contactomorphism along a smooth path.
Because the bijection mentioned above -between paths of contactomorphisms and functionsdepends on the choice of a contact form, the Shelukhin norm is not conjugation invariant.All the other mentioned norms are conjugation invariant.However, as we will see in section 3, the Shelukhin norm, the oscillation norm and the discriminant norm share the common property to be defined by minimizing some length functionals on the space of smooth paths of compactly supported contactomorphisms.Therefore it makes sense to talk about the geodesics of these three norms: they are the paths that minimize the length (see section 2 for a precise definition).
The main result that we present in this paper is the following: we show that a path of contactomorphisms generated by a function satisfying certain conditions is a geodesic for the discriminant, oscillation and Shelukhin norms.We show moreover that the norm of a contactomorphism that is the time-one of such a geodesic can be expressed in terms of the maximum of the absolute value of the corresponding function.In addition, even if the FPR norm does not a priori come from a length functional -therefore we cannot talk about its geodesics -we still can express the FPR norm of such a time-one path in terms of the maximum of the absolute value of the corresponding function.In particular we get as a corollary a new proof of the unboundedness of the discriminant norm, the oscillation norm, the Shelukhin norm and the FPR norm on Cont c 0 pR 2n ˆS1 , ξ st q.Before stating precisely these results let us define the standard contact structure and the standard contact form on R 2n ˆS1 .This will allow us to explicit the bijection between compactly supported time-dependent functions and compactly supported paths of contactomorphisms starting at the identity.
For any positive natural number n P N ą0 , the standard contact structure ξ st on R 2n`1 is given by the kernel of the 1-form α st " dz ´n ř i"1 y i dx i , where px 1 , . . ., x n , y 1 , . . ., y n , zq are the coordinate functions on R 2n`1 .The Reeb vector field R αst is given by B Bz .The 1-form α st is invariant by the action of Z on R 2n`1 given by k ¨px, y, zq " px, y, z `kq for all k P Z and px, y, zq P R 2n`1 , so it descends to the quotient R 2n ˆS1 to a 1-form we also denote α st .The kernel of this latter form, which we still denote by ξ st , is the standard contact distribution on R 2n ˆS1 , and the Reeb vector field R αst is again given by B Bz .
To any compactly supported smooth path of contactomorphisms tφ t u tPr0,1s Ă Cont c 0 pR 2n Ŝ1 , ξ st q starting at the identity one can associate the compactly supported function h : r0, 1s R2n ˆS1 Ñ R, pt, p, zq Þ Ñ h t pp, zq that satisfies α st ˆd dt φ t ˙" h t ˝φt for all t P r0, 1s.
Conversely to any time dependent compactly supported smooth function h : r0, 1sˆR 2n ˆS1 Ñ R one can associate the smooth path of vector fields X h : r0, 1s Ñ χpM q satisfying # α st pX t h q " h t dα st pX t h , ¨q " dh t `B Bz ˘αst ´dh t . ( Using the Cartan's formula one can show that the flow tφ t X h u tPR of the time dependent vector field X h , i.e. the path of diffeomorphisms such that d dt φ t X h " X t h pφ t X h q for all t P R and φ 0 X h " Id, is a smooth path of contactomorphisms.We denote by φ h :" tφ t X h u tPr0,1s the restriction of this flow to the time interval r0, 1s and we say that h is the Hamiltonian function that generates φ h . In Theorem 1.1 below ν αst S and Ą ν αst S denote the Shelukhin (pseudo-)norms on Cont c 0 pR 2n ˆS1 , ξ st q and Č Cont c 0 pR 2n ˆS1 , ξ st q respectively, and the notation L αst S stands for the Shelukhin length functional that we define in section 3. We denote by ν αst F P R and Ć ν αst F P R the FPR norms on Cont c 0 pR 2n ˆS1 , ξ st q and Č Cont c 0 pR 2n ˆS1 , ξ st q respectively.
Moreover, for any compactly supported contactomorphism ϕ P Cont c 0 pR 2n ˆS1 , ξ st q we have In Theorem 1.1 below ν d , r ν d and ν osc , Ą ν osc denote respectively the discriminant and oscillation norms on Cont c 0 pR 2n ˆS1 , ξ st q and Č Cont c 0 pR 2n ˆS1 , ξ st q respectively.The notations L d and L osc stand respectively for the discriminant length functional and the oscillation length functional that we define in section 3. Theorem 1.2.Let H : R 2n Ñ R be a smooth function with compact support such that 0 is a regular value of H in the interior of its support, i.e for all x P Interior pSupppHqq X H ´1t0u, d x H ‰ 0. Suppose that the Hessian of H satisfies max Then for any ϕ P Cont c 0 pR 2n ˆS1 , ξ st q the path ϕ ˝φh ˝ϕ´1 is a geodesic for the discriminant norms r ν d and ν d .More precisely we have If we assume moreover that H : R 2n Ñ R ě0 is non-negative (resp.H : R 2n Ñ R ď0 is nonpositive), then the path φ ˝φh ˝φ´1 is a geodesic for the oscillation norms Ą ν osc and ν osc .More precisely L osc pϕ ˝φh ˝ϕ´1 q " Ą ν osc prϕ ˝φh ˝ϕ´1 sq " ν osc pϕ ˝φ1 h ˝ϕ´1 q "tmax hu `1 presp.L osc pϕ ˝φh ˝ϕ´1 q " Ą ν osc prϕ ˝φh ˝ϕ´1 sq " ν osc pϕ ˝φ1 h ˝ϕ´1 q "t´min hu `1q.
Proof.Let B ą 0 be a positive number and consider the smooth compactly supported function Denoting by ¨ the standard Euclidean norm on R 2n , for any A ą 0 the smooth non-negative, compactly supported Hamiltonian function is independent of the S 1 -coordinate and satisfies h B,A p0, zq " max h B,A " A for all z P S 1 .Note also that h B,A does not vanish inside the interior of its support.Moreover an easy computation shows that there exists an increasing function B 0 : R ą0 Ñ R ą0 such that max ||| Hess p h B,A p¨, zq||| ă 2π for any B ě B 0 pAq and z P S 1 .So by Theorem 1.1 and Theorem 1.2, the contactomorphism φ :" φ 1 h B 0 pAq,A satifies ν osc pφq " ν d pφq " tAu `1, ν αst F P R pφq " rAs and ν αst S pφq " A. Since A can be choosen as big as we want, we deduce Corollary 1.3.Note that the diameter of the support of h B 0 pAq,A grows with A.

Unboundedness of these norms on Cont c
0 pR 2n ˆS1 , ξ st q was actually already known [11], [14], [25].The novelty of our result is that given any positive number A ą 0 one can construct an explicit Hamiltonian function such that the norm of the time-one of the path generated by this Hamiltonian function is exactly A.
Even if these norms do not a priori measure the same thing it turns out that they almost agree for the contactomorphisms described in Theorems 1.1 and 1.2.It seems then reasonnable to ask whether these norms are equivalent.Similar questions can be found in [14], [23].
Another natural question that arises from these results comes from the similarity with the geodesics of Hofer geometry.Indeed, it is well known that a time independent compactly supported function H : R 2n Ñ R with small Hessian generates a path of Hamiltonian symplectomorphisms of the standard symplectic Euclidean space pR 2n , ω st q that is a geodesic for the Hofer norm [3], [16].So the above theorems say that these geodesics can be lifted to paths of contactomorphisms that are still geodesics for the Shelukhin norm, and under some more assumptions, geodesics for the discriminant norms and the oscillation norm too.It would be interesting to know which are the geodesics of the Hofer norm that lift to geodesics of ν αst S , ν d and ν osc .
The main tool to prove Theorems 1.1 and 1.2 is the translation selector c : Cont c 0 pR 2n ˆS1 , ξ st q Ñ R ě0 constructed by Sandon using generating functions [22].The strategy for the proof is to first bound all these norms from below in terms of this translation selector (see Corollary 5.7 and Proposition 5.8) and in a second time to show that the paths we are considering realize this lower bound.
Before moving to the next section, it is interesting to note that Sandon in [22] use this translation selector to associate to any domain U Ă R 2n ˆS1 the following number cpU q :" sup tcpφq | φ P Cont c 0 pU, ξ st qu , where Cont c 0 pU, ξ st q is the set of time-one maps of smooth paths whose supports are compacts which lie inside U .The integer part of this function turns out to be a contact capacity, more precisely we have 1.cpU q ď cpV q for any U Ă V lying inside R 2n ˆS1 , 2. rcpU qs " rcpφpU qqs for any φ P Cont c 0 pR 2n ˆS1 , ξ st q and any U Ă R 2n ˆS1 .
In the same paper [22] it is shown that if a contactomorphism φ P Cont c 0 pR 2n ˆS1 , ξ st q displaces U Ă R 2n ˆS1 , i.e. φpU q X U " H, then rcpU qs ď rcpφqs `Pcpφ ´1q T . ( Corollary 5.7 and Proposition 5.8 will allow us to have similar results for the norms we are considering: Proposition 1.4.Let φ P Cont c 0 pR 2n ˆS1 , ξ st q be a contactomorphism that displaces U Ă R 2n ˆS1 , i.e. φpU q X U " H. Then rνpφqs ě 1  2 rcpU qs where ν denotes the FPR, discriminant, oscillation or Shelukhin norm.
In the next section we give the basic definitions of the group of contactomorphisms, its universal cover, conjugation invariant norms coming from length functionals and their geodesics.In the third section we recall the definition of the discriminant norm, the oscillation norm, the Shelukhin norm and the FPR norm.Then in the fourth section we give the main steps of the construction of Sandon's translation selector.Finally in the last section we give the proofs of Theorems 1.1, 1.2 and Proposition 1.4.
Acknowledgement -This project is a part of my PhD thesis done under the supervision of Sheila Sandon.I am very grateful to her for introducing me to this beautiful subject of research.I thank her for all the interesting discussions that we had without which this work would not have existed.I would also like to thank Miguel Abreu, Jean-François Barraud and Mihai Damian for their support and their useful advices.The author is partially supported by the Deutsche Forschungsgemeinschaft under the Collaborative Research Center SFB/TRR 191 -281071066 (Symplectic Structures in Geometry, Algebra and Dynamics).

Basic definitions
Let M be a connected manifold of dimension 2n `1 endowed with a co-oriented contact distribution ξ, i.e. a distribution of hyperplanes given by the kernel of a 1-form α P Ω 1 pM q such that α ^pdαq n is a volume form.We say that a diffeomorphism φ P DiffpM q is a contactomorphism if it preserves the contact distribution, i.e. d x φpξ x q " ξ φpxq for all x in M or equivalently if there exists a smooth function f : M Ñ Rzt0u such that φ ˚α " f α.We say that a r0, 1s-family of contactomorphisms tφ t u tPr0,1s is a smooth path of contactomorphisms if the map r0, 1s ˆM Ñ M , pt, xq Þ Ñ φ t pxq is smooth.From now on we denote a smooth path of contactomorphisms by tφ t u and omit the subscript t P r0, 1s.We will study the set of all contactomorphisms φ of M that can be joined to the identity by a smooth path of compactly supported contactomorphisms tφ t u, that is to say that φ 0 " Id, φ 1 " φ and Suppptφ 2. rtϕ t usrtφ t usrtϕ t us ´1 " rtϕ 1 ˝φt ˝pϕ 1 q ´1us.
Remark 2.1.By putting the strong C 8 -topology on the group of compactly supported contactomorphisms Cont c pM, ξq, the identity component of Cont c pM, ξq corresponds to Cont c 0 pM, ξq.Moreover this component is a sufficiently pleasant topological space so that its universal cover exists and can naturally be identified to Č Cont c 0 pM, ξq.For more details we refer to the book of Banyaga [1].
We will study in this paper four different norms on the groups Cont c 0 pR 2n ˆS1 , ξ st q and Č Cont c 0 pR 2n Ŝ1 , ξ st q.Three of them will be conjugation invariant.Definition 2.2.Let G be a group.We say that an application ν : G Ñ R ě0 is a norm if for all h, g P G it satisfies: 1. the triangular inequality, i.e. νpghq ď νpgq `νphq 2. symmetry, i.e. νpgq " νpg ´1q 3. non-degeneracy, i.e. νpgq " 0 if and only if g is the neutral element of the group.
If ν satisfies only the first and second property we say that ν is a pseudo-norm.Moreover we say that ν is conjugation invariant if νphgh ´1q " νpgq.Remark 2.3.Another way to study conjugation invariant norms on a group G is given by the point of view of bi-invariant metrics d : G ˆG Ñ R ě0 , i.e. metrics d such that dphg 1 , hg 2 q " dpg 1 h, g 2 hq " dpg 1 , g 2 q for all g 1 , g 2 , h P G. Indeed to any conjugation invariant norm ν : G Ñ R ě0 one can associate the bi-invariant metric dpg 1 , g 2 q :" νpg 1 g ´1 2 q.Conversely to any bi-invariant metric d : G ˆG Ñ R ě0 one can associate the conjugation invariant norm νpgq " dpId, gq.
The norms we are interested in are of two types.
The first type of norms ν on Cont c 0 pM, ξ " Kerpαqq and r ν on Č Cont c 0 pM, ξq we consider come from some length functional satisfying the following properties: 1. L is invariant by time reparamatrisation, i.e. if a : r0, 1s Ñ r0, 1s is a smooth bijective increasing function then Lptφ t uq " Lptφ aptq uq 2. Lptφ t u ¨tψ t uq ď Lptφ t uq `Lptψ t uq 3. Lptφ t uq ą 0 for any path that is non constant 4. Lptφ 1´t ˝pφ 1 q ´1uq " Lptφ t uq or Lptpφ t q ´1uq " Lptφ t uq.
The associated applications ν and r ν are then defined for any φ P Cont c 0 pM, ξqztIdu and for any rtφ t us P Č Cont c 0 pM, ξqztIdu by νpφq " inf Lptφ t uq | φ 0 " Id , φ 1 " φq ( and r νprtφ t usq " inf Lptϕ t uq | rtϕ t us " rtφ t us ( , and νpIdq " r νpIdq " 0. Because of these properties of the length functional, the associated application ν (resp.r νq is automatically a pseudo-norm whenever νpφq ă `8 for all φ P Cont c 0 pM, ξq (resp.whenever r νprtφ t usq ă `8 for all rtφ t us P Č Cont c 0 pM, ξq).
The discriminant norms ν d , r ν d and the Shelukhin norms ν α S , Ă ν α S defined on Cont c 0 pM, ξq and Č Cont c 0 pM, ξq), come respectively from the discriminant length functional L d and the Shelukhin length functional L α S that we will define in section 3 below.For this type of norms we give the following definition of a geodesic.

Definition 2.4.
A smooth compactly supported path of contactomorphisms tφ t u starting at the identity is a geodesic for ν : Following the terminology used by McDuff in [18], the second type of norms ν on Cont c 0 pM, ξq (resp.r ν on Č Cont c 0 pM, ξq) we consider come from two seminorms Ă ν `and Ă ν ´on Č Cont c 0 pM, ξq each one arising from some "semilength" functionals L `, L ´.More precisely the functionals verify the following properties: 1. they are invariant by time reparametrisation, i.e. if a : r0, 1s Ñ r0, 1s is a smooth bijective increasing function then L ˘ptφ t uq " L ˘ptφ aptq uq 2. L ˘ptφ t u ¨tψ t uq ď L ˘ptφ t uq `L˘p tψ t uq 3. if tφ t u is not the constant path then L `ptφ t uq ą 0 or L ´ptφ t uq ą 0 4. L `ptφ t uq " L ´ptφ 1´t ˝pφ 1 q ´1uq or L `ptφ t uq " L ´ptpφ t q ´1uq.
The seminorms Ă ν `and Ă ν ´coming from these functionals are defined for any element rtφ t us P Č Cont c 0 pM, ξq by From properties 2, 3 and 4 above, one can deduce that for any rtφ t us P Č Cont c 0 pM, ξq We then define the (pseudo-)norm r ν for all elements rtφ t us in Č Cont c 0 pM, ξq and the (pseudo-)norm ν for all elements φ in Cont c 0 pM, ξq by Noticing that the functional L :" maxtL `, L ´u is a genuine length functional and using again the terminology of McDuff [18] we give the following definition of a geodesic for this second type of norms.
Definition 2.5.We say that a smooth path of compactly supported contactomorphisms tφ t u starting at the identity is a geodesic for ν if νpφ 1 q " L `tφ t u ˘.
The oscillation norms ν osc , Ą ν osc and the FPR norms ν α F P R , Ć ν α F P R are defined via two seminorms.While the seminorms used to define the oscillation norm come from some semilength functionals, the seminorms of the FPR norms come from some functionals that are not invariant by time reparametrization.Therefore we are not going to talk about geodesics for the FPR norm.
Remark 2.6.For all rtφ t us P Č Cont c 0 pM, ξq note that r ν `rtφ t us ˘ď inf L `tϕ t u ˘| rtϕ t us " rtφ t us ( .This inequality has no reason a priori to be an equality.

The different norms
In this section we recall the definition of the discrimant norm [11], the oscillation norm [11], the Shelukhin norm [25] and the FPR norm [14].

The discriminant norm
Let φ be a contactomorphism of pM, ξ " Kerpαqq.A point x in M is said to be a discriminant point of φ if φpxq " x and pφ ˚αq x " α x .This definition does not depend on the choice of the contact form.Moreover a point x P M is a discriminant point of a contactomorphism φ P ContpM, ξq if and only if: ϕpxq is a discriminant point of ϕ ˝φ ˝ϕ´1 for all ϕ P ContpM, ξq.Remark 3.1.Another way to define the notion of discriminant point is to consider the symplectization of the contact manifold pM, ξq.Let us denote by S ξ pM q :" Ť T ˚M the symplectization of pM, ξq.It is a pR ˚, ¨q-principal fiber bundle on M ; the action of θ P R ˚on any element px, µq P S ξ pM q is given by θ ¨px, µq " px, θµq.Recall that pS ξ pM q, dλ ξ q is a symplectic manifold, where λ ξ is the restriction of the canonical Liouville form λ M of T ˚M to S ξ pM q, i.e. λ M pq, pqpuq " ppd pq,pq πpuqq for all pq, pq P T ˚M , u P T pq,pq T ˚X and where π : T ˚X Ñ X is the canonical projection.To any contactomorphism φ P Cont 0 pM, ξq one can associate its R ˚-lift: the symplectomorphism ψ of pS ξ pM q, dλ ξ q defined by ψpx, µq " pφpxq, µ ˝dφpxq φ ´1q , @px, µq P S ξ pM q .
So x P M is a discriminant point of a contactomorphism φ P ContpM, ξq if and only if any point px, µq P S ξ pM q is a fixed point of its R ˚-equivariant lift ψ.
For any contactomorphism φ P Cont c 0 pM, ξq we denote by DP pφq the set of all discriminant points of φ.We say that a compactly supported smooth path of contactomorphisms tφ t u tPra,bs does not have discriminant points if 1. DP ppφ s q ´1 ˝φt q " H for all s ‰ t P ra, bs, in the when case M is compact 2. DP ppφ s q ´1 ˝φt q X Interior `Suppptφ t uq ˘" H for all s ‰ t P ra, bs, in the case when M is not compact.
The discriminant length of a smooth path of contactomorphisms will count in how many pieces we have to cut this path so that each piece does not have discriminant points.More precisely, the discriminant length of a non-constant smooth path of contactomorphisms tφ t u of pM, ξq is defined by: L d ptφ t uq :" inftN P N ˚| there exists 0 " t 0 ă ... ă t N " 1, such that for all i P r0, N ´1s X N By convention we set inf H " `8.
Because of the two properties of discriminant points mentioned at the beginning of this subsection, it is straightforward that the functional L d is a length functional which is invariant under conjugation of elements in Cont

The oscillation norm
Following Eliashberg and Polterovich [13], we say that a smooth path tφ t u tPra,bs Ă Cont c 0 pM, ξq is positive (resp.negative) if 1. α φ t pxq `d dt φ t pxq ˘ą 0 presp.ă 0q, for all t P ra, bs, and for all x P M , when M is compact 2. α φ t pxq `d dt φ t pxq ˘ą 0 presp.ă 0q, for all t P ra, bs, and for all x P Interior `Supp `tφ t u ˘˘, when M is non compact.
Positivity or negativity of a path does not depend on the choice of the contact form but only on the choice of the co-orientation of ξ.One can also define the notion of non-negative (resp.non-positive) smooth path, by replacing the above strict inequalities with large inequalities.
If there exists a non-constant and non-negative contractible loop of compactly supported contactmorphisms then the contact manifold pM, ξq is called non universally orderable, in the other case we say that pM, ξq is universally orderable.When pM, ξq is universally orderable, Eliashberg and Polterovich in [13] defined a bi-invariant partial order on Č Cont c 0 pM, ξq.For more details we refer the reader to [13], [9] and to the remark 3.4 below.Remark 3.2.To prove that a contact manifold is orderable involves some "hard" symplectic/contact techniques.See for instance in [8], [9], [10], [12], [13] pseudo-holomorphic curves or generating functions techniques are used to prove the universal orderability of the unitary cotangent bundle and the 1-jet bundle of any compact manifold.We will see below (Remark 4.7) how Bhupal [2] and Sandon [22] show that pR 2n`1 , ξ st q and pR 2n ˆS1 , ξ st q are universally orderable using also generating functions techniques.Definition 3.3.Let tφ t u be a non-constant smooth path in Cont c 0 pM, Kerpαqq.We define the semilength functionals used to construct the oscillation norm by L osc `ptφ t uq :" inftN P N ˚| there exist K P N ˚, K ě N, and 0 " t 0 ă ...
L osc ´ptφ t uq :" inftN P N ˚| there exist K P N ˚, K ě N, and 0 " t 0 ă ... ă t K " 1 so that By convention we set inf H " `8.
The oscillation length of a smooth path tφ t u is then by definition As discussed in the previous section, to these seminorms we associate the application Ą ν osc :" max tĂ ν `, ´Ă ν ´u on Č Cont c 0 pM, ξq and the application ν osc on Cont c 0 pM, ξq defined by ν osc pφq " inftĄ ν osc `rtφ t usq |φ 1 " φ ( for all φ P Cont c 0 pM, ξq.Colin and Sandon in [11] showed that the applications ν osc and Ą ν osc are well defined conjugation invariant norms on Cont c 0 pM, ξq and on Č Cont c 0 pM, ξq) respectively if and only if pM, ξq is universally orderable.
Remark 3.4.An interesting property of the norm Ą ν osc is the compatibility with the bi-invariant partial order ľ defined on Č Cont c 0 pM, ξq by Eliashberg and Polterovich in [13] when pM, ξq is an universally orderable contact manifold.More precisely for any rtϕ t us ľ rtφ t us ľ Id we have Ą ν osc `rtϕ t us ˘ě Ą ν osc `rtφ t us ˘.

The Shelukhin norm
The Shelukhin length of a smooth path of compactly supported contactomorphisms tφ t u tPr0,1s in Cont c 0 pM, ξ " Kerpαqq is defined by To these length functionals we associate the applications Ă ν α S `rtφ t us ˘" inf L α S ptϕ t uq | rtϕ t us " rtφ t us ( and The application Ă ν α S is a pseudo-norm, and Shelukhin proved in [25] that the application ν α S is a norm.None of them is conjugation invariant, indeed for all ϕ P Cont c 0 pM, ξq and for all rtφ t us P Č Cont c 0 pM, ξq we have The fact that ν α S is non-degenerate is non-trivial.It is proved by Shelukhin in [25] using an energy-capacity inequality: if φ P Cont c 0 pM, ξq is not the identity then it displaces a ball, and its norm is greater than the capacity of this ball.Since the same argument cannot be applied to loops of contactomorphisms based at the identity, it may exist rtφ t us P π 1 pCont c 0 pM, ξqq ztIdu such that Ă ν α S `rtφ t us ˘" 0. For more details we refer the reader to [25].

The FPR norm
As the oscillation norm, the FPR norm [14] comes from two seminorms Ă ν α ˘and is well defined for a contact manifold pM, ξ " Kerpαqq that is universally orderable.Because the functionals

˙V
are not invariant under time reparametrization we cannot talk about semilength functionals, and so we are not going to talk about the geodesics of the FPR norm.

Generating functions and translation selector
An important ingredient to prove Theorems 1.1 and 1.2 is to be able to detect translated points of contactomorphisms and their translations.To do so and to avoid confusion we will sometimes see contactomorphisms of pR 2n ˆS1 , ξ st q as 1-periodic contactomorphisms of pR 2n`1 , ξ st q.More precisely, let Cont c 0 pR 2n`1 , ξ st q 1´per be the group of contactomorphisms φ of pR 2n`1 , ξ st q that can be joined to the identity by a smooth path tφ t u such that 1. Suppptφ t uq is contained in K ˆR, where K is a compact subset of R 2n 2. φ t px, y, z `kq " φ t px, y, zq `p0, 0, kq for any t P r0, 1s and for any k P Z.

The natural projection Cont c
0 pR 2n`1 , ξ st q 1´per Ñ Cont c 0 pR 2n ˆS1 , ξ st q is a group isomorphism whose inverse is given by lifting.Definition 4.1 ([22]).Let φ P Cont c 0 pR 2n ˆS1 , ξ st q be a contactomorphism and denote by r φ P Cont c 0 pR 2n`1 , ξ st q 1´per its lift.A point pp, rzsq P R 2n ˆS1 is a t-translated point for φ if r φpp, zq " pp, z `tq and gpp, rzsq " 0 where pp, zq P R 2n`1 is any point that projects on pp, rzsq and g denotes the conformal factor of φ with respect to α st " dz ´n ř i"1 y i dx i , i.e. φ ˚αst " e g α st .
The spectrum of φ is then defined by Spectrumpφq :" tt P R | φ has a t-translated pointu .
1.The definition of translated points depends on the choice of a contact form.
2. Let pp, rzsq P R 2n ˆS1 be a k-translated point of φ P Cont c 0 pR 2n ˆS1 , ξ st q with k P Z. Then pp, rzsq is a discriminant point of φ.In particular, Z-translated points do not depend on the choice of the contact form.In the same spirit we deduce that for all φ, ψ P Cont c 0 pR 2n ˆS1 , ξ st q Spectrumpφq X Z " Spectrumpψ ˝φ ˝ψ´1 q X Z.
Sandon in [22] constructed a function c : Cont c 0 pR 2n ˆS1 , ξ st q Ñ R ě0 φ Þ Ñ cpφq P Spectrumpφq that satisfies algebraic and topological properties that we list at the end of this section in Theorem 4.6 and that will be intensively used for the proofs of Theorem 1.1 and Theorem 1.2.
We call this function a translation selector.The rest of this section will be devoted to give the main steps of its construction that we will need for the last section concerning the proofs of Theorem 1.1, Theorem 1.1 and Proposition 5.5.For this purpose we will follow mainly [22].

The graph of a contactomorphism as a compact Legendrian of the 1-jet bundle
The aim of this paragraph is to associate to any contactomorphism φ P Cont c 0 pR 2n ˆS1 , ξ st q a Legendrian Λ φ of the 1-jet bundle of S 2n ˆS1 in such a way that there is a one-to-one correspondence between translated points of φ and intersections of Λ φ with the 0-wall of the 1-jet bundle.We give a rather explicit and detailed description of this construction since for proving the estimate of Proposition 5.5 below it will be important to make sure that all the maps involved in the process of this construction preserve not only the contact structures but also the contact forms.
First recall that for a smooth manifold X the 1-jet bundle of X is the manifold J 1 X :" T ˚X ˆR, where T ˚X is the cotangent bundle of X.This space carries a canonical contact structure ξ X given by the kernel of the 1-form α X :" dz ´λX where z is the coordinate function on R and λ X is the Liouville form of the cotangent bundle T ˚X .We denote by O X :" tpq, 0q P T ˚X | q P Xu the 0-section of the cotangent bundle T ˚X and refer to O X ˆR as the 0-wall of J 1 X.
Let φ P Cont c 0 pR 2n ˆS1 , ξ st q be a compactly supported contactomorphism and consider its 1periodic correspondent r φ P Cont c 0 pR 2n`1 , ξ st q 1´per .We write g and r g for their conformal factors with respect to α st , i.e. φ ˚αst " e g α st and r φ ˚αst " e r g α st .Then the image of the map is a Legendrian of R 2n`1 ˆR2n`1 ˆR endowed with the contact 1-form β :" α 2 st ´eθ α 1 st , where θ is the coordinate function on R and where α i st for i P t1, 2u is the pull back of α st by the projection pr i pp 1 , p 2 , θq " p i .

If tφ t u Ă Cont c
0 pR 2n ˆS1 , ξ st q is a smooth path of contactmorphisms then is a smooth path of Legendrians.
3. For any φ P Cont c 0 pR 2n ˆS1 , ξ st q, there is a one-to-one correspondence between the set of t-translated points of φ and Λ R 2n`1 r φ `R2n`1 ˘Ş O R 2n`1 ˆttu.
The aim now is to be able to replace the Euclidean space R 2n`1 of the previous lemma with a compact manifold without loosing the 3 properties above.This will be done in two steps.First using the periodicity we will replace R 2n`1 by R 2n ˆS1 .Then we will compactify the R 2n coordinates into the standard Euclidean sphere S 2n Ă R 2n`1 .
Because φ is 1-periodic in the z-direction, the map where z P R is any representant of rzs P S 1 " R{Z and pr : J 1 pR 2n`1 q Ñ J 1 pR 2n ˆS1 q the natural projection, is well defined.Note that the projection pr is a covering map that preserves the contact forms, i.e. pr ˚pα R 2n ˆS1 q " α R 2n`1 , and that the map Λ R 2n ˆS1 φ enjoys again the 3 properties of Lemma 4.3.
Since φ is compactly supported in the R 2n -direction, the map is a smooth Legendrian embedding that still enjoys the three properties of Lemma 4.3.Moreover now this Legendrian is compact.
In the case when φ is C 1 -small, Λ φ is a Legendrian section of J 1 pS 2n ˆS1 q and so it is given by the 1-jet of a function f : S 2n ˆS1 Ñ R, i.e.Λ φ pxq " px, d x f, f pxqq for all x P S 2n ˆS1 .In particular there is a one-to-one correspondence between critical points of f of critical value t and t-translated points of φ.So when φ P Cont c 0 pR 2n`1 , ξ st q z´per is C 1 -small, looking for translated points of φ is equivalent to looking for critical point of f .For the latter problem, Morse theory can be applied to ensure existence of critical points.
For a general contactomorphism φ P Cont c 0 pR 2n ˆS1 , ξ st q the map Λ φ does not need to be a section anymore, however it is smoothly isotopic to the 0-section of J 1 pS 2n ˆS1 q through a path of Legendrians thanks to Lemma 4.3.We will describe in the next paragraph how one can associate to such a Legendrian a function f : S 2n ˆS1 ˆRN Ñ R, for some N P N, such that we have again a one-to-one correspondence between critical points of f of critical value t and t-translated points of φ.Moreover a control of the behaviour of f at infinity will allow again to ensure existence of critical points of such a function f and so the existence of translated points of φ.

Generating functions
Let X be a smooth manifold.For any integer N P N, a function is said to be a generating function if 0 is a regular value of Bf Bv : X ˆRN Ñ pR N q ẘhere pR N q ˚is the set of linear forms on R N , and where Bf Bv is the derivative of f in the R N direction.It follows from the definition that Σ f :" ´Bf

¯´1
t0u is a smooth submanifold of X ˆRN of the same dimension of X whenever f : X ˆRN Ñ R is a generating function.In this case, denoting by Bf Bx the derivative of f in the X direction, the map We say that the immersed Legendrian Λ f :" j 1 f pΣ f q is generated by f .For all a P R there is a one-to-one correspondence between critical points of f with critical value a and intersections of the immersed Legendrian Λ f with O X ˆtau.
When X is a compact manifold, a sufficient condition to guarantee the existence of critical points of a smooth function f : X ˆRN Ñ R is to ask that the function f is quadratic at infinity, i.e. there exists a non-degenerate quadratic form Q : R N Ñ R and a compactly supported function g : X ˆRN Ñ R such that f px, vq " gpx, vq `Qpvq, for all px, vq P X ˆRN .The fundamental result about existence of generating functions is the following.Theorem 4.4 (Chaperon [6], Chekanov [7]).Let X be a compact manifold and tΛ t u tPr0,1s a smooth path of Legendrians in pJ 1 X, ξ X q such that Λ 0 " O X ˆt0u.Then there exists an integer N P N and a continuous path f t : X ˆRN Ñ R ( of generating functions quadratic at infinity such that f t generates the Legendrian Λ t for all t P r0, 1s.

Extracting critical values of generating functions
Classical minimax methods can be applied to extract a critical value of a function f : XˆR N Ñ R that is quadratic at infinity.Indeed, for any a P R we write X a :" tf ď au to designate the sublevel a of f .Since f is quadratic at infinity, there exists a 0 P R such that for all a ď a 0 the homotopy type of the sublevels X a and X a 0 are the same, and we write X ´8 to refer to such sublevel sets.In addition to that there exists a splitting of R N " R N `ˆR N ´for which the non-degenerate quadratic form Q is negative definite on R N ´, and so the Thom isomorphism guarantees the existence of an isomorphism where H ˚pX q is the cohomology of X with coefficient in Z 2 and H ˚pX ˆRN , X ´8q is the relative cohomology of pX ˆRN , X ´8q with coefficient in Z 2 .For any u P H ˚pX qzt0u the number cpf, uq :" inf ta P R | i åpT puqq ‰ 0u , where i a : pX a , X ´8q ãÑ pX ˆRN , X ´8q is the inclusion, is a critical value of f .Remark 4.5.Zapolski in [30] uses homology instead of cohomology to extract the wanted critical values.Let f : X ˆRN Ñ R be a generating function quadratic at infinity and α P H ˚pX qzt0u a non zero homology class.Then Cpf, αq :" infta P R | r T α P pi a q ˚pH ˚`N ´pX a , X ´8qqu, where r T : H ˚pX q Ñ H ˚`N ´pX ˆRN , X ´8q comes from the Thom isomorphism in homology, is a critical value of f .If α 0 is a generator of H n pXq, and µ 0 a generator of H n pXq, where n P N is the dimension of X, then Cpf, α 0 q " cpf, µ 0 q (see for instance [29]).
Due to the uniqueness property of generating functions quadratic at infinity proved by Viterbo [29] and Théret [26], one deduces that if f 1 and f 2 are two generating functions quadratic at infinity that generate the same Legendrian Λ Ă J 1 X that is Legendrian isotopic to O X ˆt0u, then cpf 1 , uq " cpf 2 , uq for any u P H ˚pX qzt0u.Thanks to this and to Theorem 4.4, for any Legendrian Λ that is Legendrian isotopic to O X ˆt0u and for any u P H ˚pX qzt0u the number cpΛ, uq :" cpf, uq is well defined, i.e. it does not depend on the generating function quadratic at infinity f that generates Λ. Combining this with the previous discussion in Subsection 4.2 we deduce that Λ X O X ˆtcpΛ, uqu ‰ H.
The translation selector constructed by Sandon is then given by where u 0 is a generator of H 2n`1 pS 2n ˆS1 q and Λ φ Ă J 1 pS 2n ˆS1 q is the Legendrian associated to φ constructed in Subsection 4.1.By (3) and Lemma 4.3 we come to the conclusion that cpφq P Spectrumpφq.Furthermore Sandon in [22] proved the following result.
Theorem 4.6 ([22]).The translation selector c satisfies the following properties: 1. cpφq ě 0 for any φ; 2. if tφ t u is a smooth path of compactly supported contactomorphisms starting at the identity, then t Þ Ñ cpφ t q is continuous; 3. rcpϕqs ě rcpφϕq ´cpφqs, in particular rcpφϕqs ď rcpφqs `rcpϕqs for any φ, ϕ; 4. if tφ t u and tϕ t u are smooth paths of compactly supported contactomorphisms starting at the identity such that α st `d dt φ t pxq ˘ď α st `d dt ϕ t pxq ˘for all t P r0, 1s and x P R 2n ˆS1 then cpφ 1 q ď cpϕ 1 q.
These extra properties of the translation selector will not be used for the proof of Theorem 1.1 and Theorem 1.2.However let us notice that they are used in [22], [21] to deduce (universal) orderability of pR 2n ˆS1 , ξ st q (similarly to the case of pR 2n`1 , ξ st q done by Bhupal [2]) and to show that the map T is a conjugation invariant norm.

Proof of the results
The idea of the proof of Theorem 1.1 and Theorem 1.2 is the following.First we will compute the selected translation of contactomorphisms that are generated by Hamiltonian functions satisfying the hypotheses of Theorems 1.1 and 1.2.Then we will show that all the norms we are working with are bounded from below by the translation selector.Finally a direct computation will allow to show that the length of the paths we are considering are equal to the selected translation of the time-one of these paths.We thus deduce that these paths are length minimizing paths and so are geodesics.

Computation of the selected translation
To any compactly supported time dependent function H : r0, 1s ˆR2n Ñ R, pt, pq Þ Ñ H t ppq, one can associate the smooth path of vector fields X ωst H : r0, 1s Ñ χpR 2n q defined by ι X ω st H ptq ω st " ´dH t for all t P R, where ω st " n ř i"1 dx i ^dy i .We denote by ψ t H its time t-flow for any t P R. Note that ψ t H is an Hamiltonian symplectomorphism for any t P R. Recall that from the formula (1) of the introduction one can similarly associate to a smooth compactly supported function h : r0, 1s ˆR2n ˆS1 Ñ R a path of contactomorphisms φ h .
In the next lemma for all z P R we will write rzs the corresponding element in R{Z " S 1 and debote by λ st the 1-form

Lemma 5.1. Let tφ t h u be a smooth path of contactomorphisms generated by the Hamiltonian function
where H : R 2n Ñ R is a smooth compactly supported function.Then for all pp, zq P R 2n ˆR and for all t P R φ t h pp, rzsq " `ψt H ppq, rz `F t ppqs ˘and pφ t h q ˚αst " α st where Proof.A direct computation shows that the contact vector field X h generated by the function h : R 2n ˆS1 Ñ R (see the relations (1) in the introduction) in this case is given by , for all pp, rzsq P R 2n ˆS1 .
Moreover, because the Hamiltonian function H is time-independent, H is constant along its flow, i.e.Hpψ t H ppqq " Hppq for all p P R 2n and for all t P R.So we deduce the formula (4).To prove the fact that φ t h is an exact contactomorphism for all t, i.e. it preserves not only the contact distribution ξ st but also the contact form α st " dz ´λst , we use the Cartan formula : pφ t h q ˚αst " pφ s h q ˚pι X h dα st `dι X h α st q " pφ s h q ˚p´dH `dHq " 0 , @s P R .
Since φ 0 h " Id, we deduce that pφ s h q ˚αst " `φ0 h ˘˚α st " α st for all s P R.
The hypothesis that we make about the smallness of the Hessian of H : R 2n Ñ R allows us to guarantee that the only periodic orbits of tψ t H u tPR of small periods are the constant ones.More precisely, identifying in the usual way the dual Euclidean space pR 2n q ˚with R 2n , one can see the Hessian of any smooth function H : R 2n Ñ R at a point p P R 2n as a linear map Hess p pHq : R 2n Ñ R 2n .The smallness of such maps will be measured in terms of the operator norm, i.e. if A : R 2n Ñ R 2n is a linear map then ||| Hess p pHq||| ă 2π.If there exists 0 ă T ď 1 and p P R 2n such that ψ T H ppq " p then ψ t H ppq " p for every t P R.
Proof.Let T P R ą0 and p P R 2n be such that ψ T H ppq " p and denote by γ : R{Z Ñ R 2n the loop γptq " ψ tT H ppq. The speed t Þ Ñ 9 γptq " X T H pγptqq of γ is again a smooth loop.Consider the Fourier series associated to this loop 9 γ ÿ kPZ e 2iπJt x k for all t P S 1 where x k P R 2n and J " Note that 9 γptq " JT ∇Hpγptqq for any t P S 1 , where ∇H is the Euclidean gradient of H, so : γptq " T Hess γptq Hp 9 γptqq for any t P S 1 and the Fourier series of the loop : γ is given by Using the fact that ş 1 0 9 γptqdt " γp1q ´γp0q " x 0 " 0 P R 2n and the Parseval's identity we have Moreover the above inequality is a strict inequality in the case when γ is not the constant loop.
On the other hand So combining this inequality with the inequality (5) we deduce that || 9 γ|| L 2 ă T || 9 γ|| L 2 when γ is not a constant loop, and so T ą 1.
Remark 5.3.The reader can find a similar proof in [16].
From the two previous lemmas we deduce the selected translation of the paths considered in Theorems 1.1 and 1.2 .||| Hess p pHq||| ă 2π.Then ´min H.

Proof. First let us show that
Spectrumpφ t h q " ttHppq | d p H " 0u for all t P r0, 1s.
Indeed if p P R 2n is such that d p H " 0 then using Lemma 5.1 we have that for all rzs P S 1 and t P r0, 1s φ t h pp, rzsq " pp, rz `tHppqsq and pφ t h q ˚αst " α st , and so ttHppq | d p H " 0u Ă Spectrumpφ t h q.Conversely, let t P r0, 1s and a P Spectrumpφ t h q.By definition of the spectrum and using again Lemma 5.1, this implies that there exists pp, zq P R 2n ˆR such that In particular p is a fixed point of ψ t H . Lemma 5.2 ensures that under the assumptions we made on the Hessian of H we have ψ s H ppq " p for all s P r0, 1s and d p H " 0. So we conclude that F t ppq " tHppq " a and that Spectrumpφ t h q Ă ttHppq | d p H " 0u.
Moreover, by Theorem 4.6, the map t Þ Ñ cpφ t h q P Spectrumpφ t h q is continuous, and since the set of critical values of H is a nowhere dense set due to Sard's theorem, there exists a critical point p 1 P R 2n of H such that cpφ t h q " tHpp 1 q for all t P r0, 1s.It remains to show that Hpp 1 q " max H.
For ε ą 0 small enough, Λ φ t h : S 2n ˆS1 Ñ J 1 pS 2n ˆS1 q is a Legendrian section for all t P r0, εs, and so Λ φ t h " j 1 f t is equal to the 1-jet of a generating function f t : S 2n ˆS1 Ñ R without extra coordinates.It is well known that in this case cpf t , u 0 q " max f t when u 0 denotes the generator of H 2n`1 pS 2n ˆS1 q: to see this one can combine Remark 4.5 with the arguments of Chapter 10 of [20].In addition to this, since f t is a generating function for φ t h we have We then deduce that cpφ t h q " max f t " t max H " tHpp 1 q, so H reaches its maximum at the point p 1 .Thus cpφ 1 h q " max H. Noticing that the Hamiltonian function ´h generates the path tpφ t h q ´1u " tφ ´t h u the same proof allows us to show that cppφ 1 h q ´1q " ´min H.

A lower bound for the Shelukhin norms and the FPR norms
The following proposition will allow us to find a lower bound for the Shelukhin and FPR norms in terms of the translation selector.To prove this proposition we will use Lemma 2.6 of [30].To state his lemma Zapolski in [30] fixes a non-zero homology class.We will state this lemma by fixing a non-zero cohomology class instead.By Remark 4.5 the two approaches are equivalent.
Lemma 5.6 ([30]).Let X be a closed manifold of dimension n, tΦ t u tPr0,1s be a smooth path of contactomorphisms in Cont c 0 pJ 1 X, Kerpα X qq starting at the identity and u 0 P H n pXqzt0u be the top class.Then where σ 0 : X Ñ J 1 X, x Þ Ñ px, 0, 0q is the 0-section.
In the following proof of Proposition 5.5 we use the notations and construction of Subsection 4.1.
Proof of Proposition 5.5.Let Ă φ k t P Cont c 0 pR 2n`1 , ξ st q z´per be the 1-periodic contactomorphism associated to φ t k for all t P r0, 1s and r0, 1s ˆS2n ˆS1 Ñ J 1 pS 2n ˆS1 q pt, pp, rzsqq Þ Ñ Λ φ k t pp, rzsq the associated Legendrian isotopy.The Legendrian isotopy extension theorem guarantees the existence of a smooth path of contactomorphisms tΦ t u Ă Cont c 0 `J1 pS 2n ˆS1 q, Kerpα S 2n ˆS1 q starting at the identity such that Φ t ˝ΛId " Λ φ k t for all t P r0, 1s.We claim that for all pp, rzsq P S 2n ztp 0 u ˆS1 and for all t P r0, 1s Moreover for all rzs P S 1 and for all t P r0, 1s a direct computation shows that α S 2n ˆS1 ˆd dt Φ t pΛ Id pp 0 , rzsqq ˙" 0 , (7) since Λ φ t k pp 0 , rzsq " ppp 0 , rzsq, 0q Ă J 1 pS 2n ˆS1 q is independent of time.So by Lemma 5.6 and equalities ( 6), (7) we have which proves the first inequality of Proposition 5.5.Using the fact that pφ 1 k q ´1 can be generated by the Hamiltonian function r0, 1s ˆR2n ˆS1 Ñ R pt, p, zq Þ Ñ ´k1´t pp, zq we deduce the second inequality of Proposition 5.5 exactly in the same way.
We deduce the following corollary.
Corollary 5.7.Let k : r0, 1s ˆR2n ˆS1 Ñ R be a compactly supported Hamiltonian function.Then Proof.By Proposition 5.5, By definition of the Shelukhin length functional we have max for any compactly supported Hamiltonian function k : r0, 1s ˆR2n ˆS1 Ñ R. By taking the infinimum of the length over all paths that represents φ 1 k (resp.rtφ k us) we deduce the first line of inequalities of Corollary 5.7.The second line of inequalities can be proved exactly in the same way and is left to the reader.

Computation of the Shelukhin length and proof of Theorem 1.1
The Shelukhin length of a path of contactmorphisms tφ t u Ă Cont c 0 pR 2n ˆS1 , ξ st q generated by a time independant compactly supported Hamiltonian function h : R 2n ˆS1 Ñ R is given by L αst S ptφ t uq " maxtmax h, ´min hu.Using Corollary 5.7 we then deduce that the inequalities of Proposition 5.5 are equalities in the case when tφ t u is generated by an Hamiltonian function satisfying the hypothese of Theorem 1.1 and so that it is a geodesic of the Shelukhin norm.A similar argument allows to compute the FPR norm of tφ t u in this case and conclude the proof of Theorem 1.1.Again we deduce that all these inequalities are in fact equalities.Finally using the fact the the FPR norm is conjugation invariant we get the desired result.

Proof of Theorem
The idea to prove the results for the discriminant and oscillation norm will follow the same lines: we have to show that the translation selector is a lower bound for these norms, and that the discriminant/oscillation length of the considered path realizes this lower bound.

A lower bound for the discriminant and oscillation norms
In the next proposition we formulate precisely how the translation selector gives a lower bound for the discriminant and oscillation norms.
Proposition 5.8.For any element φ P Cont c 0 pR 2n ˆS1 , ξ st q that is not the identity we have Proof.Let us first show that if tφ t u is a smooth path of compactly supported contactomorphisms starting at the identity in Cont c 0 pR 2n ˆS1 , ξ st q such that L d `tφ t u ˘" 1 then 0 ď cpφ t q ă 1 for all t P r0, 1s.Indeed, since the application t P r0, 1s Þ Ñ cpφ t q is continuous and cpφ 0 q " 0, if there exists t Ps0, 1s such that cpφ t q ě 1 then there exists t 0 Ps0, ts such that cpφ t 0 q " 1.This means that φ t 0 has a 1-translated point hence a discriminant point, which contradicts the fact that L d `tφ t u ˘" 1.So cpφ t q ă 1 for all t P r0, 1s whenever L d `tφ t u ˘" 1 and φ 0 " Id.
Let us consider an element φ P Cont c 0 pR 2n ˆS1 , ξ st qztIdu.Denote by k " ν d pφq P N ą0 its discriminant norm.Using the word metric definition of the discriminant norm, this means that there exist k smooth paths of compactly supported contactomorphisms `tφ t i u tPr0,1s ˘iPr1,ksXN such that 1. each of them starts at the identity and is of discriminant length 1 In remains to show the case when maxtcpφq, cpφ ´1 qu P N. To set the ideas down, let us assume that maxtcpφq, cpφ ´1 qu " cpφq, the case where maxtcpφq, cpφ ´1 qu " cpφ ´1q leads to the same conclusion by the same arguments.We already proved that k " ν d pφq ě cpφq, and we want to show that ν d pφq " k ą cpφq.If cpφq " 1 then using the argument of the first paragraph of this proof we know that ν d pφq ě 2 ą cpφq.So it remains to show the inequality for the cases where cpφq ě 2. Suppose by contradiction that cpφq " cp k ś i"1 φ 1 i q " k.Then rcpφ 1 i qs " 1 for all i P r1, ks, in particular cpφ 1 i q Ps0, 1r.Thanks to Theorem 4.6 we have rcp φ 1 i q ´cpφ 1 1 qs.Since cpφ 1  1 q Ps0, 1r and cpΠ k i"1 φ 1 i q P N we deduce that This leads us to the following contradiction which concludes the proof of the first inequality of the proposition.
The proof for the second inequality concerning the oscillation norm goes in the same way but will use one more argument based on the fourth point of Theorem 4.6.Let φ P Cont c 0 pR 2n Ŝ1 , ξ st qztIdu.By definition there exists rtφ t us P Č Cont c 0 pR 2n ˆS1 , ξ st q such that φ 1 " φ and ´`rtφ t us ˘) .
To set the ideas down we will assume that ν osc pφq " Ą ν osc `rtφ t us ˘" Ą ν osc ``rtφ t us ˘, the case where Ą ν osc `rtφ t us ˘" ´Ą ν osc ´`rtφ t us ˘leads to the same conclusion and is left to the reader. We If tϕ t u is non-positive by Theorem 4.6 we have cpϕ 1 q " 0. So using the triangular inequality we have again that rcpφqs ď k " Ą ν osc `prtφ t usq.
If we assume that cpφq R N then ν osc pφq ě tcpφqu `1 and we have the second inequality of the proposition.
Finally suppose that cpφq P N. If cpφq " 1 the argument from the first paragraph of this proof allows us to show that ν osc pφq ě 2 " tcpφqu `1 which proves again the second inequality of the proposition.So it remains to show the case where cpφq is an integer greater than 1.Let j " minti P r1, N s | tφ t i u is positiveu.Then thanks to Theorem 4.6 we have S c ˜N ź i"j`1 Let us assume by contradiction that cpφq " k.This implies as for the discriminant norm that for any i P r1, N s such that tφ t i u is positive, we must have cpφ 1 i q Ps0, 1r.By the fourth point of Theorem 4.6 we have k " cpφq ď cp N ś i"j 1 i q and by the triangular inequality we have cp So we deduce that cp N ś i"j φ 1 i q " k.Plugging this in the inequality (8) we obtain the following contradiction: rcpφ 1 i qs " k.

Computation of the discriminant and oscillation lengths and proof of Theorem 1.2
The next lemma will allow us to compute the discriminant and oscillation lengths of paths generated by time independent Hamiltonian functions and prove Theorem 1.2.
Lemma 5.9.Let h : R 2n ˆS1 Ñ R be a smooth compactly supported Hamiltonian function that generates the path of compactly supported contactomorphisms φ h .Then L d pφ h q " Z 1 t 0 ^`1 where t 0 " inf t ą 0 | InteriorpSuppphqq X DP pφ t h q ‰ H ( . If moreover we suppose that h : R 2n ˆS1 Ñ R is non-negative (resp.non-positive), then L d pφ h q " L osc pφ h q " Z 1 where by convention we set 1 0 " `8.

Straightforward arguments allow to show that
Supppφ h q " Suppphq , so t 0 " inf t ą 0 | InteriorpSupppφ h qq X DP pφ t h q ‰ H ( .
Suppose there exist 0 ď t ă s ď 1 such that DP ppφ t h q ´1φ s h q X InteriorpSupppφ h qq ‰ H. Since the Hamiltonian function h does not depend on time this implies that pφ t h q ´1φ s h " φ s´t h and so DP pφ s´t h q X InteriorpSupppφ h qq ‰ H.So by definition of t 0 we have that s ´t ě t 0 .
So if t 0 ą 0 and 1 t 0 is not an integer, by cutting the interval r0, 1s in , the discriminant length of φ h restricted to any of this interval is equal to one, i.e.
L d ˆtφ t h u tPri{r 1 t 0 s;pi`1q{r 1 t 0 ss ˙" 1, @i P . Moreover if we cut r0, 1s in stricly less than U intervals, then there exists at least one interval I Ă r0, 1s with length greater than t 0 and so L d `tφ t h u tPI ˘ě 2. We then deduce that when 1 ą t 0 ą 0 and 1 t 0 is not an integer we have Now let us consider the case when t 0 ą 0 and 1 t 0 is an integer.First we can see that if we cut r0, 1s into 1 t 0 intervals then at least one of them will be of length greater or equal to t 0 , and so the discriminant length of φ h restricted to this interval will be greater or equal than 2. So we deduce that L d pφ h q ě 1 t 0 `1.However if we cut r0, 1s into 1 t 0 `1 pieces such that each piece is of length 1{p 1 t 0 `1q then the same argument as before allows to show that the discriminant length of φ h restricted to any of these intervals is equal to one.So we conclude that in this case again Finally in the case t 0 " 0 it is obvious that L d pφ h q " `8.
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2.Let us first compute the t 0 coming from Lemma 5.9 : For all 0 ă t ď 1, pp, zq P DP pφ t h q if and only if pp, zq is a k-translated point for φ t h , where k is an integer.Using Lemmas 5.1 and 5.2, we know that pp, zq is a k-translated point of φ t h , for some t Ps0, 1s, if and only if p is a critical point of H and tHppq " k (see the proof of Lemma 5.4).So t 0 " inf tt ą 0 | Dpp, zq P Interior pSuppphqq , d p H " 0 and tHppq P Zu .
Since Suppphq " SupppHq ˆS1 and since 0 is a regular value of H inside of its support, we deduce that So by Lemma 5.9 we have L d pφ h q " maxttmax hu `1, t´min h`u `1u.
Finally, because the discriminant length and the discriminant norm are conjugation invariant we deduce Theorem 1.2.
The proof for the oscillation norm goes exactly the same way.

Proof of Proposition 1.4
Let us recall the statement of Proposition 1.4.

1 2
rcpU qs where ν denotes the FPR, discriminant, oscillation or Shelukhin norm.The proof of Proposition 1.4 is an immediate consequence of Corollary 5.7, Proposition 5.8 together with the inequality (2) of the introduction.Proof of Proposition 1.4.From Corollary 5.7 and Proposition 5.8 we deduce that 2 rνpφqs ě rcpφqs `Pcpφ ´1q T , where ν stands for any of the norms ν d , ν osc , ν αst F P R or ν αst S , which concludes the proof of Proposition 1.4.
ξq, one can show that this group law coincides with the concatenation of paths.More precisely let us fix a smooth bijective and increasing function a : r0, 1s Ñ r0, 1s such that all the derivatives of a at the time 0 and 1 vanish, and define for all smooth paths tφ t u, tϕ t u contained in Cont c 0 pM, ξq the concatenated path tϕ t u ¨tφ t u by tϕ t u ¨tφ t u :" Then rtφ t u ¨tψ t us " rtφ t us ˚rtψ t us.Moreover for all rtφ t us, rtϕ t us P Č Cont c 0 pM, ξq we have 1.rtφ t us ´1 " rtφ 1´t ˝pφ 1 q ´1us " rtpφ t q ´1us d and r ν d are well defined conjugation invariant norms.Another way to define the norms r ν d and ν d is to see them as word norms where the generating sets are respectively [11]tφ t us ˘" min L d ptϕ t uq | rtϕ t us " rtφ t us ( , @rtφ t us P Č Cont c 0 pM, ξqztIdu and r ν d pIdq " 0 ν d pφq " min L d ptφ t uq | φ 0 " Id and φ 1 " φ ( , @φ P Cont c 0 pM, ξqztIdu and ν d pIdq " 0.A priori the applications ν d and r ν d may take the value `8, but this is not the case.For a proof of this statement we refer the reader to[11].So for any co-oriented contact manifold pM, ξq the applications ν osc ptφ t uq " maxtL osc ˘| rtϕ t us " rtφ t us ( for all rtφ t us P Č Cont 0 pM, ξq.
[14]seminorms are defined for any rtφ t us P Č Cont c 0 pM, ξq and on Cont c 0 pM, ξq respectively.Moreover Fraser, Polterovich and Rosen showed in[14]that if the Reeb flow associated to the contact form α is periodic then Ć ν α F P R and ν α F P R are conjugation invariant norms.This comes from the fact that the fundamental group of Cont c 0 pM, ξq is included in the center of Ć Cont prtϕ t usq for any rtφ t us ľ rtϕ t us ľ Id.
The triangular inequality property that is satisfied by the translation selector (see Theorem 4.6) and the previous estimate of the translation selector on paths of discriminant length equals to 1 allow us to deduce that " ν d pφq , and so that ν d pφq ě rcpφqs.Since ν d pφ ´1q " ν d pφq we deduce that So if maxtcpφq, cpφ ´1 qu R N we proved the desired first inequality 2. φ "k ś