Skip to content
BY 4.0 license Open Access Published by De Gruyter December 1, 2016

The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

Alberto Abbondandolo , Luca Asselle , Gabriele Benedetti , Marco Mazzucchelli and Iskander A. Taimanov EMAIL logo

Abstract

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range (e0,e1) possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, e0=0 is the minimal energy of the system).

MSC 2010: 37J45; 58E05

1 Introduction

This paper is the last chapter of a work started in [3] and further developed in [2, 5, 6, 8] devoted to studying the multiplicity of periodic orbits on generic low energy levels in magnetic Tonelli Lagrangian systems on surfaces. Such a study was based on a generalization of Bangert’s waist theorem [10, Theorem 4], classically formulated for geodesic flows on S2, to the magnetic Tonelli setting. Roughly speaking, a waist is a non-constant periodic geodesic (resp. a periodic orbit in the Tonelli case) which minimizes the length (resp. the action) among nearby curves. The original waist theorem says that a Riemannian 2-sphere possesses infinitely many closed geodesics provided it possesses a waist. Such a statement is a crucial ingredient for the proof that, indeed, every Riemannian 2-sphere possesses infinitely many closed geodesics [11, 18, 19].

Let us introduce the general setting in which we will work. If M is a closed smooth manifold, a Tonelli Lagrangian L:TM is a smooth function whose restriction to any fiber of TM is superlinear with positive definite Hessian, see e.g. [20, 17, 1]. A magnetic Tonelli system is a pair (L,σ), where L:TM is a Tonelli Lagrangian and σ is a closed 2-form on M, which we refer to as the magnetic form. If π:T*MM denotes the projection of the cotangent bundle, the pair (L,σ) defines a flow on TM that is conjugated through the Legendre transformation vL to the Hamiltonian flow on (T*M,dpdq+π*σ) of the dual Tonelli Hamiltonian H:T*M, H(q,p)=max{pv-L(q,v)vTqM}, see e.g. [4, 22, 8]. A particularly relevant special case of this setting is the electromagnetic one, when the Lagrangian L is of the form L(q,v)=12gq(v,v)-U(q) for some Riemannian metric g and some smooth potential U:M. In this situation, the system (L,σ) models the motion of a particle on M with kinetic and potential energies described by L and under the further effect of a Lorentz force described by σ. When the potential U vanishes, the dynamics of the system (L,σ) is a so-called magnetic geodesic flow.

In this paper, we will focus on the case where M=S2. The energy function E:TM, E(q,v):=vL(q,v)v-L(q,v), is preserved along the motion. Therefore it is natural to study the dynamics of a magnetic Tonelli flow on a prescribed energy hypersurface E-1(e), and very different qualitative behaviors appear for different values of the energy e, see [13] and references therein. For our purposes two energy values will bear special significance: e0(L) and e1(L,σ). The former is the minimal energy e such that the corresponding energy hypersurface E-1(e)TS2 projects onto the whole S2. We postpone the precise definition of e1(L,σ) to Section 2. For now, we just mention that e1(L,σ)e0(L), and when σ is exact with primitive θ we have e1(L,dθ)=cu(L+θ), where cu(L+θ) is the Mañé critical value of the universal cover of L+θ (see e.g. [15, 1] for the definition of Mañé critical values).

The periodic orbits problem for magnetic geodesics was first studied by Novikov [21, 22] in the early 1980s. The classical least action principle for the periodic orbits with prescribed energy is not directly available in this setting, due to the potential non-exactness of the magnetic 2-form. Novikov showed how to recover the variational principle in the universal cover of the space of periodic curves, and in his celebrated “throwing out cycles” method he proposed how to exploit the corresponding deck transformation in order to detect action values of periodic orbits (for the throwing out cycles method, see also [23]). For magnetic geodesics on closed surfaces, waists were first studied by Taimanov in a series of papers [24, 25, 26]. Taimanov’s result is that, given a kinetic Lagrangian L(q,v)=12gq(v,v) and an oscillating magnetic 2-form σ on a closed two-dimensional configuration space, there exists a waist αe at the energy level e, for all e(0,e1(L,σ)) (see also [16] for a different proof). When σ is exact, Abbondandolo, Macarini, Mazzucchelli and Paternain [2] employed Taimanov’s waist αe on any energy level e belonging to a full measure subset of (0,e1(L,σ)) in order to construct a sequence of minmax families giving an infinite number of (geometrically distinct) periodic orbits with energy e. Short afterwards, Asselle and Benedetti extended the result to non-exact σ on surfaces of genus at least one [5, 6]. The results in [24, 25, 26, 2, 5, 6] have been further extended by Asselle and Mazzucchelli [8] to the general magnetic Tonelli setting. In this note we complete the picture by treating the last case remained open for the multiplicity problem: the 2-sphere. Namely, we are going to prove the following result.

Theorem 1.1

Let L:TS2R be a Tonelli Lagrangian, and σ a 2-form on S2. For almost every e(e0(L),e1(L,σ)), the Lagrangian system of (L,σ) possesses infinitely many periodic orbits with energy e.

We wish to stress that the existence of infinitely many periodic orbits on all energy values in (e0(L),e1(L,σ)) is still an open problem. In Theorem 1.1, as well as in [2, 5, 6], a negligible subset of energies must be excluded due to a lack of compactness in the variational setting that is employed. However, energy levels with only finitely many periodic orbits can be found above [27, 12] as well as below [8] the interval [e0(L),e1(L,σ)].

For closed surfaces M of genus at least one, any closed 2-form σ on M lifts to an exact 2-form on the universal cover of M. This allows to define the Mañé critical value of the universal cover cu(L,σ) for any Tonelli Lagrangian L:TM. We set e1*(L,σ):=min{e1(L,σ),cu(L,σ)} if M has positive genus, and e1*(L,σ):=e1(L,σ) if M=S2. The combination of Theorem 1.1 together with the above mentioned results in [5, 6, 8] yields the following statement about the multiplicity of periodic orbits on general closed surfaces.

Theorem 1.2

Let M be a closed surface, L:TMR a Tonelli Lagrangian, and σ a 2-form on M. For almost every e(e0(L),e1*(L,σ)), the Lagrangian system of (L,σ) possesses infinitely many periodic orbits with energy e.

Remark 1.3

The open interval (e0(L),e1*(L,σ)) is not empty for instance if M is orientable, σ is oscillating, and the Lagrangian has the form of a kinetic energy L(q,v)=12gq(v,v) for some Riemannian metric g (see [6]); in such case, e0(L)=0 is the minimal energy of the system. We recall that a 2-form σ on an orientable surface is oscillating when it satisfies σq-<0 and σq+>0 for some q-,q+M. On non-orientable surfaces, any non-zero 2-form lifts to an oscillating 2-form on the orientation double cover.

This paper is dedicated to the memory of Abbas Bahri. Bahri was interested in the problem of periodic orbits of magnetic geodesic flows. In a joint work with Taimanov [9], he established the existence of periodic magnetic geodesics with prescribed energy on closed configuration spaces of arbitrary dimension under the assumption that the analog of the Ricci curvature for the Lagrangian system is positive.

The paper is organized as follows. In Section 2 we recall the variational setting for our periodic orbits problem: we provide the definition of the action 1-form ηe, and of its global primitive Ae on the universal cover of the space of loops; at the end we will review the definition of the energy values e0 and e1, and the notion of a waist for magnetic Tonelli systems. In Section 3 we provide the proof of Theorem 1.1.

2 The Primitive of the Free-Period Action Form

2.1 The Variational Principle

Let L:TS2 be a Tonelli Lagrangian with associated energy function E(q,v)=vL(q,v)v-L(q,v), and σ a 2-form on S2. Since we will be interested in the Euler–Lagrange dynamics on a given energy hypersurface E-1(e), for some fixed e, we can modify the Tonelli Lagrangian far from E-1(e) and assume without loss of generality that each restriction L|TqM coincides with a polynomial of degree 2 outside a compact set. Let :=W1,2(𝕋;S2)×(0,), where 𝕋:=/ is the 1-periodic circle. For each energy value e, we consider the free-period action 1-form ηe on given by

η e ( γ , p ) ( ξ , q ) = d S e ( γ , p ) ( ξ , q ) + 𝕋 σ γ ( t ) ( ξ ( t ) , γ ˙ ( t ) ) d t , ( γ , p ) , ( ξ , q ) T ( γ , p ) ,

where Se: denotes the free-period action functional

S e ( γ , p ) = p 𝕋 L ( γ ( t ) , γ ˙ ( t ) / p ) d t + p e .

By the least action principle, ηe vanishes at some (γ,p) if and only if the p-periodic curve Γ(t):=γ(t/p) is an orbit of the magnetic Tonelli system of (L,σ), see e.g. [7] and references therein.

The 1-form ηe is not exact if σ is not exact. In order to work with a primitive of ηe, following Novikov [21, 22], we will lift it to the universal cover of . We see S2 as the unit sphere in 3, oriented in the usual way, and we fix the point x0=(-1,0,0)S2. We consider the universal cover

π : ~ .

As usual, we realize ~ as the space of homotopy classes relative to the endpoints of continuous paths u:[0,1] starting at u(0)=(x0,1). Here, we see x0 as the constant loop at x0. The projection map is given by π([u])=u(1). We have π*ηe=dAe, where the functional

A e : ~

is defined as follows. Given [u]~, we write u=(γ,p), where γ(s)W1,2(𝕋;S2) and p(s)(0,) for all s[0,1]. We see γ as a map of the form γ:[0,1]×𝕋S2 by setting γ(s,t):=γ(s)(t). We then set

A e ( [ u ] ) := S e ( u ( 1 ) ) + [ 0 , 1 ] × 𝕋 γ * σ .

Remark 2.1

Assume that US2 is a proper open subset, so that σ|U is exact with some primitive θ. Let 𝒰~ be a connected component of the open set of those [u]~ such that the periodic curve u(1) is contained in U. Up to an additive constant, the restriction Ae|𝒰 is equal to Seπ|𝒰, where Se: is the free-period action functional associated with the Lagrangian L+θ, i.e.

S e ( γ , p ) = p 𝕋 L ( γ ( t ) , γ ˙ ( t ) / p ) d t + γ θ + p e .

It is well known that the fundamental group of the free loop space W1,2(𝕋;S2) is isomorphic to , and therefore so is the fundamental group of . A generator [z] of π1(,(x0,1)) can be defined as follows. For each s𝕋, consider the affine plane Σs3 orthogonal to the vector (0,cos(2πs),-sin(2πs)) and passing through x0. We denote by ζ(s)W1,2(𝕋;S2) the closed curve with constant Euclidean speed whose support is precisely the intersection ΣsS2, its starting point is ζ(s)(0)=x0, and, for all s0, its orientation is such that the ordered pair sζ(s)(t),tζ(s)(t) agrees with the orientation of S2, see Figure 1.

Figure 1 
            The map ζ:𝕋→W1,2⁢(𝕋;S2)${\zeta:\mathbb{T}\to W^{1,2}(\mathbb{T};S^{2})}$.
Figure 1

The map ζ:𝕋W1,2(𝕋;S2).

We define z:=(ζ,1):𝕋. The group of deck transformations of the universal cover ~ is generated by

Z : ~ ~ , Z ( [ u ] ) = [ z * u ] ,

where z*u(s)=z(2s) for all s[0,1/2], and z*u(s)=u(2s-1) for all s[1/2,1]. The action Ae varies under such a transformation as

(2.1) A e Z ( [ u ] ) = A e ( [ u ] ) + S 2 σ .

2.2 Iterated Curves

For each v=(γ,p), we denote by vm=(γm,mp) its m-fold iterate, where γm(t)=γ(mt). The iteration map ψm:, ψm(v)=vm, is smooth. We lift this map to a smooth map of the universal cover, so that the following diagram commutes:

For instance, we can set ψ~m([u]):=[um], where

u m ( s ) = { ( x 0 , 1 + 2 s ( m - 1 ) ) if s [ 0 , 1 / 2 ] , u ( 2 s - 1 ) m if s [ 1 / 2 , 1 ] .

A remarkable property of the iteration map is given by the non-mountain pass theorem for high iterates, which was first established for electromagnetic Lagrangians in [2, Theorem 2.6], and extended to general Tonelli Lagrangians in [8, Lemma 4.3 and proof of Theorem 1.2]. As we explained in Remark 2.1, Ae coincides locally with the free-period action functional of a suitable Tonelli Lagrangian, and therefore the non-mountain pass theorem for high iterates holds for Ae as well.

Theorem 2.2

Theorem 2.2 (Non-mountain pass theorem for high iterates)

Let [v] be a critical point of Ae such that, for all mN, the critical circle of [vm] is isolated in the set of critical points of Ae. There exists m([v])N such that, for all integers m>m([v]), the following holds. There exists an (arbitrarily small) open neighborhood W of the critical circle of [vm] such that, if we set a:=Ae([vm]), the inclusion induces an injective map between path-connected components

π 0 ( { A e < a } ) π 0 ( { A e < a } 𝒲 ) .

2.3 The Critical Values of the Energy

Let us single out two significant values of the energy. The first one is e0(L):=maxE(,0), that is, the minimal energy e such that the corresponding energy hypersurface E-1(e) projects onto the whole S2. The second value e1(L,σ)e0(L), which depends also on the magnetic form σ, is defined as the supremum of the energies ee0(L) verifying the following condition: there exists a finite collection (γ1,p1),,(γn,pn) such that the γi’s are smooth pairwise disjoint loops, E(γi(),γ˙i()/pi)e for all i=1,,n, the multicurve γ1γn is the oriented boundary of a positively oriented compact embedded surface ΣS2, and we have

S e ( γ 1 , p 1 ) + + S e ( γ n , p n ) + Σ σ < 0 .

We recall that e1(L,σ) reduces to the classical Mañé critical value of L+θ in case σ is exact with primitive θ, see [6].

The proof of Theorem 1.1 will build on the following existence result, which was originally proved by Taimanov [24, 26] in the case of electromagnetic Lagrangians (see also [16] for an alternative proof), and further extended by Asselle and Mazzucchelli [8, Theorem 6.1] to the general case of magnetic Tonelli systems.

Theorem 2.3

For every energy value e(e0(L),e1(L,σ)), the Lagrangian system of (L,σ) possesses a non-self-intersecting periodic orbit (γe,pe) with energy e such that every element in π-1(γe,pe) is a local minimizer of the action functional Ae.

3 Proof of the Main Theorem

In this section we carry out the proof of Theorem 1.1. Since the case where the magnetic 2-form σ is exact is covered by [2], we focus on the case where σ is not exact, so that

(3.1) S 2 σ 0 .

3.1 Minmax Procedures

For each energy value e(e0(L),e1(L,σ)), consider the local minimizer (γe,pe) of Ae given by Theorem 2.3, and choose an arbitrary ueπ-1(γe,pe). We fix an arbitrary energy value

e * ( e 0 ( L ) , e 1 ( L , σ ) )

such that, for all m, the iterated critical point [ue*m] belongs to a critical circle that is isolated in crit(Ae*) (if there is no energy value e* with such a property, there are infinitely many periodic orbits on every energy level in the range (e0(L),e1(L,σ))). The critical points [ue*m] are still local minimizers of Ae*, as they are iterates of a local minimizer, see [3, Lemma 3.1] and Remark 2.1.

Given any subset Y~, for each m we will write

Y m := ψ ~ m ( Y ) = { [ y m ] [ y ] Y } .

The Palais–Smale condition holds locally for the free-period action functional of Tonelli Lagrangians, see [14, Proposition 3.12] or [1, Lemma 5.3]. This, together with Remark 2.1, implies that the functional Ae* satisfies the Palais–Smale condition locally as well. Therefore, a sufficiently small bounded open neighborhood 𝒲 of the critical circle of [ue*] does not contain other critical circles of Ae* and satisfies

inf 𝒲 A e * > A e * ( [ u e * ] ) ,
𝒲 m 0 Z n 1 ( 𝒲 m 1 ) = whenever ( m 0 , 0 ) ( m 1 , n 1 ) .

For any e(e0(L),e1(L,σ)), we denote by Me the closure of the set of local minimizers of Ae|𝒲. For all m0,m1 and n0,n1 such that (m0,n0)(m1,n1), we denote by

𝒫 e ( m 0 , n 0 , m 1 , n 1 )

the family of continuous paths Θ:[0,1]~ such that Θ(0)Zn0(Mem0) and Θ(1)Zn1(Mem1). We define the corresponding minmax value

c e ( m 0 , n 0 , m 1 , n 1 ) := inf { max A e Θ Θ 𝒫 e ( m 0 , n 0 , m 1 , n 1 ) } .

Lemma 3.1

There is an open neighborhood I(e0(L),e1(L,σ)) of e* such that

  1. M e is a non-empty compact set for all e I ,

  2. for each e , e I , we have max A e | M e < inf A e | 𝒲 ,

  3. for each m 0 , m 1 and n 0 , n 1 , the function e c e ( m 0 , n 0 , m 1 , n 1 ) is well defined and monotone increasing in I.

Proof.

The proof is entirely analogous to the arguments in [2, Lemmas 3.1–3.3] and it will be omitted. ∎

3.2 The Valley of Short Curves with Low Period

We equip our sphere S2 with an arbitrary Riemannian metric g, and with the Riemannian metric

(3.2) ( ξ , r ) , ( η , s ) = 𝕋 ( g ( ξ , η ) + g ( D t ξ , D t η ) ) d t + r s for all ( ξ , r ) , ( η , s ) T ( γ , p ) ,

where Dt denotes the covariant derivative associated to g. The space is not complete with respect to the Riemannian metric (3.2), nor is its universal cover equipped with the pulled-back Riemannian metric. Indeed, there are Cauchy sequences {(γn,pn)n} such that pn0. However, it turns out that this does not pose any problem while applying arguments from non-linear analysis to the functional Ae. Indeed, the functional Ae has a “valley” near the non-complete ends of , as we will review now (see [14, Section 3] and [7, Section 3] for analogous arguments in slightly different settings).

We write γ˙L2 for the L2-norm of the derivative of any curve γW1,2(𝕋;S2) measured with respect to g, i.e.

γ ˙ L 2 2 = 𝕋 g ( γ ˙ ( t ) , γ ˙ ( t ) ) d t .

We introduce the open subsets

(3.3) 𝒰 τ := { ( γ , p ) γ ˙ L 2 2 < τ p , p < τ } , τ > 0 .

If τ is small enough, 𝒰τ is connected and evenly covered by π:~. Namely, there exists a connected component 𝒱τπ-1(𝒰τ) such that π-1(𝒰τ) can be written as a disjoint union

π - 1 ( 𝒰 τ ) = n Z n ( 𝒱 τ ) .

We choose such a connected component 𝒱τ so that, for all [u]=[(γ,p)]𝒱τ with γ(1) stationary curve at some point qS2, we have

A e ( [ u ] ) = p ( 1 ) ( L ( q , 0 ) + e ) .

Lemma 3.2

For all τ>0 sufficiently small, we have

inf A e | 𝒱 τ = 0 , inf A e | 𝒱 τ > 0 .

Moreover,

lim τ 0 + ( sup A e | 𝒱 τ ) = 0 .

Proof.

We cover the sphere with two open balls D1,D2S2, and choose a primitive θi of σ on Di. Let τ>0 be sufficiently small so that for any γW1,2(𝕋;S2) with length less than τ there exists ι(γ){1,2} such that γ is entirely contained in Dι(γ). The restriction of the functional Ae to 𝒱τ takes the following form: for each [u]𝒱τ with (γ,p):=π([u]), we have

A e ( [ u ] ) = p 𝕋 L ( γ ( t ) , γ ˙ ( t ) / p ) d t + p e + γ θ ι ( γ ) .

Since we are assuming that the restriction of the Tonelli Lagrangian L to any fiber of TM is a polynomial of degree 2 outside a compact set, there exist constants 0<h1<h2 such that, for all (q,v)TM, we have

L ( q , v ) L ( q , 0 ) + v L ( q , 0 ) v + h 1 g q ( v , v )
- E ( q , 0 ) + v L ( q , 0 ) v + h 1 g q ( v , v )
(3.4) - e 0 ( L ) + v L ( q , 0 ) v + h 1 g q ( v , v )

and

(3.5) L ( q , v ) h 2 ( g q ( v , v ) + 1 ) .

We denote by λ the 1-form on S2 given by vL(,0). The lower bound (3.4) implies that, for all [u]𝒱τ with (γ,p):=π([u]), we have

A e ( [ u ] ) h 1 γ ˙ L 2 2 p + ( e - e 0 ( L ) ) > 0 p - | γ ( λ + θ ι ( γ ) ) |
h 1 γ ˙ L 2 2 p + ( e - e 0 ( L ) ) p - 1 4 d λ + d θ ι ( γ ) σ L γ ˙ L 2 2

where the latter inequality follows from [1, Lemma 7.1]. This readily implies that Ae>0 on 𝒱τ provided

h 1 τ > 1 4 d λ + σ L .

Assume now that [u]𝒱τ. If p=τ, we have

A e ( [ u ] ) > ( e - e 0 ( L ) ) τ > 0 .

If p<τ, then γ˙L22=pτ, and therefore

A e ( [ u ] ) h 1 τ - 1 4 d λ + σ L τ 2 > 0 .

Overall, this proves that infAe|𝒱τ>0.

Inequality (3.5) implies that, for all [u]𝒱τ with (γ,p):=π([u]), we have

A e ( [ u ] ) h 2 γ ˙ L 2 2 p + h 2 p + e p + γ θ ι ( γ )
h 2 γ ˙ L 2 2 p + h 2 p + e p + 1 4 σ L γ ˙ L 2 2
h 2 τ + h 2 τ + e τ + 1 4 σ L τ 2 ,

where, as before, the second inequality follows from [1, Lemma 7.1]. This readily implies that supAe|𝒱τ0 as τ0+, which, together with the fact that Ae>0 on 𝒱τ, also implies that infAe|𝒱τ=0. ∎

3.3 Essential Families

Let us fix an energy value eI. We say that a union of critical circles

crit A e { A e = c e ( m 0 , n 0 , m 1 , n 1 ) }

is an essential family for 𝒫e(m0,n0,m1,n1) when for every neighborhood 𝒰 of there exists a path Θ𝒫e(m0,n0,m1,n1) whose image Θ([0,1]) is contained in the union 𝒰{Ae<ce(m0,n0,m1,n1)}.

We denote by Idiscr the subset of those e(e0(L),e1(L,σ)) such that the set of critical points crit(Ae) is a union of isolated critical circles (that is, the periodic orbits with energy e are isolated). Notice that every energy level e(e0(L),e1(L,σ))Idiscr contains infinitely many periodic orbits. The existence of essential families can be guaranteed on generic energy levels in Idiscr. The precise statement is the following.

Lemma 3.3

There is a subset II of full Lebesgue measure such that, for all eIIdiscr, m0,m1N, and n0,n1Z with (m0,n0)(m1,n1), the space of paths Pe(m0,n0,m1,n1) admits an essential family.

Proof.

The proof goes along the lines of the one of [2, Lemma 3.5], but the fact that we are working on the universal cover of with the functional Ae requires some variations of the original argument, and therefore we provide full details for the reader’s convenience.

For all m0,m1 and n0,n1 such that (m0,n0)(m1,n1), we denote by I(m0,n0,m1,n1) the subset of those eI such that the function

(3.6) e c e ( m 0 , n 0 , m 1 , n 1 )

is differentiable at e. By Lemma 3.1 (iii), the function (3.6) is monotone increasing in e, and therefore I(m0,n0,m1,n1) is a full measure subset of I. We define the subset I of the statement as

I := ( m 0 , n 0 ) ( m 1 , n 1 ) I ( m 0 , n 0 , m 1 , n 1 ) .

Being a countable intersection of full Lebesgue measure subsets of I, the subset II has full Lebesgue measure as well.

Now, we fix eIIdiscr and two distinct (m0,n0),(m1,n1)×. In order to simplify the notation, we will just write ce and 𝒫e for ce(m0,n0,m1,n1) and 𝒫e(m0,n0,m1,n1), respectively. We choose an arbitrary strictly decreasing sequence {eαα}I such that eαe as α, and we set ϵα:=eα-e. By definition of I, there exists k0=k0(e)>0 such that

| c e α - c e | k 0 ϵ α for all α .

For all [u]=[(γ,p)]~ such that Ae([u])ce-ϵα and Aeα([u])ceα+ϵα, the period p(1) of the curve u(1) can be bounded as

p ( 1 ) = A e α ( [ u ] ) - A e ( [ u ] ) ϵ α c e α + ϵ α - c e + ϵ α ϵ α k 0 + 2 = : k 1 ,

while the action Ae([u]) can be bounded as

A e ( [ u ] ) A e α ( [ u ] ) c e α + ϵ α c e + ( k 0 + 1 ) ϵ α c e + k 1 ϵ α .

We introduce the subspaces

𝒳 r := { [ u ] = [ ( γ , p ) ] ~ p ( 1 ) r } , r > 0 .

By the definition of the minmax value ceα and by the estimates that we have just provided, for each α there exists a path Θα𝒫eα such that

Θ α ( [ 0 , 1 ] ) { A e c e - ϵ α } ( 𝒳 k 1 { A e c e + k 1 ϵ α } ) .

We recall that, by the definition of the spaces of paths 𝒫eα, we have that

Θ α ( i ) Z n i ( M e α m i ) Z n i ( 𝒲 m i ) , i = 0 , 1 .

Lemma 3.1 (ii) readily implies that we can attach two suitable tails to the path Θα: we can find two continuous paths

Φ α : [ 0 , 1 ] Z n 0 ( 𝒲 m 0 ) { A e A e ( Θ α ( 0 ) ) } ,
Ψ α : [ 0 , 1 ] Z n 1 ( 𝒲 m 1 ) { A e A e ( Θ α ( 1 ) ) } ,

such that Φα(0)Zn0(Mem0), Φα(1)=Θα(0), Ψα(0)=Θα(1), and Ψα(1)Zn1(Mem1); see [2, Lemma 3.2] for a proof of this elementary fact. Since the open set 𝒲 is bounded, there exists k2>k1 large enough such that

Z n 0 ( 𝒲 m 0 ) Z n 1 ( 𝒲 m 1 ) 𝒳 k 2 .

We define the continuous path

Υ α : [ 0 , 1 ] { A e c e - ϵ α } ( 𝒳 k 2 { A e c e + k 1 ϵ α } ) ,
Υ α ( s ) := { Φ α ( 3 s ) if s [ 0 , 1 / 3 ] , Θ α ( 3 ( s - 1 / 3 ) ) if s [ 1 / 3 , 2 / 3 ] , Ψ α ( 3 ( s - 2 / 3 ) ) if s [ 2 / 3 , 1 ] .

Notice that Υα𝒫e, and maxAeΥαce as α.

We claim that crit(Ae)Ae-1(ce)𝒳k2+2 is an essential family for 𝒫e. Let 𝒰~ be an arbitrary open set such that

𝒰 crit ( A e ) = crit ( A e ) A e - 1 ( c e ) 𝒳 k 2 + 2 .

Our goal for the remainder of the proof is to deform one of our paths Υα, away from its endpoints, so that the modified path will have image inside {Ae<ce}𝒰. Notice that, since eIdiscr, if μ>0 is small enough we have

(3.7) 𝒰 crit ( A e ) = crit ( A e ) A e - 1 [ c e - μ , c e + μ ] 𝒳 k 2 + 2 ,

and 𝒰 contains at most finitely many critical circles of Ae. In particular, we can find a smaller open neighborhood 𝒰𝒰 of 𝒰crit(Ae) and some >0 such that every smooth path Θ:[0,1]𝒰¯ with Θ(0)𝒰 and Θ(1)𝒰 has length at least . Here, the length is the one measured with respect to the pull-back of the Riemannian metric (3.2) to the universal cover ~.

Consider the open subsets 𝒰τ introduced in (3.3), and the selected connected components of their preimage 𝒱τπ-1(𝒰τ). Since eIdiscr, the set Me is the union of finitely many critical circles of Ae. In particular, there exists τ2>0 small enough such that

{ Υ α ( 0 ) , Υ α ( 1 ) α } π - 1 ( 𝒰 τ 2 ) = .

If needed, we reduce τ2>0 so that the open subset 𝒰τ2 is connected and evenly covered by π:~. By Lemma 3.2, there exist δ>0 and 0<τ1<τ2 such that, for all n,

(3.8) inf A e | ( Z n ( 𝒱 τ 2 ) ) - sup A e | Z n ( 𝒱 τ 1 ) δ .

Finally, we fix an index α large enough so that

(3.9) k 1 ϵ α < min { μ , δ } .

In the following, we will denote by the Riemannian norm induced by the Riemannian metric (3.2). With a slight abuse of notation, we will denote by also the Riemannian norm that is pulled-back to the universal cover ~. Fix τ0(0,τ1) and introduce a vector field on ~ of the form V:=fAe, for some suitable smooth function f:~[-1,0], such that

  1. V ( [ u ] ) 2 for all [u]~,

  2. supp ( V ) A e - 1 [ c e - ϵ α - 1 , c e + k 1 ϵ α - 1 ] π - 1 ( 𝒰 τ 0 ) ,

  3. d A e ( [ u ] ) V ( [ u ] ) - min { A e ( [ u ] ) 2 , 1 } for all [u]~π-1(𝒰τ1) such that Ae([u])[ce-ϵα,ce+k1ϵα].

We denote by ϕt:~~ the flow of V. This flow is complete. Indeed, since the vector field V is uniformly bounded, the flow lines that may not be defined for all positive times are those that enter all sets 𝒳r, for r>0 arbitrarily small. Since V is non-negatively proportional to -Ae, its flow lines are non-negative reparametrizations of those of -Ae. Finally, if a flow line of -Ae is not defined for all positive times, then it must enter the set π-1(𝒰τ0) (see [7, Proposition 3.1 (2)] for a proof of this fact), but this latter set is outside the support of V. Actually, since V2, we have

ϕ 1 ( 𝒳 k 2 ) 𝒳 k 2 + 2 .

The free-period action form ηe satisfies a generalized Palais–Smale condition on subsets of where the period is bounded from above and bounded away from zero, see [7, Theorem 2.1 (2)]. Moreover, for each sequence {(γn,pn)n} such that pn0 and ηe(γn,pn)0 as n, we have γ˙nL22/pn0 as n, see [7, Theorem 2.1 (1)]. In particular, (γn,pn) belongs to 𝒰τ0 for n large enough. This, together with (3.7), implies that there exists a constant ν(0,1) such that

(3.10) A e ( [ u ] ) ν for all [ u ] A e - 1 [ c e - μ , c e + μ ] 𝒳 k 2 + 2 ( π - 1 ( 𝒰 τ 0 ) 𝒰 ) .

We fix an index βα large enough so that k1ϵβ<min{ν,ν2}, which together with (3.9) implies

(3.11) k 1 ϵ β < min { μ , δ , ν , ν 2 } .

The composition ϕ1Υβ belongs to 𝒫e. We claim that its image ϕ1Υβ([0,1]) is contained in {Ae<ce}𝒰, which sets our goal for the proof. First of all, since Ae does not increase along the flow lines of ϕt, we have

(3.12) A e ( ϕ t Υ β ) A e ( Υ β ) c e + k 1 ϵ β for all t [ 0 , 1 ] .

There are three possible cases to consider:

  1. If ϕtΥβ(s)𝒰 for some t[0,1] and ϕ1Υβ(s)𝒰, equations (3.10), (3.11), and (3.12) imply that

    A e ( ϕ 1 Υ β ( s ) ) = A e ( ϕ t Υ β ( s ) ) + t 1 d A e ( ϕ r Υ β ( s ) ) V ( ϕ r Υ β ( s ) ) d r
    c e + k 1 ϵ β - ν t 1 V ( ϕ r Υ β ( s ) ) d r
    c e + k 1 ϵ β - ν < c e .

  2. If ϕtΥβ(s)π-1(𝒰τ1) for some t[0,1], then, since ϕtΥβ(s)π-1(𝒰τ2), equations (3.8), (3.11), and (3.12) imply

    A e ( ϕ 1 Υ β ( s ) ) A e ( ϕ t Υ β ( s ) ) c e + k 1 ϵ β - δ < c e .

  3. If ϕtΥβ(s)𝒰π-1(𝒰τ1) for all t[0,1], then property (iii) in the definition of V above, together with equations (3.10), (3.11), and (3.12), implies

    A e ( ϕ 1 Υ β ( s ) ) = A e ( Υ β ( s ) ) + 0 1 d A e ( ϕ r Υ β ( s ) ) V ( ϕ r Υ β ( s ) ) d r
    = c e + k 1 ϵ β - 0 1 d A e ( ϕ r Υ β ( s ) ) 2 d r
    c e + k 1 ϵ β - ν 2 < c e .

Overall, we showed that, for an arbitrary s[0,1], if ϕ1Υβ(s) is not contained in 𝒰, then it is contained in the sublevel set {Ae<ce}.∎

Lemma 3.4

For each eIIdiscr and [v]crit(Ae), there exists a constant m([v])N with the following property. Consider the critical circle C of a critical point Zn([vm]), where nZ and m>m([v]). If E is an essential family containing C, then EC is an essential family for the same space of paths as well.

Proof.

We set am,n:=Ae(Zn([vm])), where m and n. By Theorem 2.2, there exists m([v]) such that, for all integers m>m([v]), the following statement holds. There exists an (arbitrarily small) open neighborhood 𝒲 of the critical circle of [vm] such that the inclusion induces an injective map between path-connected components

π 0 ( { A e < a m , 0 } ) π 0 ( { A e < a m , 0 } 𝒲 ) .

For every n, we denote by 𝒲n:=Zn(𝒲) the corresponding neighborhood of the critical circle 𝒞 of Zn([vm]). Clearly, the inclusion induces an injective map

(3.13) π 0 ( { A e < a m , n } ) π 0 ( { A e < a m , n } 𝒲 n ) .

Now, assume that 𝒞 belongs to an essential family for 𝒫e(m0,n0,m1,n1). In particular, we have am,n=ce(m0,n0,m1,n1).

We require the neighborhood 𝒲 to be small enough so that for all neighborhoods 𝒲 of 𝒞 sufficiently small, we have 𝒲¯n𝒲¯=. The existence of such a disjoint 𝒲¯ is guaranteed by the fact that the set of critical points of Ae comes in isolated critical circles. Since, by Lemma 3.3, 𝒫e(m0,n0,m1,n1) admits an essential family, there exists a continuous path Θ𝒫e(m0,n0,m1,n1) whose image is contained in the union

𝒲 n 𝒲 { A e < c e ( m 0 , n 0 , m 1 , n 1 ) } .

Notice that

(3.14) max { A e ( Θ ( 0 ) ) , A e ( Θ ( 1 ) ) } < c e ( m 0 , n 0 , m 1 , n 1 ) .

Indeed, Θ(0) and Θ(1) belong to distinct critical circles that are isolated local minimizers of Ae, and this latter functional satisfies the Palais–Smale condition locally.

By (3.14) and since the map (3.13) is injective, there exists another path Θ𝒫e(m0,n0,m1,n1) whose image is contained in the union

𝒲 { A e < c e ( m 0 , n 0 , m 1 , n 1 ) } .

Therefore, 𝒞 is also an essential family for 𝒫e(m0,n0,m1,n1). ∎

Now, let Ifinite be the (possibly empty) subset of those energy values eIdiscr such that there are only finitely many (non-iterated) periodic orbits with energy e. In order to prove Theorem 1.1, all we need to do is to prove that the intersection IIfinite is empty. We will show this in Theorem 3.7, after exploring what would happen on energy values in IIfinite.

Lemma 3.5

For each energy level eIIfinite and compact interval [a0,a1]R, there exists a finite union of critical circles Ecrit(Ae) such that, for all m0,m1N and n0,n1Z with ce(m0,n0,m1,n1)[a0,a1], E contains an essential family for Pe(m0,n0,m1,n1).

Proof.

Let (γ1,p1),,(γr,pr) be the only non-iterated periodic orbits with energy e, where r is some natural number, and choose [vi]π-1(γi,pi) for all i=1,,r. Consider the constants m([vi]) given by Lemma 3.4, so that if we remove the critical circle of any Zn([vim]) with n and m>mmax from an essential family contained in Ae-1(ce(m0,n0,m1,n1)), the result is still an essential family for the same space of paths. We set

m max := max { m ( [ v 1 ] ) , , m ( [ v r ] ) } .

By equations (2.1) and (3.1), we infer that there exists nmax such that Ae(Zn([vim]))[a0,a1] for all i{1,,r}, m, and n with mmmax and |n|>nmax. We claim that the statement of the lemma holds taking

:= { Z n ( [ v i m ] ) i { 1 , , r } ,  1 m m max , | n | n max } .

Indeed, consider m0,m1 and n0,n1 such that ce(m0,n0,m1,n1)[a0,a1]. Let be an essential family for 𝒫e(m0,n0,m1,n1), whose existence is guaranteed by Lemma 3.3. By Lemma 3.4, if we remove from all the critical circles of periodic orbits of the form Zn([vim]) for m>mmax, the resulting set is still an essential family for 𝒫e(m0,n0,m1,n1). Therefore, is an essential family for 𝒫e(m0,n0,m1,n1). ∎

Let m0 and n0. For all m1, and n1 we know that ce(m0,n0,m1,n1) is bounded from below by minAe|Zn0(Mem0). Hence, the following quantity is a well-defined real number:

c e ( m 0 , n 0 ) := inf { c e ( m 0 , n 0 , m 1 , n 1 ) m 1 , n 1 with ( m 1 , n 1 ) ( m 0 , n 0 ) } .

The deck transformation Zk induces a homeomorphism between the spaces of paths 𝒫e(m0,n0,m1,n1) and 𝒫e(m0,n0+k,m1,n1+k), and we have

c e ( m 0 , n 0 + k , m 1 , n 1 + k ) = c e ( m 0 , n 0 , m 1 , n 1 ) + k S 2 σ for all k .

This readily implies

(3.15) c e ( m 0 , n 0 + k ) = c e ( m 0 , n 0 ) + k S 2 σ for all k .

The infimum in the definition of ce(m0,n0) is actually attained provided eIIfinite.

Lemma 3.6

If eIIfinite, for all (m0,n0)N×Z there exists (m1,n1)N×Z such that (m0,n0)(m1,n1) and ce(m0,n0)=ce(m0,n0,m1,n1).

Proof.

Let us fix (m0,n0)×, and set

a 0 := min A e | Z n 0 ( M e m 0 ) c e ( m 0 , n 0 , m 0 + 1 , n 0 ) = : a 1 .

Notice that ce(m0,n0)[a0,a1]. By Lemma 3.5, there exists a finite union of critical circles crit(Ae) such that, whenever ce(m0,n0,m1,n1)[a0,a1], contains an essential family for 𝒫e(m0,n0,m1,n1). We introduce the finite set of critical values

F := { A e ( [ w ] ) [ w ] } .

The value ce(m0,n0) is the infimum of those ce(m0,n0,m1,n1) belonging to the finite set F, and therefore it is a minimum. ∎

3.4 The Main Multiplicity Result

Theorem 1.1 is an immediate consequence of the following more precise statement.

Theorem 3.7

The set IIfinite is empty. Namely, for all energy values eI, there are infinitely many periodic orbits with energy e.

In the proof of Theorem 3.7, we will need the following abstract lemma established in [2, Lemma 2.5] for the free-period action functional. Being a local statement, such a lemma holds for the functional Ae as well (see Remark 2.1).

Lemma 3.8

Every isolated critical circle Ccrit(Ae)Ae-1(c) has an arbitrarily small open neighborhood U such that the intersection U{Ae<c} has only finitely many connected components.

Proof of Theorem 3.7.

We assume by contradiction that there exists eIIfinite. We set

a 0 := 0 < | S 2 σ | = : a 1 .

Lemma 3.5 provides a finite union of critical circles

= 𝒞 1 𝒞 s crit ( A e )

such that, whenever ce(m0,n0,m1,n1)[a0,a1], contains an essential family for 𝒫e(m0,n0,m1,n1). By equation (3.15) and Lemma 3.6, for each m there exist nm and (mm,nm)× such that

a 0 c e ( m , n m ) = c e ( m , n m , m m , n m ) < a 1 .

In particular, contains an essential family for 𝒫e(m,nm,mm,nm). For each i=1,,s, we consider an open neighborhood 𝒰i of the critical circle 𝒞i given by Lemma 3.8. We define

:= i = 1 , , s { 𝒱 𝒱 is a connected component of 𝒰 i { A e < A e ( 𝒞 i ) } } .

Notice that has finite cardinality according to Lemma 3.8. For each m, there exists 𝒱m with the following property: there exist a path Θm𝒫e(m,nm,mm,nm) and sm[0,1] such that the restriction Θm|[0,sm] is contained in the sublevel set {Ae<ce(m,nm,mm,nm)}, and Θm(sm)𝒱m. Since is finite, by the pigeonhole principle there exist distinct m1,m2 such that 𝒱m1=𝒱m2. In particular, ce(m1,nm1,mm1,nm1)=ce(m2,nm2,mm2,nm2).

Consider the path Θ:[0,1]~ obtained by concatenation of three paths: the restricted path Θm1|[0,sm1], some path connecting Θm1(sm1) with Θm2(sm2) within 𝒱m1, and the restricted path Θm2|[0,sm2] traversed in the opposite direction. By construction, Θ𝒫e(m1,nm1,m2,nm2). However,

max A e Θ < c e ( m 1 , n m 1 , m m 1 , n m 1 ) = c e ( m 1 , n m 1 ) c e ( m 1 , n m 1 , m 2 , n m 2 ) ,

which contradicts the definition of ce(m1,nm1,m2,nm2). ∎


Dedicated to the memory of Abbas Bahri (1955–2016)



Communicated by Paul Rabinowitz


Award Identifier / Grant number: AB 360/2-1

Award Identifier / Grant number: SFB 878

Award Identifier / Grant number: WKBHJ (ANR-12-BS01-0020)

Award Identifier / Grant number: COSPIN (ANR-13-JS01-0008-01)

Funding statement: A.A. and L.A. are partially supported by the DFG grant AB 360/2-1 “Periodic orbits of conservative systems below the Mañé critical energy value”. G.B. is partially supported by the DFG grant SFB 878. M.M. is partially supported by the ANR grants WKBHJ (ANR-12-BS01-0020) and COSPIN (ANR-13-JS01-0008-01). Part of this project was carried out while M.M. was visiting the Sobolev Institute of Mathematics in Novosibirsk (Russia), under the Program “Short-Term Visits to Russia by Foreign Scientists” of the Dynasty Foundation; M.M. wishes to thank the Foundation and Alexey Glutsyuk for providing financial support, and Iskander A. Taimanov for the kind hospitality.

References

[1] Abbondandolo A., Lectures on the free period Lagrangian action functional, J. Fixed Point Theory Appl. 13 (2013), no. 2, 397–430. 10.1007/s11784-013-0128-1Search in Google Scholar

[2] Abbondandolo A., Macarini L., Mazzucchelli M. and Paternain G. P., Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level, preprint 2014, https://arxiv.org/abs/1404.7641; to appear in J. Eur. Math. Soc. (JEMS). 10.4171/JEMS/674Search in Google Scholar

[3] Abbondandolo A., Macarini L. and Paternain G. P., On the existence of three closed magnetic geodesics for subcritical energies, Comment. Math. Helv. 90 (2015), no. 1, 155–193. 10.4171/CMH/350Search in Google Scholar

[4] Arnold V. I., Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR 138 (1961), 255–257. 10.1007/978-3-642-01742-1_13Search in Google Scholar

[5] Asselle L. and Benedetti G., Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1525–1545. 10.1007/s00526-015-0834-1Search in Google Scholar

[6] Asselle L. and Benedetti G., Periodic orbits in oscillating magnetic fields on T2, preprint 2015, https://arxiv.org/abs/1510.00152. Search in Google Scholar

[7] Asselle L. and Benedetti G., The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles, J. Topol. Anal. 8 (2016), no. 3, 545–570. 10.1142/S1793525316500205Search in Google Scholar

[8] Asselle L. and Mazzucchelli M., On Tonelli periodic orbits with low energy on surfaces, preprint 2016, https://arxiv.org/abs/1601.06692. 10.1090/tran/7185Search in Google Scholar

[9] Bahri A. and Taimanov I. A., Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2697–2717. 10.1090/S0002-9947-98-02108-4Search in Google Scholar

[10] Bangert V., Closed geodesics on complete surfaces, Math. Ann. 251 (1980), no. 1, 83–96. 10.1007/BF01420283Search in Google Scholar

[11] Bangert V., On the existence of closed geodesics on two-spheres, Internat. J. Math. 4 (1993), no. 1, 1–10. 10.1142/S0129167X93000029Search in Google Scholar

[12] Benedetti G., Magnetic Katok example on the two-sphere, preprint 2016, https://arxiv.org/abs/1507.05341; to appear in Bull. Lond. Math. Soc. 10.1112/blms/bdw050Search in Google Scholar

[13] Cieliebak K., Frauenfelder U. and Paternain G. P., Symplectic topology of Mañé’s critical values, Geom. Topol. 14 (2010), no. 3, 1765–1870. 10.2140/gt.2010.14.1765Search in Google Scholar

[14] Contreras G., The Palais–Smale condition on contact type energy levels for convex Lagrangian systems, Calc. Var. Partial Differential Equations 27 (2006), no. 3, 321–395. 10.1007/s00526-005-0368-zSearch in Google Scholar

[15] Contreras G. and Iturriaga R., Global Minimizers of Autonomous Lagrangians, 22o Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1999. Search in Google Scholar

[16] Contreras G., Macarini L. and Paternain G. P., Periodic orbits for exact magnetic flows on surfaces, Int. Math. Res. Not. IMRN 2004 (2004), no. 8, 361–387. 10.1155/S1073792804205050Search in Google Scholar

[17] Fathi A., Weak KAM Theorem in Lagrangian Dynamics. Preliminary Version Number 10, Cambridge University Press, Cambridge, 2008. Search in Google Scholar

[18] Franks J., Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. 10.1007/BF02100612Search in Google Scholar

[19] Hingston N., On the growth of the number of closed geodesics on the two-sphere, Int. Math. Res. Not. IMRN 1993 (1993), no. 9, 253–262. 10.1155/S1073792893000285Search in Google Scholar

[20] Mather J. N., Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169–207. 10.1007/BF02571383Search in Google Scholar

[21] Novikov S. P., Variational methods and periodic solutions of equations of Kirchhoff type. II, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 37–52, 96. 10.1007/BF01106155Search in Google Scholar

[22] Novikov S. P., The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49, 248. Search in Google Scholar

[23] Taimanov I. A., The principle of throwing out cycles in Morse–Novikov theory, Dokl. Akad. Nauk SSSR 268 (1983), no. 1, 46–50. Search in Google Scholar

[24] Taimanov I. A., Non-self-itersecting closed extremals of multivalued or not everywhere positive functionals, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 2, 367–383. Search in Google Scholar

[25] Taimanov I. A., Closed extremals on two-dimensional manifolds, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 143–185, 223. 10.1070/RM1992v047n02ABEH000880Search in Google Scholar

[26] Taimanov I. A., Closed non-self-intersecting extremals of multivalued functionals, Sibirsk. Mat. Zh. 33 (1992), no. 4, 155–162, 223. Search in Google Scholar

[27] Ziller W., Geometry of the Katok examples, Ergodic Theory Dynam. Systems 3 (1983), no. 1, 135–157. 10.1017/S0143385700001851Search in Google Scholar

Received: 2016-08-12
Accepted: 2016-11-04
Published Online: 2016-12-01
Published in Print: 2017-02-01

© 2017 by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 27.11.2022 from frontend.live.degruyter.dgbricks.com/document/doi/10.1515/ans-2016-6003/html
Scroll Up Arrow