Normal forms for strong magnetic systems on surfaces: Trapping regions and rigidity of Zoll systems

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily small rescalings.


Introduction
Let M be a closed, oriented surface. A magnetic system on M is a pair (g, b), where g is a Riemannian metric on M and b : M → R is a function, which we refer to as the magnetic function. A (g, b)-geodesic is a curve γ : R → M which is parametrised by arc-length and solves the equation where κ γ is the geodesic curvature of γ. The term "magnetic system" refers to the fact that a solution of (1.1) describes a trajectory of a particle γ with unit charge and speed under the effect of the Lorentz force generated by a stationary magnetic field. To fix ideas, if M is embedded in the euclidean three-dimensional space R 3 and B : R 3 → R 3 is a magnetic field in the ambient space, then g is the restriction of the euclidean metric on M and b is the inner product of B with the unit normal to M in R 3 .
The tangent lifts (γ,γ) of (g, b)-geodesics yield the trajectories of a flow Φ (g,b) : SM × R → SM on the unit sphere bundle SM , whose dynamical properties have been the subject of an intensive research since the where π : T M → M is the canonical projection, π(q, v) = q. Notice that λ is the pull-back of the Liouville one-form on T * M by means of the metric g. We define the symplectic form ω (g,b) ∈ Ω 2 (T M ) on T M via ω (g,b) := dλ − π * (bµ) and the kinetic Hamiltonian function via where | · | is the norm associated with g. We denote by X ω (g,b) Hg the Hamiltonian vector field of H g with respect to ω (g,b) . Let S r M := {(q, v) ∈ T M | |v| q = r} be the sphere bundle of radius r > 0. Then, S r M is invariant under the Hamiltonian flow and X ω (g,b) Hg | Sr M is a nowhere vanishing section of the characteristic distribution ker(ω (g,b) | Sr M ) ⊂ T (S r M ).

A Hamiltonian normal form.
We will find now a Hamiltonian normal form for Φ (g,ǫ −1 b1) over an open set U ⊂ M on which b 1 is nowhere vanishing and the circle bundle π : SU → U admits a section. Such a normal form can be seen as the Hamiltonian upgrade of a vector-field normal form for Φ (g,ǫ −1 b1) which is due to Arnold [3,Theorems 2 and 3], and on which we will comment more at the end of this subsection.
Thus, let W : U → SU be a section and write θ : SU → S 1 for the angle between a unit tangent vector and W , measured according to the Riemannian metric and the given orientation on M . We refer to θ as the angular function associated with W .
If b 1 is constant, then we can choose Ψ ǫ so that where K : M → R is the Gaussian curvature of g.

Remark 1.2.
There is an analogous normal form over open sets U of M , whose tangent bundle is not trivial. This happens exactly when U = M and M = T 2 . In this situation, ω ǫ | SM = dα ǫ for a path of contact forms α ǫ on SM . Integrating a suitable vector field Z ǫ , one finds a path of diffeomorphisms Ψ ǫ : SM → SM such that Ψ * ǫ α ǫ = (1 − H ǫ )α 0 , where H ǫ is of the form (1.2) or (when b 1 is a non-zero constant) (1.3).
The normal form up to order ǫ 2 contained in Equation (1.2) was already proved by Castilho in [9, Theorem 3.1] and used to show the existence of KAM tori under certain assumptions on b 1 , see Section 1.2 for more details. A similar normal form was also developed by Raymond and Vũ Ngo . c in [17] to study the semi-classical limit of magnetic systems.
Our main contribution in Theorem 1.1 is, therefore, to push Castilho's normal form to order ǫ 4 when the magnetic function b 1 is constant, see equation (1.3). We shall mention that our method of proof is slightly different from his, as we construct the vector field Z ǫ generating the isotopy Ψ ǫ at once using Moser's method [16]: Z ǫ is the unique vector field contained in the distribution H tangent to the level sets of θ satisfying for some function c ǫ : SU → R that we will suitably choose. Let us now comment on the first dynamical implications of Theorem 1.1. This result tells us that we can read off (g, ǫ −1 b 1 )-geodesics as trajectories of a time-dependent Hamiltonian flow on U ′ . To this purpose, let us write by H ǫ,θ : U ′ → R, θ ∈ S 1 , the function H ǫ,θ (q) := H ǫ (e iθ W (q)), where e iθ : SU → SU is the fiberwise rotation by angle θ. We define the θ-dependent vector field X H ǫ,θ on U ′ by b 1 µ(X H ǫ,θ , · ) = −dH ǫ,θ (1.4) and the vector field X Hǫ on SU ′ by whereX H ǫ,θ denotes the lift of X H ǫ,θ tangent to the level sets of the angular function. If Φ t Hǫ is the flow of X Hǫ on SU ′ , then for all θ 0 ∈ S 1 and θ 1 ∈ R ϕ θ0,θ1 We see that X Hǫ spans the kernel of −π * (b 1 µ) + d(H ǫ dθ). Therefore, Ψ ǫ sends trajectories of the flow Φ Hǫ to tangent lifts of (g, ǫ −1 b 1 )-geodesics, up to reparametrization. More precisely, let z ∈ SU ′ and consider From the Formulae (1.2) and (1.3) for H ǫ , we see that (g, ǫ −1 b 1 )-geodesics (i) follow for a long time non-degenerate level sets of b 1 with a drift velocity proportional to ǫ 2 |db 1 |b −3 1 ; (ii) follow for a long time non-degenerate level sets of K, if b 1 is a non-zero constant, with a drift velocity proportional to ǫ 4 |dK|b −2 1 . Such a dichotomy follows already from Arnold's normal form for vector fields cited above [3], which shows that the magnetic function b 1 or the Gaussian curvature K, if b 1 is constant, are adiabatic invariants for the flow Φ (g,ǫ −1 b1) . Roughly speaking, a quantity I is an adiabatic invariant if, for some α > 0 and any threshold δ > 0, there exists an ǫ δ > 0 such that for all ǫ < ǫ δ the change of I along solutions with parameter ǫ is smaller than δ for a time-interval of length ǫ −α (see [2, Section 52] for more details). We remark, however, that the normal form for vector fields is not enough for the two applications considered in this paper and we crucially need the Hamiltonian normal form contained in Theorem 1.1.
We conclude this subsection observing that the theory of adiabatic invariants for magnetic fields in threedimensional euclidean space plays an important role in plasma physics as it can be applied to confine charged particles in some region of space; see [14] and the references therein. We refer to [8] for a mathematical treatment of magnetic adiabatic invariants in euclidean space and to [13] for their applications to the spectrum of the magnetic Laplacian.

Application I: trapped motions.
Throughout this subsection, we denote by (g, b 1 ) a magnetic system on a closed, oriented surface M such that b 1 : M → (0, ∞) is positive. We also denote by µ the area form given by g and the orientation on M . To write the statements in a more compact form, we define the function (1.8) Moreover, we will denote by L c0 a non-empty connected component of a regular level set {ζ = c 0 } for some c 0 ∈ R. In particular, L c0 is an embedded circle belonging to some family of embedded circles c → L c with c ∈ (c 0 − δ 0 , c 0 + δ 0 ) for a suitable δ 0 > 0. There are action-angle coordinates (I, ϕ) ∈ (I − , I + ) × T around L c0 for the symplectic form µ such that ζ =ζ • I for some functionζ : Combining the Hamiltonian normal form in Theorem 1.1 with Moser twist theorem [15], one can show that (g, ǫ −1 b 1 )-geodesics are trapped in neighborhoods of non-resonant circles.
Theorem 1.3. Let L c0 be non-resonant and let U be a neighborhood of L c0 . Then, there exists ǫ 0 > 0 and a neighborhood U 1 ⊂ U of L c0 such that, for every ǫ ∈ (0, ǫ 0 ), all (g, ǫ −1 b 1 )-geodesics with starting point in U 1 remain in U for all times.
Following Castilho [9, Corollary 1.3], one can use Theorem 1.3 to prove trapping of (g, ǫ −1 b 1 )-geodesics around circles L of minima (or maxima) for ζ. The key idea is that L is approximated on either sides by non-resonant circles. We now want to give some criteria for the existence of non-resonant circles. One classical idea is to look for such circles around a non-degenerate local minimum (maximum) point q * ∈ M of ζ by means of the Birkhoff normal form. Birkhoff normal form tells us that ζ can be written in a suitable Darboux chart centered at q * as ζ(q) = ζ(q * ) ± 1 2 det Hess ζ(q * ) r 2 + a(q * )r 4 + o(r 4 ), where r is the radial coordinate and a(q * ) ∈ R is a real number. Here det Hess ζ (q * ) is the determinant of the Hessian of ζ in Darboux coordinates, which coincides with the determinant of the Hessian with respect to g. Since I = 1 2 r 2 is the action variable in Darboux coordinates, we see that as soon as a(q * ) = 0. In this case, small circles L c around q * will be non-resonant. We mention here an equivalent way to check (1.10) using coordinates (x, y) around q * where ζ = ζ(q * ) + 1 2 r 2 , which exist thanks to Morse lemma. Let ρ be the only function satisfying µ = ρdx ∧ dy. We will see in Lemma 4.7 that (1.10) holds if and only if the Laplacian ∆ρ(q * ) of ρ at q * in the (x, y)-coordinates does not vanish.
On the other hand, when the local minimum (maximum) point q * is isolated but degenerate, we will see that the existence of non-resonant circles accumulating at q * automatically follows. Corollary 1.5. Let q * ∈ M be an isolated local minimum (maximum) point for ζ which is either degenerate or is non-degenerate and satisfies a(q * ) = 0, equivalently ∆ρ(q * ) = 0. Then for all neighborhoods U of q * , there exist ǫ 0 > 0 and a neighborhood U 1 ⊂ U of L such that, for all ǫ ∈ (0, ǫ 0 ), all (g, ǫ −1 b 1 )-geodesics with initial starting point in U 1 remain in U for all times.
We now present two other situations in which non-resonant circles can be found. The first one uses a saddle point for ζ. Theorem 1.6. Suppose that ζ has a non-degenerate saddle critical point q * ∈ M such that there are no critical values of ζ in the interval (ζ(q * ), ζ(q * ) + δ) for some δ > 0. Then, there exists a sequence of embedded circles L cn such L cn is non-resonant and lim n→∞ dist(q * , L cn ) = 0.
The second one works when the closed, oriented surface M is the two-sphere and the function ζ has exactly one minimum and one maximum point which, by Corollary 1.5, can be both assumed to be non-degenerate.
Theorem 1.7. Suppose that M = S 2 and that ζ has exactly one non-degenerate minimum at q min ∈ S 2 and one non-degenerate maximum at q max ∈ S 2 . If det Hess ζ(q min ) = det Hess ζ(q max ), then there is a non-resonant circle L c . Here the Hessian is taken with respect to the metric g. The only known Zoll pairs (g, b), which are different from the trivial examples (g con , b con ), or the purely Riemannian examples with b = 0 on S 2 , were constructed in [4]. These exotic Zoll pairs are defined on the torus T 2 with angular coordinates (x, y). Their metric g is flat and their magnetic function b depends on the x-variable only.
Unlike the trivial and the purely Riemannian examples, such Zoll pairs do not remain Zoll if we rescale b by an arbitrary constant r > 0. Therefore, we would like to understand, if there are examples of magnetic systems (g, b) such that (g, r −1 b) is Zoll for different values of r or, more generally, for values of r belonging to some given set. This corresponds to asking that the Hamiltonian flow of H g on the twisted tangent bundle (T M, ω (g,b) ) is Zoll at several energy levels 1 2 r 2 . To better handle this question, we introduce the following definition.
Definition 1.8. Let M be a family of magnetic systems on a closed, oriented surface M and let R be a subset of (0, ∞). We say that M is Zoll-rigid at R provided the following holds: If (g, b) ∈ M is a magnetic system such that (g, r −1 b) is Zoll for every r ∈ R, then g has constant curvature and b is constant, or M = S 2 , g is a Zoll metric and b is identically zero. If M is not Zoll-rigid at R, we say that M is Zoll-flexible at R.
In [5], first rigidity phenomena were discovered: i) Magnetic systems on T 2 are Zoll-rigid at any set R which accumulates to zero and infinity; ii) Magnetic systems on surfaces with genus at least two and with Mañé critical value equal to c are Zoll-rigid at any R ⊂ (0, √ 2c) which accumulates to √ 2c. In this paper we push the study of rigidity further and prove the following statement. Theorem 1.9. Magnetic systems on a closed, oriented surface M are Zoll-rigid at any set R which accumulates to zero. Namely, if (g, b 1 ) is a magnetic system on M such that (g, ǫ −1 n b 1 ) is Zoll for some sequence ǫ n → 0, then: Either b 1 = 0, M = S 2 and g is a Zoll metric, or b 1 is a non-zero constant and g is a metric of constant curvature.
The proof of Theorem 1.9 hinges on the normal form proved in Theorem 1.1. Roughly speaking, if b 1 is not constant, we can suppose, up to changing the sign, that b 1 has a positive maximum and apply the normal form to regions U = U 1 = {b 1 > δ 1 } and U ′ = U 2 = {b 1 > δ 2 } for some positive δ 1 < δ 2 in the interval (min b, max b). Using a theorem of Ginzburg [10], we can find U 3 such that there exists q 0 ∈ U 3 with the following property: the flow line of Φ Hǫ starting at W (q 0 ) has period 2π. If now (g, ǫ −1 b 1 ) is Zoll, a topological argument shows that for all q ∈ U 3 the flow line of Φ Hǫ starting at W (q) has period 2π. However, since b 1 is not constant, the normal form tells us that there must be q 1 ∈ U 3 such that the flow line of Φ Hǫ starting at W (q 1 ) drifts with speed of order ǫ 2 along a regular level set of b 1 inside U 3 . Hence, this flow line cannot close up in time 2π if ǫ is small enough. Thus, (g, ǫ −1 b 1 ) cannot be Zoll for ǫ small enough, when b 1 is not constant. If now b 1 is a non-zero constant but the Gaussian curvature K of g is non-constant, we can apply a similar argument to regions U 1 = {K > δ 1 } and U 2 = {K > δ 2 } to some δ 1 < δ 2 in the interval (min K, max K) to get that (g, ǫ −1 b 1 ) cannot be Zoll for ǫ small.
An intriguing problem is to understand exactly which sets guarantee rigidity and which one flexibility. The threshold between the two behaviors can be quite subtle as the next result shows. We first define the sets where J 1 is the first Bessel function. The set R * is discrete, bounded away from zero and unbounded from above. In particular, R N is a countable dense subset of (0, ∞).
Theorem 1.10. Let M be the set of rotationally invariant magnetic systems on T 2 with average 2π. Namely, The family M is (a) Zoll-rigid at every set R which is unbounded from above and not contained in R N ; (b) Zoll-flexible at each of the sets 1 k R * , k ∈ N.
Remark 1.11. By [7], there are no rotationally invariant Zoll systems on T 2 with zero average. Thus, up to rescaling b by a constant, there is no loss of generality in assuming that the average is 2π. Theorem 1.10 is rather surprising and represents the first instance where the structure of the space of Zoll magnetic systems is influenced by the topology of the surface. Indeed, since the Mañé critical value of magnetic systems with non-zero average is infinite, we can reformulate Theorem 1.10, b) by saying that magnetic systems on T 2 with average 2π are Zoll-flexible for a particular set accumulating at the Mañé critical value. This is in sharp contrast with the rigidity for surface with genus at least two described in the item ii) above [5].
Structure of the paper.
In Section 2 we present some classical facts about the geometry of surfaces that serve as preliminaries to the Hamiltonian normal form contained in Theorem 1.1 which will be proved in Section 3. In Section 4 we establish the existence of trapping regions and prove Theorem 1.3 and its Corollaries 1.4 and 1.5. The criteria contained in Theorem 1.6 and Theorem 1.7 for the existence of non-resonating circles will also be discussed there. Section 5 deals with the proof of Theorem 1.9 about the Zoll-rigidity of strong magnetic fields, while Section 6 shows Theorem 1.10 about the rigidity versus flexibility behaviour for rotationally symmetric magnetic fields on the two-torus.

Preliminaries from the differential geometry of surfaces
In this section, we recall some facts from the Riemannian geometry of surfaces that will be useful later on. Let U ⊂ M be an open set and assume that we have a section W : U → SU of π : SU → U . We denote by e iθ : SU → SU the flow of fiberwise rotations and use the shorthand (·) ⊥ := e iπ/2 (·). We denote by ∂ θ the vector field generating the flow of rotations. Moreover, we denote by H := ker dθ ⊂ T (SU ) for the distribution tangent to the level sets of θ. We have If u ∈ T U , we denote byũ the unique element in H such that dπ[ũ] = u.
A key object in our computations will be the 1-form α ∈ Ω 1 (U ) given by where ∇ is the Levi-Civita connection of g. The rotations commute with the connection, so that for every θ ∈ S 1 we have Letλ be the pullback on SM of λ. Our first task is to compute dλ on SU in terms of α.
Proof. Let v = e iθ W (q) for some θ ∈ S 1 and q ∈ U . The formula for (dλ) v has to be checked for the pairs We can suppose without loss of generality that u 1 , u 2 are defined in a neighborhood of q and satisfy [u 1 , u 2 ] = 0 there. We compute first where in the second equality we used the fact that v ⊥ = e iθ W ⊥ and the identity g(∇ · (e iθ W ), e iθ W ) = 1 2 d|e iθ W | 2 = 1 2 d(1) = 0, and Identity (2.2) in the third equality.
The value of dλ(ũ 1 ,ũ 2 ) is now the antisymmetrization of the expression above since [u 1 , u 2 ] = 0. Taking into account that ∇ is symmetric we obtain For the pair ∂ θ ,ũ we similarly get We now link the Gaussian curvature K : U → R of g to the function In particular, there exists a function a : SU → R such that which is enough to prove for θ = 0. Using the symmetry of ∇, we get To compute the Gaussian curvature, we use now the classical identity Kµ = dα, so that Let us now set θ := θ(v). Then, the vector fields e iθ W and e iθ W ⊥ extend the vectors v and v ⊥ on the whole U . Therefore, we can expand the right-hand side of (2.6) using (2.2): so that we arrive at (2.4). Fixing q ∈ U and averaging Identity (2.4) over π −1 (q), we get It follows that the function has zero average along the circle π −1 (q) for all q ∈ U . Because of this fact, the function is well-defined and it is immediate to check that ∂ θ a = f 1 , as required.

The construction of the normal form
This section is entirely dedicated to the proof of Theorem 1.1. Recalling thatλ is the restriction to SM of λ we writeω ǫ := ǫλ − π * (b 1 µ) for the pullback of ω ǫ to SM .
We consider U ⊂ M an open set on which we have a section W of π : SU → U with associated angular function θ. We further assume that b 1 is nowhere vanishing on U . We take an open set U ′ such that U ′ ⊂ U . Then, there exists a positive number r > 0 such that any curve γ : [t 0 , t 1 ] → M with γ(t 0 ) ∈ U ′ and length less than r is entirely contained in U .
Let Ψ ǫ : SU ′ → SU denote an isotopy for ǫ in some interval [0, ǫ 0 ] such that Ψ 0 is the standard inclusion SU ′ ֒→ SU . Up to shrinking ǫ 0 further, by the definition of r > 0 we see that Ψ ǫ is obtained integrating an ǫ-dependent vector field Z ǫ on SU . We aim at finding Z ǫ such that Taking the derivative of (3.1) in ǫ and using that ∂ ǫωǫ =λ, we see that (3.1) is equivalent to Thanks to Cartan formula, this equation can be rewritten as We require now that Z ǫ belong to the horizontal distribution H tangent to the level sets of θ. By taking the pull-back by (Ψ ǫ ) −1 on both sides, we see that (3.2) is solved if there exists a function c ǫ : SU → R witĥ This equation can be decomposed in the horizontal and vertical component at v ∈ SU : For any function c ǫ , Equation (3.4a) determines uniquely the vector field Z ǫ , and hence the isotopy Ψ ǫ , sincê ω ǫ | H is a non-degenerate bilinear form for ǫ small enough. Then, Equation (3.4b) determines h ǫ , and hence the function H ǫ , uniquely. We expand Z ǫ and h ǫ to the first order in ǫ: We choose c ǫ = ǫ 2 c ′ ǫ for some function c ′ ǫ to be determined later. Then, we evaluate Equations (3.4a) and (3.4b) at ǫ = 0. From (3.4a) we getω By the definition ofω 0 andλ, this equation can be rewritten as Second, from (3.4b) we get that h 0 = 0.
Dividing (3.4a) and (3.4b) by ǫ, we arrive at the equivalent equationŝ Evaluating (3.5a) at ǫ = 0 and using the expression for dλ obtained in Lemma 2.1, we find the equation from which it follows that which is equivalent to , which yields the desired expansion of H ǫ to the second order in ǫ.
Let us now go to higher order under the hypothesis that b 1 is a non-zero constant. We write and substitute these expressions in (3.5a) and (3.5b). After dividing by ǫ, we get the new set of equationŝ In this case, evaluating (3.6b) at ǫ = 0, we get h ′′ 0 = 0. Thus, we can substitute h ′′ ǫ = ǫh ′′′ ǫ in (3.6b) and dividing this equation by ǫ, we find In order to determine the first summand on the right, we evaluate (3.6a) at ǫ = 0 and plug inṽ: Here we have used that b 1 is constant. Recalling the definition of the function f from (2.3), we arrive at . Let a : SU → R be the function given by Lemma 2.2 and choose c ′′ ǫ = c ′′ 0 = −b −1 1 a. Thanks to (2.5) and the fact that b 1 is constant, we see that (3.7) is equivalent to h ǫ = −ǫb −1 1 − ǫ 3 (2b 1 ) −1 K and the formula for H ǫ follows. This finishes the proof of Theorem 1.1.

Trapping regions for the magnetic flow
We start by recalling the celebrated twist theorem of Moser [15]. We give here the version for flows which can be easily deduced from the original one for mappings.   and I + 0 ∈ (I 0 , I + ) arbitrarily close to I 0 (which can also be taken to be non-degenerate), we can find ǫ 0 such that for ǫ ∈ [0, ǫ 0 ] all the trajectories of Φ Hǫ starting in the region U ′ between the tori T ǫ I − 0 and T ǫ I + 0 will remain in U ′ for all times.
Let us see how we can apply Moser's theorem for local perturbations of an autonomous Hamiltonian H 1 : M → R on some symplectic surface (M, ω). Suppose that L c0 is a connected component of a regular level set {H 1 = c 0 } which is diffeomorphic to a circle. Then, there is a tubular neighborhood U of L c0 foliated by embedded circles c → L c for c ∈ (c − , c + ) so that H 1 (L c ) = c. By [2, Section 50], there are action-angle coordinates on the neighborhood of L c0 foliated by the circles L c . The action variable I can be written as a compositionĨ • H 1 , whereĨ is the strictly monotonically increasing function given bỹ I(c) = Ac 0 ,c ω and A c0,c denotes the annular region between L c0 and L c . We have that where η is any one-form on U satisfying ω = η ∧ dH 1 . For example we can take where the gradient and the norm are taken with respect to an arbitrary metric. With this choice, we see that there exists a positive constant δ > 0 not depending on c such that where ds is the arc-length of L c . Ifh 1 is the inverse function ofĨ, we conclude that H 1 =h 1 •Ĩ and   Remark 4.5. The non-resonance condition for a circle L c is a property that is invariant upon multiplying the symplectic form by a non-zero constant. In view of the applications to magnetic fields, we need to consider the case in which ω also depends on H 1 in a certain way. More specifically, suppose that H 1 is positive and that ω = H −1 1 ω 1 for some symplectic form ω 1 . Then, , where we put the symplectic form in subscript to distinguish the two cases. Differentiating in c, we get so that a level set of H 1 is non-degenerate with respect to H −1 1 ω 1 if and only if it is non-degenerate as a level set of 1 2 H 2 1 with respect to ω 1 .

Let us consider a family of time-depending Hamiltonians on U having the form
where h 0 (ǫ) is some constant, k is a positive integer and the dependence in the time θ ∈ T is periodic. Using the action-angle coordinates on U around L c0 described above, we can bring the Hamiltonian in the form (4.1). If now L c0 is a non-resonant circle, the following statement follows directly from Moser twist Theorem 4.1 and the subsequent Remark 4.2: for all neighborhoods U ′ of L c0 , there exists an ǫ 0 > 0 and a neighborhood U ′′ ⊂ U ′ of L c0 with the property that for all ǫ ∈ [0, ǫ 0 ], every trajectory of Φ Hǫ starting in U ′′ × T will stay in U ′ × T for all times. We can now use the above discussion to prove trapping for strong magnetic fields.
Proof of Theorem 1.3. Suppose that L = L c0 is a non-resonant circle for ζ = b −2 1 with respect to the symplectic form µ on M . By Remark 4.5, L is a non-resonant circle for H 1 := − 1 2 b −1 1 with respect to ω = b 1 µ. Let U be an arbitrary neighborhood of L. Upon shrinking it, we can assume that U is a tubular neighborhood of L so that SU admits an angular function θ. We apply Theorem 1.1 to U and a further tubular neighborhood U ′ of L withŪ ′ ⊂ U : There exists ǫ 0 such that for all ǫ ∈ (0, ǫ 0 ] there is a Hamiltonian H ǫ,θ = h 0 (ǫ) + ǫ 2 H 1 + o(ǫ 2 ) with h 0 (ǫ) = 0 such that the Hamiltonian flow Φ Hǫ is conjugated via the map Ψ ǫ to the magnetic flow Φ (g,ǫ −1 b1) , up to time reparametrization. Applying Moser twist theorem, we see that, upon shrinking ǫ 0 , there is a neighborhood U ′′ of L in U ′ such that for all ǫ ∈ (0, ǫ 0 ) a trajectory of Φ Hǫ starting in SU ′′ will stay in SU ′ for all times. This means that all (g, ǫ −1 b 1 )-geodesics with initial velocity vector in Ψ ǫ (SU ′′ ) will stay in π(Ψ ǫ (SU ′ )) for all times. Upon shrinking ǫ 0 we see that there exists a neighborhood U ′′′ of L with SU ′′′ ⊂ Ψ ǫ (SU ′′ ) and that π(Ψ ǫ (SU ′ )) ⊂ U . This means that any (g, ǫ −1 b 1 )-geodesics starting in U 1 := U ′′′ will stay inside U for all times.
This finishes the proof of Theorem 1.3 for the function ζ = b −2 1 . The proof of Theorem 1.3 when b 1 is a positive constant and the function ζ is the Gaussian curvature K is analogous and is left to the reader.
We now give some criteria for the existence of non-degenerate circles for H 1 . The first criterion is due to Castilho [9,Corollary 1.3]. We present the short proof here for the convenience of the reader. By inequality (4.3), we get the lower bound As c tends to c 0 , length(L c ) is bounded away from zero, while max Lc |∇H 1 | converges to zero. The limit in (4.4) follows.
Proof of Corollary 1.4. We recall that by Remark 4.5, a circle L is non-resonant for with respect to µ. Let U be an arbitrary neighborhood of the critical circle L c0 . We can assume that U is a tubular neighborhood of L. We apply Theorem 1.1 to U and a neighborhood U ′ of L such thatŪ ′ ⊂ U and get a diffeomorphism Ψ ǫ and a function H ǫ,θ = h 0 (ǫ) + ǫ 2 H 1 + o(ǫ 2 ) with h 0 (ǫ) = 0 for all ǫ ∈ [0, ǫ 0 ]. By Lemma 4.6 there are non-resonant circles L − and L + for H 1 inside U ′ on either side of L c0 . Applying Moser twist Theorem 4.1 to L − and L + we see that, for a possibly smaller ǫ 0 , there is a neighborhood U ′′ of L such that for all ǫ ∈ [0, ǫ 0 ] every trajectory of Φ Hǫ with initial condition in SU ′′ remains in SU ′ for all times. Taking an even smaller ǫ 0 , we get π(Ψ ǫ (SU ′ )) ⊂ U and we find a neighborhood U ′′′ of L such that SU ′′′ ⊂ Ψ ǫ (SU ′′ ). Thus, any (g, ǫ −1 b 1 )-geodesics starting in U 1 := U ′′′ will be contained in U for all times.
The proof of Corollary 1.4 for ζ = b −2 1 is completed. For ζ = K and b 1 a positive constant, the proof is analogous and we omit it.
Let us analyze the situation in a neighborhood of an isolated minimum or maximum q * ∈ U for H 1 . Let c 0 := H 1 (q * ), so that we have a family of circles L c for c in some interval (c 0 , c + ) (or (c − , c 0 )) converging uniformly to q * as c → c 0 . Lemma 4.7. Let q * ∈ M be an isolated local minimum or maximum for H 1 . If q * is degenerate, then If q * is non-degenerate, then where the Hessian is taken in Darboux coordinates around q * , and ρ is a function with the property that ω = ρ dx ∧ dy in coordinates (x, y) around q * such that H 1 = c 0 + 1 2 (x 2 + y 2 ). Proof. We deal only with the case of a local minimum and leave the case of a local maximum to the reader. We make the preliminary remark that, up to an additive constant, we havẽ where {H 1 ≤ c} is a sublevel set of H 1 in a neighborhood of q * .
Assume first that q * is degenerate. There exists Darboux coordinates (x, y) around q * such that H 1 = c 0 + 1 2 (ux) 2 + o(r 2 ) for some constant u ≥ 0. In particular, we find a constant C > 0 such that 3 . We distinguish two cases.
If u = 0, then we readily see that Since the last term diverges as c tends to c 0 , we get (4.5).
If u = 0, then Therefore, arguing as above we obtain that Since the rightmost term diverges as c tends to c 0 , we again get (4.5).
From the lemma above we deduce immediately the following result which, together with Remark 4.4, yields Theorem 1.7. 1 with respect to µ. Let U be an arbitrary neighborhood of q * which we can suppose to be diffeomorphic to a ball. We apply Theorem 1.1 to U and a neighborhood U ′ of q * such thatŪ ′ ⊂ U . We obtain a diffeomorphism Ψ ǫ and a function H ǫ,θ = h 0 (ǫ) + ǫ 2 H 1 + o(ǫ 2 ) with h 0 (ǫ) = 0 for every ǫ ∈ [0, ǫ 0 ]. By Lemma 4.7 there is a non-degenerate circle L for H 1 inside U ′ . Using Moser twist Theorem 4.1 on L, we can find, by choosing a smaller ǫ 0 , a neighborhood U ′′ of q * with the property that, for all ǫ ∈ [0, ǫ 0 ], every flow line of Φ Hǫ starting in SU ′′ stays in SU ′ forever. After shrinking ǫ 0 again, we have π(Ψ ǫ (SU ′ )) ⊂ U and there is a neighborhood U ′′′ of q * such that SU ′′′ ⊂ Ψ ǫ (SU ′′ ). We deduce that any (g, ǫ −1 b 1 )-geodesics passing through U 1 := U ′′′ will stay in U forever. We have thus showed Corollary 1.5 for ζ = b −1 1 . The proof for ζ = K when b 1 is a positive constant is completely analogous. The last criterion deals with a saddle point of H 1 and together with Remark 4.4 yields Theorem 1.6. Proof. There is a chart U ′ → (−δ, δ) 2 centered at q * such that H 1 has the form H 1 = c 0 + xy in the corresponding coordinates. Since there are no critical values in (c 0 , c 1 ), there is a smooth family of circles c → L c , c ∈ (c 0 , c 1 ) with the property that where in the last inequality we used that It is now easy to see that the last integral diverges for c tending to c 0 .
The set Q ǫ plays a central role in the proof of Theorem 1.9 and now we want to study it better. To this purpose, let us consider the following construction. Let z : [0, T ] → SU 1 be a curve. We defineθ z : [0, T ] → R to be any real lift of the function θ • z : [0, T ] → S 1 . We define In particular, if z(T ) = z(0), we see that ∆θ z = 2kπ for some integer k. Let us specialize this construction further and consider for any v ∈ SU 1 the (g, ǫ −1 b 1 )-geodesic γ v : R → M withγ(0) = v. We define the set Then, the function ∆θ : is well-defined and continuous since the function dθ d dtγ v is continuous in v. Lemma 5.1. A point q ∈ U 3 belongs to Q ǫ if and only if the (g, ǫ −1 b 1 )-geodesics with initial velocity vector Ψ ǫ (W (q)) has some period ℓ, it is contained in Ψ ǫ (SU 2 ) and ∆θ(Ψ ǫ (W (q)), ℓ) = 2π.
We now show that A ǫ has a critical point by checking that A ǫ has an interior maximum. Indeed, by (5.1), there exists C > 0 such that which has an interior minimum in U 3 because K has an interior maximum in U 3 . In Lemma 5.3, we take δ 4 to be a regular value of K in the interval (δ 3 , max K) and use again (5.7) instead of (5.1).
This finishes the argument for b 1 constant and K non-constant and proves Theorem 1.9.

Rotationally invariant Zoll systems
In this section we prove that rotationally invariant Zoll magnetic systems on T 2 with average 2π are Zollrigid at unbounded sets not contained in R N but are Zoll-flexible at any of the unbounded discrete sets 1 k R * for some k ∈ N. To this end, we require some preliminaries referring to [4] for the details. For any c > 0, we write T c := R/cZ, We consider T 2 = T 1 × T L for some L > 0 as the rectangle [0, 1] × [0, L] with identified sides, and denote by (x, y) the usual angular coordinates. A rotationally invariant magnetic system (g, b) on T 2 is then given by where a : T 1 → (0, +∞) and b : T 1 → R are smooth functions. Up to rescaling the y-variable (x, y) → (x, cy), which changes L to cL and a to a/c, we can assume that a integrate to 1 over [0, 1]. Requiring the average of b to be 2π, then amounts to having b = T1 a(x)b(x) dx = 2π.
If r is a positive constant, the equations of motion of a (g, r −1 b)-geodesic (x, y) : R → T 2 whose tangent vector makes an angle θ : R → T 2π with ∂ x read             ẋ = cos θ, y = sin θ a(x) , Being invariant under the flow of the vector field ∂ y , the system admits by Noether's theorem an additional first integral, which is given by where B ′ (x) = a(x)b(x). Let us now take g = g flat , namely a ≡ 1. Then, (6.1) and (6. . From this formulae we see that x is 2π-periodic in θ, while there is a shift in the y coordinate when θ goes from −π/2 to π/2 given by the shift function ∆ r : T 2πr −1 → R defined as ∆ b r (I 0 ) := π/2 −π/2 dy dθ dθ = r π/2 −π/2 sin θ · v r(sin θ − I 0 ) dθ = r 2 π/2 −π/2 cos 2 θ · v ′ r(sin θ − I 0 ) dθ, (6.3) where we used integration by parts. Clearly, (g, r −1 b) is Zoll if and only if the shift function ∆ r vanishes identically. In order to check this condition, it is convenient to rewrite ∆ r slightly by making the change of variables u = sin θ: The function ∆ b r has zero average and therefore it vanishes if and only if its Fourier coefficients ∆ b r (k) for k ∈ Z \ {0} vanish. We claim that ∆ b r (k) = −iπrJ 1 (kr) · v(−k), (6.4) where J 1 : (0, ∞) → R is the first Bessel function Indeed, we have ∆ r (k) = r 2 From the properties of J 1 , we know that R * is discrete, bounded from below and unbounded from above, so that R N is countable and dense in (0, ∞). If r / ∈ R N , then the only real function v satisfying (6.4) for all k ∈ Z is the constant function. Therefore, On the other hand, for every k ∈ N, let us implicitly define b k : T 1 → (0, ∞) with b k = 2π by specifying v k (u) := 1 2π + ǫ sin(ku), for some ǫ ∈ − 1 2π , Then, v k (k ′ ) = 0 for |k ′ | = 0, k and using (6.4) we conclude that ∆ b k r = 0 if and only if r ∈ 1 k R * . In other words, we have the equivalence (g flat , r −1 b k ) is Zoll ⇐⇒ r ∈ 1 k R * . (6.6) We are now in position to prove Theorem 1.10.
Proof of Theorem 1.10. Let R be any subset of (0, ∞) which is unbounded from above and not contained in R N . Let us suppose that (g, b) is a rotationally symmetric magnetic system on T 2 with average 2π such that (g, r −1 b) is Zoll for every r ∈ R. Then, [5, Proposition 2.1] implies that g = g flat is a flat metric. From (6.5) applied to r ∈ R \ R N , we deduce that b is constant. This shows Statement (a). Statement (b) follows by considering the function b k constructed above and using (6.6).