Suspension of the Billiard maps in the Lazutkin's coordinate

In this paper we proved that under the Lazutkin's coordinate, the billiard map can be interpolated by a time-1 flow of a Hamiltonian $H(x,p,t)$ which can be formally expressed by \[ H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\quad(x,p,t)\in\T\times[0,+\infty)\times\T, \] where $V(\cdot,\cdot,\cdot)$ is $C^{r-5}$ smooth if the convex billiard boundary is $C^r$ smooth. Benefit from this suspension we can construct transitive trajectories between two adjacent caustics under a variational framework.


Introduction
Let's concern the following reflective mechanism: For a convex domain Ω of R 2 with a sufficiently smooth boundary ∂Ω, there exists a inner partical which makes uniform motion in straight line. When it hits ∂Ω, the impact angle is the same as the reflected angle (see figure  1). The Billiard map is just defined by the correspondence of two adjacent reflective points, i.e. φ : P 0 → P 1 . According to this mechanism, if there exists a closed convex curve Γ lying  in Ω, such that any tangent of Γ remains a tangent line after once reflection, then we call it a caustic (see figure 2). If we formalize the arc length of ∂Ω by 1, then each reflective point can be fixed by the coordinate (s, v), and where d(·, ·) is the Euclid distance of R 2 , and The twist property is implied by −∂ 12 h > 0 once the boundary is strictly convex [12]. With this variational approach, a caustic will correspond to an invariant curve in A. J. Mather gave a short and smart proof on the nonexistence of caustics: Theorem 1.1. [8] If ∂Ω has a flat point, that is if there is a point, where the curvature vanishes, then the billiard map φ has no invariant curve.
On the other side, Lazutkin succeeded in applying the averaging method to prove the existence of plenty of caustics which are close to the boundary ∂Ω: Theorem 1.2. [6] ∀ small 0 < a < 1, fixed constant σ ≥ 3 and A > 0, we denote by E(a) the set of all the (A, σ)−Diophantine number in [0, a], i.e. ∀η ∈ E(a), there exists a constant A, such that |nη − m| ≥ A|m|n 0.5−σ , ∀m ∈ Z, n ∈ Z + .
For C r smooth boundary ∂Ω with r ≥ 553, there exists a constant b * ∈ (0, a) depending only on ∂Ω C r , σ and A, such that all the quasi-periodic curves persist with the frequency in E(a) [0, b * ].

Remark 1.3.
Recall that E(a) is a sparse set in [0, a] but has a positive Lebesgue measure: mes(E(a)) ≥ a − c 1 a σ−0.5 , where c 1 depends only on σ and A, and as σ → ∞, c 1 → 0. In another paper [4], Douady improved the smoothness of boundary by r ≥ 6.
Lazutkin's prove is due to the following procedure: for sufficiently small reflected angle v, the billiard map can be expressed by where with ρ(s) being the curvature of ∂Ω and s being the arc parameter. Then he found an exact symplectic transformation by and then (4) becomes with the invariant symplectic form by dx ∧ dy 2 /2. If we take l = y 2 /2, then is exactly standard symplectic with the generating function Formally we can find (7) becomes a nearly integrable map for small l > 0, that's why several steps KAM iterations can be applied and he can get the persistence of positive measure caustics.
As now the Lazutkin's coordinate proves to be a powerful tool in exploring the properties of the billiard maps in the nearing boundary domain, we can naturally make a comparison with the convex nearly integrable Hamiltonian flow. That's because the latter is more flexible and developed in practical applications, e.g. we can get resonant normal forms by the KAM iterations (related with the existence of resonant cuastics), or use the Aubry Mather theory or weak KAM theorem get the properties for all kinds of invariant set. This motivation urges us to prove the following conclusion: There exists a smooth Hamiltonian H(x, l, t) defined for (x, l, t) ∈ T × [0, ∞) × T, of which the billard map (7) can be interpolated by the time-1 flow φ 1 H for 0 ≤ l 1. Moreover, H(x, l, t) can be formally expressed by We should confess that J. Moser is the first mathematician making a connection between monotones twist maps and time-1 periodic flow of Hamiltonians, and in [11] he gave a precise prove for how to suspend general monotone twist maps by time-periodic Hamiltonians. But (7) is just of the type excluded by his paper, because from (9) you can find a singular kinetic part 2 √ 2 3 l 3/2 , with the C 2 norm blowing up as l → 0. So we need to cover this case by preciser quantitative evaluation.
The heuristic idea of the proof is the following: first, suspend (7) by a straight flow Hamiltonian, which is unnecessarily to be time periodic; then we can evaluate the kinetic part and perturbation part, and strictly separate they two under the · C 2 norm; at last, due to the evaluation we slightly modify the Hamiltonian to be time periodic (see Section 2 for more details).
Due to the Legendre transformation, we can get the conjugated Lagrangian of (9) by which conforms to the basic setting of [9]. So we can use the variational approach of (10) to prove the following application: Corollary 1.5. Any two adjacent caustics can be connected by one special billiard reflection plan; in other words, for any two adjacent invariant curves Γ 0 , Γ 1 of (9) and any two small open neighborhoods U 0 , U 1 of each, there exists one trajectory φ t H (x * , l * ) passing by these two neighborhoods in turns.
We postpone the proof of this Corollary to Section 3, where we list some special applications of the Aubry Mather theory for the billiard maps as well.

Proof of the main Theorem
Recall that (4) is explicit only for sufficiently small v, and the expansion of (8) can be estimated by: for sufficiently small ∆x = x + − x. So we just need to care the dynamic behavior of the Lazutkin's map (7). Then we can slightly changeh(x, X) into with P (x + 1, X + 1) = P (x, X), x ≤ X and x ∈ R. Here ρ(·) ∈ C r (R, R) satisfying is just a smooth transitional smooth function with ρ (t) ≤ 1 ε and ρ (t) ≤ 1 √ ε . We can always take ε 1 sufficiently small such that there exists a constant c depending only on ε and Remark 2.1. Actually, this modified generating function conform with (7) only for the domain Due to the conclusion of [6], We can always find a list of KAM tori T ω = {(x, l ω (x))|x ∈ T} with the Diophantine frequency ω ∈ E(ε). We can always pick one T ω which encloses an invariant domain Λ with T 0 = {(x, 0)|x ∈ T} as the lower bound. Θ ⊆ Λ and (7) is still the map in Λ.
This proof for this Theorem is outlined as follows: We first suspend (7) by a smooth but non-periodic Hamiltonian H(x, l, t), then slightly modified it into a periodic but only piecewise continuousĤ(x, l, t). The last step, we polishĤ(x, l, t) toH(x, l, t) which becomes smooth of time t again, t ∈ T. Due to (7),H(x, l, t) can be formally expressed as (9). 2.1. Suspension.
Proof. Actually, the selection of L(x, v, t) is rather arbitrary, so we can specially choose the one with linear Euler-Lagrange equation, i.e.
Differentiate both sides with respect to v variable, we should have By characteristic method we can solve previous P.D.E by where G(·, ·) is a designated function later on. Then This is just a formal deduction, and we can specially choose the boundary conditions by L(x, 0, t) = L v (x, 0, t) = 0. Conversely, these trivial boundary conditions constraint if we take it back into (12). By aware that −∂ 12 h(x, X) > 0 only for X > x, and G(x, 0) = 0. That means the twist index decays to 0 for X − x → 0, which is quite different from the case considered by J. Moser in [11]. Finally we can solve the Lagrangian by By the Legendre transformation we get Remark 2.3. We can generalize (12) to a rescaled version: By taking the following variational principle holds: Accordingly, and easily prove that For sufficiently small 0 < κ < 1 5 , we can define a modified map which still ensures the strictly twist property ∂ 2 f (x, l) > 0. Moreover, we can find S ϕ (x, l) satisfying This is because the following Lemma: We can apply this Lemma with M = T and restrict on the upper semi-part {(x, l) ∈ T * T|l > 0}. Once we get S ϕ (x, l), we can get the generating function by because X = f (x, l) and the twist property. Actually, there exists a constant c > 0 depending only on κ and ε, such that all converge to 0 as X → x. That means h ϕ (x, X) can be at least C 2 −smoothly extended to the domain {X ≥ x|x ∈ R}.
As we have already got the generating function h ϕ (x, X), we can apply Lemma 2.2 one more time and get a modified Hamiltonian H (x, l, t) such that ϕ t H is the interpolating flow with ϕ 1 H = ϕ and ϕ 0 H = id, t ∈ [0, 1], (x, l) ∈ T × [0, ∞). In other words, φ can be suspended by the following modified flow (20) which is generated by the periodic Hamiltonian with (x, l, t) ∈ T × [0, ∞) × T. Later we will see that for sufficiently small l 1, Unfortunately, there comes out two discontinuities ofĤ(x, l, t) at t = κ, 1 − κ. Later we will polishĤ(x, l, t) into a smoothH(x, l, t) with flowφ t satisfyingφ 0 = id andφ 1 = φ.

Smoothness.
To find a smooth time-periodicH(x, l, t) being the suspension for φ, the following tools are necessary: H (x, l, t). The analysis of this subsection is based on a simple fact: the Euler-Lagrange flow of Lemma 2.2 has a constant velocity, i.e.

Remark 2.5. Recall that from (15) of previous remark, (21) now becomeŝ
We can see that the kinetic part is always 2 √ 2 3 l 3/2 for all t ∈ [0, 1], which is much greater than the perturbation part l 5/2 New generating function. Here we involve a new type generating function S(X, l, t) := Xl + w(X, l, t) which corresponds to the flow map φ t H of the Hamiltonian H(x, l, t), where (x, l, t) ∈ T × [0, ∞) × [0, 1] (H(x, l, t) is not necessarily time-periodic). Recall that φ t H is exact symplectic, that means By deriving of variable t on both sides,
In this way we remove the discontinuity at {t = κ}, and by the same argument we can remove the discontinuity at {t = 1 − κ} and get a totally smooth HamiltonianH(x, l, t) which is periodic-1 of time t ∈ [0, 1] andH ll > 0 for (x, t) ∈ T 2 and 0 < l 1. Moreover, from aforementioned argument and (39),H can be finally established by (9).

Aubry Mather theory for the Billiard maps and applications
Let's first make a brief introduction of some elementary definitions and theorems. We concerns the following C 2 −smooth Tonelli Lagrangian L(x, v, t) with (x, v, t) ∈ T M × T, which satisfies these assumptions [9]: • positively definite the Hessian matrix L vv is positively definite for any (x, v, t) ∈ T M × T; Due to the positive definiteness and super linearity, both of these two functions are convex and superlinear, and where the equality holds only for c ∈ D + β(h) and h ∈ D + α(c) (sub-derivative set). We denote by M(c) ⊂ T M × T the closure of the union for all the supports of the minimizng measures of (41), which is the so called Mather set. Its projection to M × T is the projected Mather set M(c). From [9] we know that π −1 M(c) : M × S 1 → T M × S 1 is a Lipschitz graph, where π is the standard projection from T M × T to M × T. Remark 3.1. Actually, aformentioned definition can be applied for the set of closed probability measures M c , instead of M L . We can get a closed probability measure from a closed loop of M due to the Birkhoff ergodic theorem: where T c is the periodic of the loop γ c , andγ − (T c ) =γ + (0) may be the case. This point is firstly proposed by Mañé in [7], and we can still get the same α(c) and β(h) with this neww setting.
Follow the setting of [1], we define where t, t ∈ R with t < t , and where τ, τ ∈ S 1 . Then a curve γ : for all t, t ∈ R and τ = t mod 1, τ = t mod 1. A semi static curve γ is called c-static if Before we apply the Aubry Mather theory to (9), let's first expand the Hamiltonian to the whole cotangent space, i.e.
Correspondingly, the Lagrangian becomes symmetric as well: But we should keep in mind that {x ∈ T, l = 0, t ∈ T} forms a rigid 'wall' which separates the phase space T * T × T into two disconnected parts, which are mirror images of each other. The reason we did so is to ensure the globally super-linear and positively definite, as we know V (x, |l|, t) = 0 for |l| > from the setting of Section 2. And the completeness is also followed because the velocity v is dominated in a compact region.  Proof. This is a direct corollary from [10]. As the configuration space M = T, so α(c) is actually C 1 smooth. There would be a unique w = α (c) corresponding to the rotational number of M(c). If w is irrational, then M(c) contains only one uniquely ergodic measure, soÃ(c) = N (c) due to [1]. Recall thatÃ(c) has graph property, then N (c) {t = 0} T. Otherwise, N (c) forms an invariant curve which separates Γ 0 and Γ 1 . This leads to a contradiction.
If w = p/q is rational, there exists a maximal flat [c 1 , c 2 ] of α(c), such that c ∈ [c 1 , c 2 ] (it may happen that [c 1 , c 2 ] collapse to a single point c). If c ∈ (c 1 , c 2 ), N (c) will has a unified rotational number p/q (irreducible fraction); If c = c 1 (or c 2 ), then N (c) contains p/q periodic orbits and p/q − heteroclinic orbits (or p/q + heteroclinic orbits) [10]. This order-preserving structure prevents N (c) {t = 0} = T just like before. This Lemma is trivial because N (c) is closed and due to Lemma 3.3, we can always cover N 0 (c) with finitely many disconnected open intervals The following variational principle will help us to construct a connecting orbit which passes by A(c(s)) and A(c(s 0 )).
Then the set {γ i } is pre-compact in C 1 (R, M ). Let γ : R → M be an accumulation orbit of If we denote by C η,µ,ρ the set of all the minimizers of (49), then any orbit γ(t) : R → M in it conforms to the Euler Lagrange equation as long as we take s sufficiently close to s 0 . Moreover, γ is a heteroclinic orbit connecting A(c(s 0 )) and A(c(s)).
Proof. This is a direct citation of conclusions in [2,3], so here we just give a sketch of the strategy: Recall that L η,µ,ρ is also superlinear and positively definite, because we just add a linear term of v to the original L(x, v, t). So we can get the compactness of {γ i } from the superlinearity. In turns we get the existence of accumulation orbits and C η,µ,ρ = ∅. Moreover, C η,µ,ρ is a upper semi-continuous set valued function of the additional term {µ, ρ}, which is due to the pre-compactness as well. Recall that N 0 (c(s 0 )) ⊂ k i=1 U i and suppµ(x) ( k i=1 U i ) = ∅, there must exist a small time 0 < δ < 1 such that N (c(s 0 )) {t ∈ [0, δ]} ⊂ k i=1 U i , then we use the upper semi-continuity of C η,µ,ρ , for sufficiently small |c(s) − c(s 0 )|, C η,µ,ρ {t ∈ [0, δ]} ⊂ k i=1 U i as well. This is because C η,0,0 = N (c(s 0 )). On the other side, only for t ∈ [0, δ] we have L η,µ,ρ = L, whereas ∀γ ∈ C η,µ,ρ , suppµ(x) ( k i=1 U i ) = ∅ makes γ| [0,δ] conforms to the same Euler Lagrange equation as (51). Remark 3.6. This locally connecting mechanism was initially used to solve the a priori unstable Arnold diffusion problems. Similar skill was also revealed independently in [1]. Corollary 1.5 actually tells us that, in geometrical optical meaning, adjacent caustics will form a closed, but instable region (so called 'caustic tube' in [6]).  It's trivial that β(h) ≥ 0, so we get the degeneracy of β(h) at zero by lim h→0 β(h) h 2+ = 0, ∀ ∈ (0, 1).
Because α(c) is C 1 smooth, so α (0) = h = 0. Then we use the continuity of α (c) get that as c → 0, h c → 0 as well. where ρ is a constant.