Analysis of Hamiltonian Dynamics: Perturbation Theory and Viscosity Solutions, Papers of Health sciences

Download Analysis of Hamiltonian Dynamics: Perturbation Theory and Viscosity Solutions and more Papers Health sciences in PDF only on Docsity! M P E JJ Mathematical Physics Electronic Journal ISSN 1086-6655 Volume 12, 2006 Paper 3 Received: Nov 2, 2005, Revised: May 8, 2006, Accepted: Jul 15, 2006 Editor: K. Khanin PERTURBATION THEORY AND DISCRETE HAMILTONIAN DYNAMICS DIOGO A. GOMES AND CLAUDIA VALLS Departamento de Matemática Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa, Portugal Abstract. In this paper we discuss a weak version of KAM the- ory for symplectic maps which arise from the discretization of the minimal action principle. These maps have certain invariant sets, the Mather sets, which are the generalization of KAM tori in the non-differentiable case. These sets support invariant measures, the Mather measures, which are action minimizing measures. We generalize viscosity solution methods to study discrete systems. In particular, we show that, under non-resonance conditions, the Mather sets can be approximated uniformly, up to any arbitrary order, by finite perturbative expansions. We also present new re- sults concerning the approximation of Mather measures. 1. Introduction In this paper, we discuss perturbation methods for symplectic maps that arise from the discretization of the minimal action principle. The motivation for this work is the following: the phase space of close to in- tegrable Hamiltonian system can be split into regular parts, composed of invariant tori in which the system displays very simple periodic or quasi-periodic behaviour, and in another part where the system may This work was partially supported by the Center for Mathematical Ana- lysis, Geometry, and Dynamical Systems and FCT - POCTI-POCI/FEDER, POCI/MAT/55745/2004, and SFRH/BPD/14404/2003. 1 2 D. Gomes, C. Valls exhibit irregular behaviour. As the system gets farther from the inte- grable, the invariant tori do not disappear altogether but shrink to cer- tain sets which have a characteristic action minimizing property that allows to prove their existence by a variational argument. In these sets, one can define certain action minimizing measures, the Mather measures. To observe these sets and measures numerically one must do some type of discretization. In this paper, we discuss a time dis- cretization scheme for autonomous Hamiltonian systems, and study the stability of the action minimizing sets and measures under small perturbations. The Aubry-Mather theory [Mat89a], [Mat89b], studies invariant mea- sures of Lagrangian systems which have special minimizing properties. In its origin was the study of discrete systems such as area-preserving twist diffeomorphisms [Mat79], [Mat81], [Mat82], [Mat91]. Recently, the techniques of viscosity solutions [Fat97a], [Fat97b], [Fat98a], [Fat98b], [E99], [EG01] have been used with success to study continuous La- grangian systems and can be appropriately adapted to study discrete systems [Gom02]. One motivation to study symplectic maps instead of flows comes from the fact that by discretizing the minimal action principle from classical mechanics, one obtains certain maps, which are symplectic, and have the form { pn+1 − pn = hDxH(pn+1,xn) xn+1 − xn = −hDpH(pn+1,xn), where h is the time step and H the Hamiltonian. Those maps are discrete analogs of Hamilton’s equations { ṗ = DxH(p,x) ẋ = −DpH(p,x). It is very important to notice that this discretization comes from vari- ational principles, as we discuss in section 2, and has interesting geo- metric and analytic features as will be pointed out later in the paper. If the Hamiltonian H only depends on p, the dynamics is very simple since pn = p0, and xn = x0 − nhDpH(p0). In this case, the system is called integrable. We would like to study Hamiltonians that are close to integrable. That is, when H is of the form H(p, x) = H0(p) +O(²), for ² a small parameter. There are certain measures, the Mather measures, characterized by a variational principle (see section 2), which are invariant under the discrete Hamilton’s equations. Our main objective is to understand the Perturbation theory 5 Euler-Lagrange equations (2), the other is to discretize the variational principle. They are not equivalent, and this last approach has sev- eral advantages. In fact, in the continuous setting there are certain invariant sets, the Mather sets, which are obtained using a variational principle. Using this discretization of the variational principle, one can construct Mather measures, see for instance [Gom02]. Furthermore, the map that is obtained this way has better geometrical properties as it preserves the symplectic structure. The discretization of the variational problem can be done by means of the Euler method for the ODE ẋ = v(t). This yields the discrete dynamics xn+1 = xn + hvn, in which h is the time-step. The corresponding variational problem consists in minimizing the action h N ∑ n=0 L(xn,vn), among all choices of vn, 0 ≤ n ≤ N , with fixed endpoints x0 and xN . The analog of the Euler-Lagrange equations is (3) −DvL(xn+1,vn+1)−DvL(xn,vn) h +DxL(xn+1,vn+1) = 0. In the continuous case, (2) can be written in a Hamiltonian form: { ṗ = DxH(p,x) ẋ = −DpH(p,x), for p = −DvL(x, ẋ), and the Hamiltonian H(p, x) = sup v [−p · v − L(x, v)] . Similarly, (3) can be written in the equivalent form [Gom02] (4) { pn+1 − pn = hDxH(pn+1,xn) xn+1 − xn = −hDpH(pn+1,xn), with (5) pn+1 = −DvL(xn,vn). Note that the dynamics (4) is semi-explicit, that is, implicit in p and explicit in x. This may therefore constraint the size of h for which (4) defines a discrete flow, depending on bounds for the derivatives of H. 6 D. Gomes, C. Valls We consider the case in which the Lagrangian L²(x, v) : T n × Rn → R, is a perturbation of an integrable one L0(v). More precisely, (6) L² = L0(v) + ²L1(x, v), in which ² is a small parameter. When ² = 0, the dynamics (4) is very simple since vn is constant. The Hamiltonian corresponding to (6) has an expansion of the form H(p, x) = H0(p) + ²H1(p, x) + ² 2H2(p, x) + · · · This expression is a straightforward application of the implicit function theorem for the Legendre transform p = −DvL². As in the case of continuous flows (see [Arn89], [AKN97] or [Gol80]), one can use generating functions to change coordinates, and, in particu- lar, there is a version of the Hamilton-Jacobi integrability for maps. In the next proposition we prove the main result on generating functions and change of coordinates, which serves to motivate our methods. Theorem 2.1. Let (x, p) ∈ R2n be the original canonical coordinates and (X,P ) ∈ R2n be another coordinate system. Suppose there is a smooth function S(x, P ) such that (7) p = DxS(x, P ) X = DPS(x, P ) defines a global change of coordinates (this function is called a gener- ating function). Additionally, assume that D2xPS is non-singular, and suppose there is a smooth function H(P,X) such that, for h sufficiently small, S(x, P )− S(x̂, P̂ ) + hH(DxS(x̂, P̂ ), x)−(8) − hDxS(x̂, P̂ )DpH(DxS(x̂, P̂ ), x)+ + hDPS(x̂, P̂ )DXH(P,DPS(x̂, P̂ )) = = hH(P,DPS(x̂, P̂ )), in which (9) P̂ −P = hDXH(P,DPS(x̂, P̂ )) x̂−x = −hDpH(DxS(x̂, P̂ ), x). In the new coordinate system, the equations of motion (4) are (10) { Xn+1 −Xn = −hDPH(Pn,Xn+1) Pn+1 −Pn = hDXH(Pn,Xn+1) Perturbation theory 7 In particular, if H does not depend on X, these equations simplify to (11) { Xn+1 −Xn = −hDPH(Pn) Pn+1 −Pn = 0. Remark. 1. When we perform this change of coordinates we obtain a new dynamics that is semi-explicit as (4), but this time is implicit in X and explicit in P . Remark. 2. In the continuous case, given a Hamiltonian H and the generating function S, the new Hamiltonian is fully determined since H(p, x) = H(P,X). However, in our case, that is not immediate since (8) is in fact a partial differential equation for H. Remark. 3. In this paper we will use mostly this theorem when (x, p) ∈ Tn × Rn. In this case, S is identified with a function in the tangent space of the universal covering of Tn. Proof. For simplicity, we will set h = 1 in the proof, by absorbing it into H (note however that we need h sufficiently small in order for the implicit expressions to be defined and smooth). Define Ŝ = S(x̂, P̂ ), and we use the convention DxŜ = (DxS)(x̂, P̂ ), as well as DP Ŝ = (DPS)(x̂, P̂ ), to simplify the notation. By differentiating (8) with respect to x we obtain, DxS −DxŜDxx̂−DP ŜDxP̂ +DpHDx ( DxŜ ) +DxH −Dx ( DxŜ ) DpH− −DxŜDx (DpH) +Dx ( DP Ŝ ) DXH +DP ŜDx ( DXH ) = = DXHDx ( DP Ŝ ) . Then, by canceling some terms we have DxS −DxŜDxx̂−DP ŜDxP̂ +DxH− −DxŜDx (DpH) +DP ŜDx ( DXH ) = 0. Observe that Dxx̂ = I −Dx(DpH), and DP ŜDx ( −P̂ +DXH ) = 0. Therefore (12) DxS −DxŜ +DxH = 0. 10 D. Gomes, C. Valls Motivated by the formal change of variables that was discussed in the previous section, we would like to relate this weak solution with the dynamics (4) - the next theorem makes this connection. Theorem 3.3. Let u be a viscosity solution of (1). Then: - For each P ∈ Rn, there exists at least one subset of Tn×Rn, called Mather set, which is invariant under the dynamics (4) and is contained in the graph (x, p) = (x, P +Dxu(x)). - There exists a probability measure µ(x, p) on Tn × Rn (discrete Mather measure) invariant under (4) supported on this invariant set. - This measure minimizes (16) ∫ L(x, v) + Pvdν, with v = −DpH(p̂, x), and p̂ − p = hDxH(p̂, x), over all probability measures ν on Tn × Rn that satisfy ∫ φ(x+ hv)− φ(x)dν = 0, for all continuous function φ : Tn → R. Furthermore (17) −H = ∫ L(x, v) + Pvdµ, where H is the unique number for which (1) admits a periodic viscosity solution. One of the main points in the previous theorem is that one can translate properties of viscosity solutions into properties of Mather sets or measures and vice-versa. In the next proposition we discuss some of the properties of viscosity solutions, and its relations with the dynamics (4). Proposition 3.4. Suppose (x, p) is a point in the graph G = {(x, P +Dxu(x)) : u is differentiable at x}. Then, for all n ≥ 0, the solution (xn,pn) of (4) with initial conditions (x, p) belongs to G. A further result that we need, taken also from [Gom02], is a repre- sentation formula for H as a minimax. This is the discrete analog of the minimax formula for flows proved in [CIPP98]. Proposition 3.5. For each P ∈ Rn, (18) H(P ) = inf ϕ sup (x,v) [ ϕ(x)− ϕ(x + hv) h − L(x, v)− Pv ] , in which the infimum is taken over continuous periodic functions ϕ. Perturbation theory 11 Suppose that L²(x, v) = L0(v) + ²L1(x, v). The last estimates in this section show that, even without non-resonance conditions, the viscosity solution of (1) has good bounds for the semiconcavity and semiconvexity constants of u. These estimates should be seen as weak estimates for the second derivatives of u. Roughly speaking we have “D2xxu = O( √ ²)” in the Mather set. Proposition 3.6. Suppose L² is as in (6), with L0(v) smooth, strictly convex, with bounded second derivative and coercive, and L1(x, v) is smooth, superlinear, strictly convex in v, periodic in x and with uni- formly bounded second derivatives. Let u be a viscosity solution of (1). Then, for any x and y we have u(x+ y)− 2u(x) + u(x− y) ≤ C √ ²|y|2. Proof. Let (xn,vn), with xn+1 = xn + hvn, 0 ≤ n ≤ N − 1 be an optimal trajectory with x0 = x, such that u(x) = u(xN) + h N−1 ∑ n=0 [L0(vn) + ²L1(xn,vn) + Pvn +H(P )]. Then, u(x± y) ≤ u(xN) + h N−1 ∑ n=0 [ L0 ( vn ∓ y Nh ) + ²L1 ( xn ± N − n N y,vn ∓ y Nh ) + +P ( vn ∓ y Nh ) +H(P ) ] . Therefore, u(x+ y)−2u(x) + u(x− y) ≤ ≤ h N−1 ∑ n=0 [ L0 ( vn − y Nh ) − 2L0(vn) + L0 ( vn + y Nh )] + + h² N−1 ∑ n=0 [ L1 ( xn + N − n N y,vn − y Nh ) − 2L1(xn,vn)+ +L1 ( xn − N − n N y,vn + y Nh )] . Note that, since D2L0 is bounded, L0 ( vn − y Nh ) − 2L0(vn) + L0 ( vn + y Nh ) ≤ C |y| 2 N2h2 . 12 D. Gomes, C. Valls Also L1 ( xn + N − n N y,vn − y Nh ) − 2L1(xn,vn)+ + L1 ( xn − N − n N y,vn + y Nh ) ≤ C ( 1 + 1 N2h2 ) |y|2. Consequently, for h small, u(x+ y)− 2u(x) + u(x− y) ≤ C [ 1 Nh + ²hN + ² hN ] |y|2. By choosing N = O ( 1 h √ ² ) we obtain u(x+ y)− 2u(x) + u(x− y) ≤ C √ ²|y|2. ¥ Proposition 3.7. Suppose L² is as in (6), with L0(v) smooth, strictly convex, with bounded second derivative and coercive, and L1(x, v) is smooth, superlinear, strictly convex in v, periodic in x and with uni- formly bounded second derivatives. Let u be a viscosity solution of (1). Then, if x is in the Mather set and y is arbitrary, we have u(x+ y)− 2u(x) + u(x− y) ≥ −C √ ²|y|2. Proof. Since x belongs to the Mather set, there is a trajectory (xn,vn), with xn+1 = xn+hvn, 0 ≤ n ≤ N −1 with xN = x, such that u(x0) = u(x) + h N−1 ∑ n=0 [L0(vn) + ²L1(xn,vn) + Pvn +H(P )]. Note that this identity implies that this trajectory achieves the mini- mum in (14). Thus, u(x0) ≤ u(x± y) + h N−1 ∑ n=0 [ L0 ( vn ± y Nh ) + ²L1 ( xn ± n N y,vn ± y Nh ) + +P ( vn ± y Nh ) +H(P ) ] , Perturbation theory 15 Therefore, the inverse function theorem implies Pn+1 −Pn = O(²N + |Pn − P0|N). Then X̂ = Xn+1 +O(² N + |Pn − P0|N−1). So, equation (23) reads Xn −Xn+1 = DP H̃²N(Pn) +O(²N + |Pn − P0|N−1). ¥ The Linstead method consists in constructing solutions of (1) by using an iterative procedure that yields an expansion ũ²N of the solution u² and H̃²N of the Hamiltonian H ² , as a power series in ² and (P −P0). Then, ũ²N and H̃ ² N satisfy (19). Of course, there are some conditions that have to be satisfied in order to construct the approximated solution. These can be expressed in terms of the Diophantine properties of the vector P0. We say that a vector ω ∈ Rn is Diophantine if (24) ∀k ∈ Zn\{0}, m ∈ Z, |ω · k −m| ≥ C|k|s , for some C, s > 0. We will assume that the vector ω0 = DPH0(P0) is Diophantine. We look for an expansion of the form ũ²N(x, P ) = ²v1(x, P0) + ²(P − P0)DPv1(x, P0) + ²2v2(x, P0) + 1 2 ²(P − P0)2D2PPv1(x, P0) + ²2(P − P0)DPv2(x, P0) + . . . = N−1 ∑ j=1 j ∑ i=1 1 (j − i)!² i(P − P0)j−iDj−iP j−ivi(x, P0), (25) with the notation that, for N = 1, ũ²1(x, P ) = 0. Furthermore, H̃²N(P ) =H̃0(P0) + ²H̃1(P0) + (P − P0)DP H̃0(P0)+ + ²2H̃2(P0) + ²(P − P0)DP H̃1(P0) + (P − P0)2 2 D2PP H̃0(P0) + · · · We will try to choose the functions vj in such a way that, formally, u²(x, P )− ũ²N(x, P ) = O(²N + |P − P0|N), by matching powers ² and (P − P0) in both sides of (19). The first term arises from taking ² = 0, and P = P0 in (19). Then, H̃0(P0) = H0(P0), and the solution ũ²0 = 0. The first order terms in ² yield v1(x, P0)− v1(x− hDpH0(P0), P0) + hH1(P0, x) = hH̃1(P0), 16 D. Gomes, C. Valls and this equation determines v1(x, P0) and H̃1(P0). Furthermore DP H̃0(P0) = DPH0(P0). The function v2(x, P0) and H̃2(P0) are determined by solving the equa- tion v2(x, P0)− v2(x− hDpH0(P0), P0)+ + h 2 (Dxv1(x− hDpH0(P0), P0))2D2ppH0(P0)+ + hDpH1(P0)Dxv1(x− hDpH0(P0), P0)+ + hH2(P0, x) = hH̃2(P0). To obtain DPv1(x, P0) and DP H̃1(P0) we consider the equation DPv1(x, P0)−DPv1(x− hDpH0(P0), P0)+ + hDpH1(P0, x) = hDP H̃1(P0). In general, we will have to solve equations of the form Gω0u = f + λ, in which the operator G is given by (26) Gω0u = u(x)− u(x− ω0), the function f can be computed in terms of functions that are already known, λ is the unique constant for which (26) has a solution, and ω0 = DPH0(P0). This operator can be analyzed by using Fourier coefficients. Note that, Gw0e 2πikx = e2πikx(1− e2πikw0). Thus, if η(x) = ∑ k ηke 2πikx and u(x) = ∑ k uke 2πikx, the equation (27) Gω0u(x) = η(x), reduces formally to uk = ηke −2πikx 1− e2πikω0 . If ω0 is non-resonant, the equation in (27) can be solved formally in Fourier coefficients, since no denominator vanishes, except for k = 0. Moreover, in order for (27) to have a solution, we need η0 = 0. So this implies (28) λ = ∫ f(x)dx. Perturbation theory 17 Furthermore, under Diophantine conditions on ω0, the series for the solution u converges, in appropriate function spaces, as long as η is smooth enough. From (28) we have H̃1(P0) = ∫ H1(P0, x)dx, DP H̃1(P0) = ∫ DPH1(P0, x)dx, and H̃2(P0) = ∫ 1 2 (Dxv1(x− hDpH0(P0), P0))2D2ppH0(P0)dx+ + ∫ DpH1(P0)Dxv1(x− hDpH0(P0), P0) +H2(P0, x)dx. Therefore, by computing these expansions we obtain, formally, that H ² (P ) = H̃²N(P ) +O(² N + |P − P0|N ). This identity will be made rigorous in the next section. 5. Estimates for the Effective Hamiltonian In this section we prove that H̃²N(P ) is an asymptotic expansion to H²(P ), therefore proving rigorously some of the results from the previous section. Proposition 5.1. Let u² and H ² be solutions of (1). Assume that there is an approximate solution ũ²N of u ² and H̃²N of H ² satisfying (19). Then, (29) H ² (P ) ≤ H̃²N(P ) +O(²N + |P − P0|N). Proof. We set h = 1 for simplicity. The inf sup formula (18) implies that H ² (P ) ≤ sup (x,v) [ũ²N(x, P )− ũ²N(x+ v, P )− L(x, v)− Pv] . So, the optimal point v∗ is given by Dxũ ² N(x + v ∗, P ) = −Lv(x, v∗)− P, and so L(x, v∗) = −H(P +Dxũ²N(x+ v∗, P ), x)− (P +Dxũ²N(x + v∗, P ))v∗. 20 D. Gomes, C. Valls Furthermore, sup x |u²(x, P²)− ũ²N(x, P²)| = O(²M), in which u²(x, P²) is any viscosity solution of u²(x, P²)− u²(x̂, P²) + hH(P² +Dxu²(x̂, P²), x)−(32) − hDxu²(x̂, P²)DpH(P² +Dxu²(x̂, P²), x) = hH ² (P²), with x̂− x = −hDpH(P² +Dxu²(x̂, P²), x), and ũ²N(x, P²), H̃ ² N(P²) satisfy (19), to which an appropriate constant has been added for normalization. Remark. Note that this theorem is still valid even if there is no uniqueness for viscosity solution of (32). Proof. Define P² by solving the equation ω0 = DP H̃ ² N(P²), that is ω0 = DP H̃0(P0) + ²DP H̃1(P0) + (P² − P0)D2PP H̃0(P0) + · · · , in which the expansion in terms of P − P0 and ² is taken up to order N − 1. Under the strict convexity assumption for H0(P ) = H̃0(P ) = H0(P ), we have detD2PP H̃0(P0) 6= 0, and so, the implicit function theorem yields a unique solution of the form P² = P0 + ²P1 + · · · , for ² small enough, and P1 = − [ D2PP H̃0(P0) ]−1 DP H̃1(P0). Define the new coordinates (P,X) by (7), that is (33) { p = P +Dxũ ² N(x, P ) X = x +DP ũ ² N(x, P ). To simplify the notation we denote X = φ(x, P ). Let x0 be the point given by Lemma 5.3. Let (x0,p0) = (x0, P² +Dxu ²(x0, P )) be the initial conditions for a trajectory (xn,pn) of (4). In the new coordinates, we have P0 = P². Perturbation theory 21 Also the dynamics is transformed into (34) { Xn+1 −Xn = −hDP H̃²N(Pn) +O(²N + |Pn − P0|N−1) Pn+1 −Pn = O(²N + |Pn − P0|N). From this equation, it is clear that Pn stays close to P² for long times. The next lemma proves a quantitative estimate that reflects this idea. Lemma 6.2. sup 0≤n≤ 1 ²N/4 |Pn − P²| ≤ O(²N/8). Proof. Note that for any δ > 0, we have |Pn+1 − P²|2 − |Pn − P²|2 = |Pn+1 + Pn − 2P²||Pn+1 −Pn| ≤ ≤ δ|Pn+1 + Pn − 2P²|2 + 1 δ |Pn+1 −Pn|2 ≤ ≤ 2δ [ |Pn+1 − P²|2 + |Pn − P²|2 ] + 1 δ |Pn+1 −Pn|2. Moreover, |Pn+1 −Pn|2 ≤ C²2N + C²2N |Pn − P²|2, as long as (35) |Pn − P²|2N−2 ≤ C²2N . We will show that this inequality is always satisfied for large N , and for the range of values n that will be used in our estimates. By choosing δ = ²N and setting an = |Pn − P²|2, we have an+1 − an ≤ C²N (an+1 + an) + C²N . To obtain the final estimate, we need an auxiliary lemma: Lemma 6.3. Suppose an is a sequence such that a0 = 0, and (1− C²N )an+1 ≤ (1 + C²N)an + C²N . Then, for all 0 ≤ n ≤ 1 3C²N and ² small enough, we have an ≤ √ C²N/2ne2n √ C²N/2. Proof. Set α = C²N . We will proceed by induction over n. For n = 0, the estimate is clear. Therefore, we assume it holds for some n and we will prove it for n+ 1. We have an+1 ≤ 1 + α 1− αan + α 1− α ≤ (1 + 3α)an + 2α, 22 D. Gomes, C. Valls for α sufficiently small. Then, by the induction hypothesis, an+1 ≤ (1+3α) √ αne2 √ αn+2α ≤ √ αe2 √ α(n+1)+ √ α(ne2 √ αn+2 √ αne2 √ αn). Since 1 + 2 √ α ≤ e2 √ α, we get an+1 ≤ √ α(n+ 1)e2 √ α(n+1). We should note that both the proof and the result of the previous lemma are not sharp, however they are sufficient for our purposes. In fact, this previous lemma implies that |Pn − P²| ≤ C²N/8, for all 0 ≤ n ≤ C ²N/4 , and therefore (35) is also satisfied. Observe that X = ψ(x) ≡ φ(x, P²) is a diffeomorphism, for small ². Let U(X) = u²(ψ−1(X), P²)− ũ²N(ψ−1(X), P²). Define X̃n = ψ(xn). Then, we have U(X̃n+1)− U(X̃n) = = u²(xn+1, P²)− u²(xn, P²)− ũ²N(xn+1, P²) + ũ²N(xn, P²) = = −hH²(P²) + hH(P² +Dxu²(xn+1, P²),xn)− − hDxu²(xn+1, P²)DpH(P² +Dxu²(xn+1, P²),xn)+ + hH̃²N(P²)− hH(P² +Dxũ²N(x̃n+1, P²),xn)+ + hDxũ ² N(x̃n+1, P²)DpH(P² +Dxũ ² N(x̃n+1, P²),xn)+ + ũ²N(x̃n+1, P²)− ũ²N(xn+1, P²) +O(²N), in which x̃n+1 − xn = −hDpH(P² +Dxũ²N(x̃n+1, P²),xn). Therefore U(X̃n+1)− U(X̃n) ≤ ≤ C|Dxu²(xn+1, P²)−Dxũ²N(xn+1, P²)|+ C|x̃n+1 − xn+1|+O(²N). Now, observe that pn+1 = Pn+1 +Dxũ ² N(xn+1,Pn+1), together with pn+1 = P² +Dxu ²(xn+1, P²), Perturbation theory 25 Since h [DpH(P² +Dxu ²(x̂, P²), x)−DpH(P² +Dxũ²N(x̃, P²), x)] = x̃− x̂, and −ũ²N(x̃, P²)−Dxũ²N(x̃, P²)(x̂− x̃) ≥ −ũ²N (x̂, P²)− C²|x̃− x̂|2, with |x̃− x̂|2 ≤ C|Dxũ²N(x̃, P²)−Dxu²(x̂, P²)|2, we obtain, O(²N) ≥ ũ²N(x, P²)− ũ²N(x̂, P²)− u²(x, P²) + u²(x̂, P²)+ + γ̃|Dxũ²N(x̃, P²)−Dxu²(x̂, P²)|2, for some constant γ̃, as long as ² is small enough. Therefore, the previous theorem yields |Dxũ²N(x̃, P²)−Dxu²(x̂, P²)|2 ≤ O(²M). ¥ We should point out that this last theorem shows that the Mather sets can be approximated through a perturbative method. In fact, since the Mather set is supported in (x,−DpH(P² +Dxu²(x̂, P²), x), the result implies that this graph is approximated uniformly by (x,−DpH(P² +Dxũ²N(x̃, P²), x). 7. Approximate Mather measures For integrable Hamiltonian systems, as discussed in Theorem 2.1, one can change coordinates by (7). In these new coordinates, the Lebesgue measure dX is invariant by the flow. Thus, by the change of coordinates formula, the measure det(I +D2xPu)dx is invariant for the original dynamics. The objective of this section is to show that Mather measures can be approximated by det(I +D2xP ũ ² N)dx. As the Mather measure in general is singular, we do not assert that it has a density approximated by the above expression but we claim that the mass in small boxes are comparable. 26 D. Gomes, C. Valls Theorem 7.1. Let M > 0. Assume ω0 = DpH0(P0) is Diophan- tine, and there exists functions ũ²N and H̃ ² satisfying (19) for N large enough. Let µ² be any Mather measure corresponding to P² (see theo- rem 6.1) . Consider push-forward ν² of µ² by the map (33). Let k be an arbitrary integer and ² sufficiently small. Then, there exists a partition of Tn in boxes {B+η } of size 1k such that ν²(B + η ) ≥ kn (k + 1)n |B+η |, for all η. Additionally, there exists another partition of Tn in boxes {B−η } of size 1 k such that ν²(B − η ) ≤ kn (k − 1)n |B − η |, for all η. In both cases, k can be taken polynomially large in ². Proof. To prove the first estimate, let k be an integer and consider a partition of Tn in boxes of size 1 k+1 (for the partition we use semi- open boxes, in such a way that they are pairwise disjoint). Obviously, for some box B, we have ν²(B) ≥ |B|. Now consider an additional partition of Tn in boxes {B+η } of size 1k such that the center of one of the boxes coincides with the center of B. As in Theorem 6.1, consider the ergodization time for the linear flow associated with a covering of T n of radius smaller that 1 2k2 . For ² sufficiently small and every η, there exists a time Tη, smaller than the ergodization time T , such that the image of B under the map (34) lies in the interior of the corresponding box B+η . Therefore, by the invariance of the Mather measure under this map, we have ν²(B + η ) ≥ |B| = kn (k + 1)n |B+η |. To justify that k can be taken polynomially large in ², suppose k = ²−α. The ergodization time for the linear flow corresponding to ²2α is of order T = O(²−2αr). For sufficiently large N and small ², the image of the box B under time Tη < T lies in the interior of B + η . The proof of the second estimate is similar: one considers a cover of Tn in boxes of size 1 k−1 . One of these boxes B has ν²(B) ≤ |B|. Consider an additional partition of Tn in boxes {B−η } of size 1k such that the center of one of the boxes coincides with the center of B. As Perturbation theory 27 before, we obtain ν²(B − η ) ≤ |B| = kn (k − 1)n |B − η |. ¥ As a corollary to this theorem we can state: Corollary 7.2. Let M > 0, J be a positive integer and f be a Lipschitz function on Tn. Assume ω0 = DpH0(P0) is Diophantine, and there exists functions ũ²N and H̃ ² satisfying (19) for N large enough. Let µ² be any Mather measure corresponding to P² (see theorem 6.1). Consider push-forward ν² of µ² by the map (33). Then, ∫ Tn f(X)dX = ∫ Tn f(X)dν² +O(² J). Proof. We can assume f to be a positive function. Consider a partition in boxes B+η as in the previous theorem. Then we have ∫ fdν² ≥ ∑ η ν²(B + η )min B+η f ≥ k n (k + 1)n ∑ η |B+η |min B+η f ≥ O(1 k ) + kn (k + 1)n ∑ η ∫ B+η f, since f is Lipschitz. Therefore ∫ fdν² ≥ O( 1 k ) + kn (k + 1)n ∫ Tn f ≥ ∫ f +O(²J), taking k = ²−J , and using the fact that f is bounded. The other inequality is similar, using the partition B−η given by the previous theorem. ¥ From this corollary we conclude that for any Lipschitz function f(x, p) and any positive J we can choose N large enough such that ∫ Tn f(x, P²+Dxu ²)dµ² = ∫ Tn f(x, P²+Dxũ ² N) det(I+D 2 xP ũ ² N)dx+O(² J), as ²→ 0. References [AKN97] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics. Springer-Verlag, Berlin, 1997. Trans- lated from the 1985 Russian original by A. Iacob, Reprint of the original