## Action minimizing invariant measures for positive definite Lagrangian systems.(English)Zbl 0696.58027

Let $$M$$ be a compact smooth manifold and let $$L: TM\times {\mathbb{R}}\to {\mathbb{R}}$$ be a time dependent Lagrangian of class $$C^ 2$$ which is periodic of period one in the time variable (i.e. $$L(\xi,z+1)=L(\xi,t))$$, satisfies the Legendre condition (in local coordinates $$L_{\dot x\dot x}>0)$$, and has superlinear growth along the fibers of $$TM$$ (i.e. $$L(\xi,t)/\| \xi \| \to +\infty$$ as $$\| \xi \| \to +\infty$$, for $$\xi\in TM$$, $$t\in {\mathbb{R}})$$. Let $$\Phi_ L$$ denote the associated Euler-Lagrange flow. Assume that $$\Phi_ L$$ is complete.
For every $$\Phi_ L$$-invariant probability measure $$\mu$$ on TM$$\times {\mathbb{R}}/{\mathbb{Z}}$$ define the average action as $$A(\mu)=\int L\, d\mu$$ and the rotation vector $$\rho (\mu)\in H_ 1(M,{\mathbb{R}})$$ by $$\langle[\lambda],\rho (\mu)\rangle=\int \lambda\, d\mu$$, where $$\lambda : TM\to {\mathbb{R}}$$ is a closed 1-form on $$M$$ and $$[\lambda]\in H^ 1(M,{\mathbb{R}})$$ is its cohomology class. The rotation vector $$\rho$$ ($$\mu)$$ exists whenever $$A(\mu)<\infty.$$
Theorem: For every $$h\in H_ 1(M,{\mathbb{R}})$$, there exists a $$\Phi_ L$$- invariant probability measure $$\mu$$ such that $$A(\mu)<\infty$$ and $$\rho (\mu)=h$$. Moreover, $$A$$ takes a minimum value $$\beta(h)$$ on the set of $$\Phi_ L$$-invariant probability measures $$\mu$$ such that $$A(\mu)<\infty$$ and $$\rho (\mu)=h$$. The function $$\beta$$ is convex and has superlinear growth.
For $$c\in H^ 1(M,{\mathbb{R}})$$, let $${\mathfrak M}_ c$$ denote the set of $$\Phi_ L$$-invariant probability measures $$\mu$$ which minimize $$A_ c(\mu)=A(\mu)-\langle,\rho (\mu)\rangle$$. Let $$\text{supp }{\mathfrak M}_ c$$ denote the support of $${\mathfrak M}_ c$$. Let $$\pi : TM\times {\mathbb{R}}/{\mathbb{Z}}\to M\times {\mathbb{R}}/{\mathbb{Z}}$$ denote the projection.
Theorem. Supp $${\mathfrak M}_ c$$ is compact. The restriction of $$\pi$$ to supp $${\mathfrak M}_ c$$ is injective. The inverse mapping from $$\pi (\text{supp } {\mathfrak M}_ c)$$ to $$\text{supp }{\mathfrak M}_ c$$ is Lipschitz.
An application of these results is given to $$C^ 1$$ small perturbations to a Hamiltonian system having a KAM torus satisfying a positive definiteness condition in the normal direction. Also, it is shown how basic results concerning area preserving twist maps follow from these results.
Reviewer: J.N.Mather

### MSC:

 37J50 Action-minimizing orbits and measures (MSC2010) 37C10 Dynamics induced by flows and semiflows 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37E40 Dynamical aspects of twist maps 28D20 Entropy and other invariants

### Keywords:

Lagrangian system; Euler-Lagrange flow; probability measure
Full Text:

### References:

 [1] Aubry, S., Le Daeron, P.Y.: The discrete Frenkel-Kontorova model and its extensions I. Physica D8, 381–422 (1983) · Zbl 1237.37059 [2] Akheizer: The calculus of variations, New York: Blaisdell 1962 [3] Ball, J., Mizel, V.: One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal.90, 325–388 (1985) · Zbl 0585.49002 [4] Bangert, V.: Mather sets for twist maps and geodesics on tori. Dyn. Rep.1, 1–56 (1988) · Zbl 0664.53021 [5] Bangert, V.: Minimal geodesics. Preprint (1987) [6] Bernstein, D., Katok, A.: Birkhoff periodic orbits for small perturbations of completely integrable systems with convex Hamiltonians. Invent. Math.88, 225–241 (1987) · Zbl 0642.58040 [7] Caratheodory, C.: Variationsrechnung und partielle Differentialgleichung erster Ordnung. Leipzig-Berlin: B.G. Teubner 1935 [8] Cartan, E.: Lecons sur les invariants integrals. Paris: Herman 1922 · JFM 48.0538.02 [9] Dacorogna, B.: Direct methods in the calculus of variations. Appl. Math. Sic. vol. 78, Berlin Heidelberg New York: Springer 1989 · Zbl 0703.49001 [10] Denzler, J.: Mather sets for plane Hamiltonian systems. J. Appl. Math. Phys. (ZAMP)38, 791–812 (1987) · Zbl 0641.70014 [11] Douady, R.: Stabilité ou instabilité des points fixes elliptiques. Ann. Sci. Éc. Norm. Supér., IV. Sér.21, 1–46 (1988) · Zbl 0656.58020 [12] Hedlund, G.: Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. Math., II. Ser.33, 719–739 (1932) · JFM 58.1256.01 [13] Herman, M.R.: Existence et non existence de tores invariants par des diffeomorphismes symplectiques. Preprint (1988) · Zbl 0664.58005 [14] Katok, A.: Minimal orbits for small perturbations of completely integrable Hamiltonian systems. Preprint (1988) · Zbl 0762.58024 [15] Kryloff, N.M., Bogoliuboff, N.N.: La théorie générale de la mesure et son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. Math., II. Sér.38, 65–113 (1937) · Zbl 0016.08604 [16] Lanford, O.: Selected Topics in Functional Analysis. In: DeWitt, Stora (eds.): Statistical Mechanics and Quantum Field Theory. Proceedings of the Summer School of Theoretical Physics, Les Houches 1970, pp. 109–214. New York: Gordon and Breach 1971 [17] Mather, J.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology21, 457–467 (1982) · Zbl 0506.58032 [18] Mather, J.: Minimal Measures. Comment. Math. Helv.64, 375–394 (1989) · Zbl 0689.58025 [19] Mather, J.: Minimal action measures for positive definite lagrangian systems. In: Simon, B. et al. (eds.): IX International Congress on Mathematical Physics. Bristol-New York: Adam Hilger 466–468 (1989) · Zbl 0850.70195 [20] Moser, J.: Monotone Twist Mappings and the Calculus of Variations. Ergodic Theory Dyn. Syst.6, 401–413 (1986) · Zbl 0619.49020 [21] Moser, J.: On the construction of almost periodic solutions of ordinary differential equations. Proc. Int. Conf. Funct. Anal. and Rel. Top., Tokyo 60–67 (1969) [22] Nemytskii, V.V., Stepanov, V.V.: Qualitative Theory of Differential Equations. Princeton, N.J.: Princeton University Press 1960 · Zbl 0089.29502 [23] Rockafellar, R.T.: Convex Analysis. Princeton Math. Ser., vol. 28. Princeton: Princeton University Press 1970 · Zbl 0193.18401 [24] Salamon, D.: The Kolmogorov-Arnold-Moser theorem. Forschungsinstitut für Mathematik, ETH-Zentrum, Preprint (1986) · Zbl 1136.37348 [25] Salamon, D., Zehnder, E.:KAM theory in configuration space. Comment. Math. Helv.64, 84–132 (1989) · Zbl 0682.58014 [26] Schwartzman, S.: Asymptotic cycles. Ann. Math. II. Ser.,66, 270–284 (1957) · Zbl 0207.22603 [27] Weinstein, A.: Lagrangian submanifolds and Hamiltonian systems. Ann. Math. II. Ser.98, 377–410 (1973) · Zbl 0271.58008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.