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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
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