## 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
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### References:

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