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A generic property of families of Lagrangian systems. (English) Zbl 1175.37067
Let $$M$$ be a compact smooth manifold, and $$\mathbb{T}:= \mathbb{R}/\mathbb{Z}$$ or $$\{0\}$$. A Tonelli Lagrangian will be a $$C^2$$ real function $$L$$ on $$\mathbb{T}\times TM$$ such that:
$$\bullet$$ for each $$(t,m)\in\mathbb{T}\times M$$, $$L_{(t,m)}$$ is convex on $$T_mM$$ and $$\lim_{|\theta|\to\infty} L_{(t,m)}(\theta)/|\theta|= +\infty$$;
$$\bullet$$ the flow $$\varphi$$ $$(\mathbb{R}\times \mathbb{T}\times TM\to \mathbb{T}\times TM)$$ defined by the Euler-Lagrange equation $${d\over ds}L_\theta= L_m$$ is complete;
$$\bullet$$ the previous properties hold as well for any $$(L-u)$$ instead of $$L$$, $$u$$ summing $$C^\infty(\mathbb{T}\times M,\mathbb{R})$$.
Let $$m(L)$$ denote the set of $$\varphi$$-invariant probability measures $$\mu$$ on $$\mathbb{T}\times TM$$ which minimize the action $$\int_{\mathbb{T}\times TM} L\,d\mu$$.
Let $$A$$ be a finite-dimensional space of Tonelli Lagrangians.
Then the main theorem asserts that there exists a dense subset $$O$$ of $$C^\infty(\mathbb{T}\times M,\mathbb{R})$$, countable intersection of open subsets, such that $u\in O,\;L\in A\rightarrow\dim m(L-u)\leq \dim A;$ in other words, there exist at most $$(1+\dim A)$$ ergodic minimizing measures of $$(L-u)$$.
This result generalizes a result of R. Mañé [Nonlinearity 9, No. 2, 273–310 (1996; Zbl 0886.58037)], which corresponds to $$\dim A= 1$$. The proof is mainly made in an abstract context.
The authors notice finally that $$u$$ can be replaced as well as by cohomology class of 1-forms on $$M$$.

##### MSC:
 37J50 Action-minimizing orbits and measures (MSC2010) 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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