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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 \(M\).

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