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Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values (1998)

Contreras, G. ; Iturriaga, R. ; Paternain, G.P. ; [et al.]

Springer

Geometric and functional analysis 8 (1998), S. 788-809

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ISSN: 1420-8970

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract. Let $\Bbb L$ be a convex superlinear Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value of the lift of $\Bbb L$ to a covering of N equals the infimum of the values of k such that the energy level k bounds an exact Lagrangian graph in the cotangent bundle of the covering. As a consequence, we show that up to reparametrization, the dynamics of the Euler-Lagrange flow of $\Bbb L$ on an energy level that contains supports of minimizing measures with non-zero rotation vector can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of $\Bbb L$ on the energy level k is Anosov, then k must be strictly bigger than the critical value c u ( $\Bbb L$ ) of the lift of L to the universal covering of N. It follows that given k 〈 c u ( $\Bbb L$ ), there exists a potential $\psi$ with arbitrarily small C 2-norm such that the energy level k of $\Bbb L + \psi$ possesses conjugate points. Finally we show the existence of weak KAM solutions for coverings of N and we explain the relationship between Fathi's results in [F1,2] and Mañé's critical values and action potentials.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/s000390050074

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Critical values of autonomous Lagrangian systems (1997)

Paternain, G. ; Paternain, M.

Springer

Commentarii mathematici Helvetici 72 (1997), S. 481-499

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ISSN: 1420-8946

Keywords: Key words. Lagrangian systems, minimizing measures, critical values.

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract. Let M be a closed manifold and $ L : TM \rightarrow \bf {R} $ a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Mañé in [5]. Let c u (L) denote the critical value of the lift of L to the universal covering of M and let c a (L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, $ c_{u}(L) \leq c_{a}(L) $ . Let c 0 (L) denote the strict critical value of L defined as the smallest critical value of $ L - \omega $ where $ \omega $ ranges among all possible closed 1-forms. We show that c a (L) = c 0 (L). We also show that if there exists k such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all $ k'\geq k $ , then $ k 〉 c_{u}(L) $ . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy $ k 〈 c_{a}(L) $ , thus answering in the negative a question raised by Mañé. This example also shows that the inequality $ c_{u}(L) \leq c_{a}(L) $ could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not have minimizing measures. Finally, we describe a large class of Lagrangians for which c u (L) is strictly bigger than the maximum of the energy restricted to the zero section of TM.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/s000140050029

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The Palais-Smale Condition and Mañé's Critical Values (2000)

Contreras, G. ; Iturriaga, R. ; Paternain, G.P. ; [et al.]

Springer

Annales Henri Poincaré 1 (2000), S. 655-684

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ISSN: 1424-0661

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics , Physics

Notes: Abstract. Let $ {\Bbb L} $ be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [23]. We define energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of $ {\Bbb L} $ to any covering of N equals the infimum of the values of k such that the energy level t satisfies the Palais-Smale condition for every t 〉 k provided that the Peierls barrier is finite. When the static set is not empty, the Peierls barrier is always finite and thus we obtain a characterization of the critical value of $ {\Bbb L} $ in terms of the Palais-Smale condition.¶We also show that if an energy level without conjugate points has energy strictly bigger than c u ($ {\Bbb L} $) (the critical value of the lift of $ {\Bbb L} $ to the universal covering of N), then two different points in the universal covering can be joined by a unique solution of the Euler-Lagrange equation that lives in the given energy level. Conversely, if the latter property holds, then the energy of the energy level is greater than or equal to c u ($ {\Bbb L} $). In this way, we obtain a characterization of the energy levels where an analogue of the Hadamard theorem holds. We conclude the paper showing other applications such as the existence of minimizing periodic orbits in every non-trivial homotopy class with energy greater than c u ($ {\Bbb L} $) and homologically trivial periodic orbits such that the action of $ {\Bbb L} $ + k is negative if c u ($ {\Bbb L} $) 〈 k 〈 c a ($ {\Bbb L} $), where c a ($ {\Bbb L} $) is the critical value of the lift of $ {\Bbb L} $ the abelian covering of N. We also prove that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange flow in the given energy level.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/PL00001011

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