The critical value $c(L)$ was introduced by Mane as the infimum of $k \in {\Bbb R}$ such that, for some $q \in M$, $\Phi\sb {k}(q,q) > -\infty$.

The authors show that such a critical value can also be characterized (whenever the Peierls barrier $h\sb {c}(q\sb {1}, q\sb {2})$ is finite) as the infimum of the levels $k$ such that the functional ${\scr A}\sb {L} \colon {\Bbb R} \times \{x \in H\sp {1}(0,T; M)\vert x(0) = q\sb {1}, x(T) = q\sb {2}\}\to {\Bbb R}$, ${\scr A}\sb {L}(b, x) = b\int\sb {0}\sp {1} L(x(t), \dot{x}(t)/b) \,dt$, satisfies the (PS) condition at level $k$. The condition on the Peierls barrier is shown to hold whenever the static set is not empty, in particular whenever $M$ is compact.

If $M$ is the universal covering of $N$ and $q\sb {0} \in N$ has no conjugate point, then it is also shown in the paper that for every $q \in M$ there is a unique solution with energy $k$ joining $q\sb {0}$ and $q$ provided $k > c(L)$, and, conversely, that the existence of such a unique solution for every $q \in M$ implies that $k \geq c(L)$.

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The introduction presents classes of interesting examples to which the theory developed in the book applies: mechanical Lagrangians, magnetic Lagrangians and twisted geodesic flows. In Section 4-2 relations to Finsler geometry are treated via the Maupertuis principle. Chapters 2 and 3 form the core of the book. Here the authors present complete proofs for existence and properties of minimizing measures, minimizing orbits, Mather's minimal average action and Mane's critical value, Peierls's energy barrier, etc. Chapter 4 treats the relation to weak solutions of the Hamilton-Jacobi equation as developed by A. Fathi [C. R. Acad. Sci. Paris Ser. I Math. 324 (1997), no. 9, 1043--1046; MR 98g:58151] and to G. P. and M. Paternain's work [Math. Z. 217 (1994), no. 3, 367--376; MR 95k:58117] on Anosov energy levels. The final chapter is based on work by Mane and by the authors and investigates how the theory is simplified and sharpened if one assumes the Lagrangian to be generic.

A non-expert will often have to refer to the original literature in order to find more motivation for concepts and results. Apart from this the book is a very useful and almost everywhere reliable source of information.

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\{For the entire collection see MR 2001e:00023.\}

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\{The papers, all in Spanish, are being reviewed individually.\}

Cited in: 1 787 550 1 787 545 1 787 543 2001g:37067 2001f:57008 2001f:37073

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They give necessary and sufficient conditions for these bundles to define an Anosov splitting for the flow, extending the previously known results in the case of geodesic flows.

By comparing the Green bundles with the splitting provided by Oseledets and Pesin theory, they obtain a formula for the metric entropy of the Liouville measure, generalizing the results of A. Freire and R. Mane [Invent. Math. 69 (1982), no. 3, 375--392; MR 84d:58063] for the case of geodesic flows.

In the dual language of Lagrangian flows, Mane [Nonlinearity 9 (1996), no. 2, 273--310; MR 97d:58118] proved that for any Lagrangian $L\colon TM \toR$, there is a generic set of potentials $\Phi$ for which the Lagrangian flow of $L+\Phi$ has a unique minimizing measure. In the paper under review, the authors prove the following conjecture of Mane: potentials for which the support of the unique minimizing measure is a periodic orbit can be arbitrarly approximated by potentials for which the minimizing measure is still supported by a periodic orbit, and with the following additionnal properties: the periodic orbit is hyperbolic, and its stable and unstable manifolds have only transverse intersections.

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The proof goes as follows. It is well known that in the $C\sp {k}$ topology the set ${\scr A}\sb {e}$ is open and the set ${\scr B}\sb {e}$ is closed. G. P. Paternain and M. Paternain [Math. Z. 217 (1994), no. 3, 367--376; MR 95k:58117] showed that ${\scr A}\sb {e}$ is always contained in ${\scr B}\sb {e}$, hence to prove their theorem the authors need to show that an interior point of ${\scr B}\sb {e}$ lies in ${\scr A}\sb {e}$. For this they take a system without conjugate points which is not Anosov, then they show that they can make a $C\sp {2}$ small perturbation in a neighborhood of an orbit segment to end up with a system that has conjugate points. This is achieved by showing that the index form of the perturbed system is not positive definite on that segment.

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Cited in: 98i:58092

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In a seminal paper [Math. Z. 207 (1991), no. 2, 169--207; MR 92m:58048], J. N. Mather studied periodic Lagrangians on closed Riemannian manifolds, introducing concepts such as the homology of an invariant measure, minimizing measures, etc. Here the author considers a Lagrangian system $(d/dt)(\partial L/\partial v)(x,x',y)=(\partial L/\partial x)(x,x',y)$, where the parameter $y$ belongs to a compact Riemannian manifold $N$ and varies according to an autonomous system $y'=X(y)$. This setting covers for instance the case where $N$ is a torus and $X$ an irrational linear flow (the quasiperiodic case) or when the flow on $N$ is hyperbolic. In the latter, the results come closer to those of Mather.

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(c) 2001, American Mathematical Society