Abstracts

Lagrangian Flows: The Dynamics of Globally Minimizing Orbits II.
We prove most of the Theorems of Ricardo Mañé's unfinished work
Lagrangian Flows: The Dynamics of Globally Minimizing Orbits. dvi file and ps file available.

Convex Hamiltonians without conjugate points
We generalize some of results of the theory of geodesics whithout conjugate points: existence of Green bundles, hiperbolicity from the transversality of these bundles and an index form. We use these tools to proof that for generic Lagrangians, minimizing measures supported in periodic orbits or critical points are hyperbolic. Theorem stated by Ricardo Mañé in his unfinished work. We also derive a formula for the metric entropy. dvi file and ps file available.

On the creation of conjugate points for Hamiltonian systems
For a fixed Hamiltonian H on the cotangent bundle of a compact manifold M and a fixed energy level e, we prove that the set of potentials on M such that the Hamiltonian flow of H plus the potential is Anosov, is the interior in the C^2 topology of the set of potentials such that the flow has no conjugate points. dvi file and ps file available.

Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values
Let L be a convex superlinear Lagrangian on a closed connected manifold M. We consider critical values of Lagrangians as defined by R. Mañé. We show that the critical value of the lift of L to a covering of M 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 L on an energy level that contains minimizing measures with nonzero homology can be reduced to Finsler metrics. We also show that if the Euler-Lagrange flow of L on the energy level k is Anosov, then k must be strictly bigger than the critical value c_u(L) of the lift of L to the universal covering of M. It follows that given k less than c_u(L), there exists a potential g with arbitrarily small C^2-norm such that the energy level k of L+g possesses conjugate points. dvi file and ps fileavailable.

Connecting orbits between static classes for generic Lagrangian systems.
Let L be a smooth convex superlinear Lagrangian on a closed manifold M. We show that if the number of static classes is finite, then there exist chains of semistatic orbits that connect any two given static classes. Using this property we show that if there is only one static class, then the homoclinic orbits to the set of static orbits generate over R the relative homology of the pair (M,U), where U is a sufficiently small neighborhood of the set of static orbits in M.

We show that generically in the sense of Mañé the set of semistatic orbits coincides with the support of a uniquely minimizing measure, therefore generically, the homoclinic orbits to the support of the minimizing measure generate over R the relative homology of the pair (M,U), where U is a sufficiently small neighborhood of the projection of the support of the measure to M.

This last result was obtained -with a different proof- by S. Bolotin, assuming the existence of a C^{1+Lipschitz} function f:M->R such that L+c-df >= 0, where c is the critical value of L.

Finally we obtain two consequences. The first one says that if M is a closed manifold with first Betti number >= 2 then there exists a generic set O of smooth functions M->R such that if f is in O, the Lagrangian L+f has a unique minimizing measure and this measure is uniquely ergodic. When this measure is supported on a periodic orbit, this orbit is hyperbolic and the stable and unstable manifolds have transverse homoclinic intersections.

The second consequence says that if M is a closed manifold with first Betti number different from zero and if L is a symmetric Lagrangian, then there exists a generic set O of smooth functions M->R such that if g is in O, then L+g has a unique minimizing measure and this measure is supported on a hyperbolic fixed point whose stable and unstable manifolds have transverse homoclinic intersections.

Lyapunov minimizing measures for expanding maps of the circle.
We consider the set F_{a+} of C^{1+b} self-maps f of the circle with b>a and which are covering maps of degree D, expanding , f'(x) >1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f-invariant probability measure m minimizing integral of ln(f') over all f-invariant probability measures. We show there exists a subset G_+ of F_{a+}, open and dense in the C^{1+a}-topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that every f not in G_+ is a limit in the C^{1+a}-topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.

We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian Systems, the analog equation was introduced by R. Mañé and extended to a sub-cohomological equation to all the configuration space by A. Fathi.

We will also present some results on the set of f-invariant measures m maximizing the integral of A for a fixed C^1-expanding map f and a general potential A, not necessarily equal to -ln(f') . LaTeX file, dvi file and ps.gz file available.

Version for subshifts of finite type. LaTeX file, dvi file and ps.gz file available.

The Palais-Smale condition and Mañé's critical values.
Let L be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé. We define energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of 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 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 (the critical value of the lift of 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. 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 and homologically trivial periodic orbits such that the action of L+k is negative if c_u<k<c_a, where c_a is the critical value of the lift of 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. LaTeX file, dvi file and ps.gz file available.

The Asymptotic Maslov index and its Applications.

Let N be a 2n-dimensional manifold equipped with a symplectic structure w and G(N)$ be the Lagrangian Grassmann bundle over N. Consider a flow f^t on N that preserves the symplectic structure and a f^t-invariant connected submanifold S. When there exists a continuous section S->G(N), we can associate to any finite, f^t-invariant measure with support in S, a quantity: The Asymptotic Maslov Index, that describes the way Lagrangian planes are asymptotically wrapped in average around the Lagrangian Grassmann bundle. A particular attention is paid to the case when the flow is derived from an optical Hamiltonian and when the invariant measure is the Liouville measure on compact energy levels. The situation when the energy levels are not compact is discussed in an appendix.   LaTeX file and ps.gz file available.    

Action Potential and weak KAM solutions.

 We characterize the Peierl's barrier and Fathi's weak KAM solutions in terms of Mañé's  action potential and the static classes for convex superlinear lagrangians on compact manifolds M. We show how to construct weak KAM solutions for lagrangians on compact manifolds,   non-compact manifolds when the Peierls barrier is finite and when it is infinite. In the non-compact case, we compactify  the parameter space manifold M by adding ``extended static classes at infinity'' using weak KAM solutions and give a characterization of those solutions using ``Busemann weak KAM solutions''. LaTeX and ps.gz files available.    

Weak solutions of the Hamilton-Jacobi equation for Time Periodic Lagrangians.

  In this work we generalize to periodic Lagrangians several results -originally stated for autonomous Lagrangians- including the existence of a Mañé's critical value, its characterization in terms of smooth subsolutions of the Hamilton Jacobi equation, and the existence of Fathi's weak KAM solutions.  

Partially Hyperbolic geodesic flows are Anosov.

  We prove that if a $\Z$ or $\re$-action by symplectic linear maps on a symplectic vector bundle $E$ has a weakly dominated invariant splitting $E=S\oplus U$ with $\dim U=\dim S$, then the action is hyperbolic. In particular, contact and geodesic flows with a dominated splitting with $\dim S=\dim U$ are Anosov. LaTeX, dvi, dvi.gz, PS, PDF, ps.gz, pdf.gz files available.  

Genericity of geodesic flows with positive topological entropy on S2 .

  We show that the set of smooth riemannian metrics on S2 or RP2 whose geodesic flow has positive topological entropy is open and dense in the C2 topology. LaTeX, dvi, dvi.gz, PS, PDF, ps.gz, pdf.gz files available.  

Periodic orbits for exact magnetic flows on surfaces.
with Leonardo Marcarini and Gabriel Paternain.

  We show that any exact magnetic flow on a closed surface has periodic orbits in all energy levels. Moreover, we give homological and homotopic properties of these periodic orbits in terms of the Mañé's critical values of the corresponding Lagrangian. We also prove that if M is not the 2-torus the energy level k is of contact type if and only if k>Co, where Co is Mañé's strict critical value. When M is the 2-torus we give examples for which the energy level Co is of contact type. PS, PDF, ps.gz files available.  

The Palais-Smale condition for contact type energy levels for convex lagrangian systems.

  In this paper we continue the study of the Morse theory of the free time action functional for convex lagrangian systems that we did in~\cite{CIPP2}. This time we try to aboard the case of lower energy levels, where very little is known. The main problem with the free time action functional is that it may fail to satisfy the Palais-Smale condition, usually required for variational methods. We also prove that when an energy level projects onto the whole configuration space M, the set of closed loops has a mountain pass geometry. An adaptation of an argument by Struwe to the mountain pass geometry shows the existence of convergent Palais-Smale sequences for almost all energy levels. This implies that for almost all energy levels which project onto M the Euler-Lagrange flow has a periodic orbit, closed orbit loops starting at any x in M and conjugate points if the energy is below Ma\~n\'e's critical value of the universal cover. The same holds for an energy level which satisfies the Palais-Smale condition. In particular, for contact type energy levels. In particular, for exact magnetic flows, almost all energy levels have a periodic orbit. dvi, pdf, ps.gz files available.  

C2-densely the 2-sphere has an elliptic closed geodesic.

 We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by one whose geodesic flow has an elliptic closed geodesic. PS, PDF, ps.gz files available.  

Geodesic flows with positive topological entropy, twist maps and dominated splittings.

  We prove a perturbation lemma for the derivative of geodesic flows in high dimension. This implies that a C2 generic riemannian metric has a non-trivial hyperbolic basic set in its geodesic flow. PDF, file available.  

A generic property of families of Lagrangian systems.
with Patrick Bernard.

  We prove that a generic lagrangian has finitely many minimizng measures for every cohomology class. dvi and PDF, files available.  

Homogenization on Arbitrary Manifolds.
with Renato Iturriaga and Antonio Siconolfi.

  We describe a setting for homogenization of convex hamiltonians on abelian covers of any compact manifold. In this context we also provide a new simple variational proof of standard homogenization results. PDF file.  

Ground States are generically a Periodic Orbit.

  We prove that for an expanding transformation the maximizing measures of a generic Lipschitz function are supported on a single periodic orbit. PDF file.  

Last modified: July 1, 2013.