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.
On the creation of conjugate points for Hamiltonian systems
Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values
Convex Hamiltonians without conjugate points
Average Linking Numbers
Finsler Metrics and Action Potentials
The Palais Smale condition and Mañé's Critical Values
We also show that if an energy level without conjugate points
satisfies the Palais-Smale condition, then
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.
If an energy level without conjugate points has energy
bigger than c_u(L) (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(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 homotopy class with energy greater than c_u(L) and
homologically trivial periodic orbits such that the action of L+k is
negative if c_u(L)
A Minimax Selector for a Class Of Hamiltonians on a
Cotangent Bundle
A Geometric proof of the existence of
the Green Bundles
Topological Shocks in Burgers Turbulence.
The Asymptotic Maslov index and its Applications.
Weak solutions of the Hamilton Jacobi equation for
periodic Lagrangians
Burgers Turbulence and Random
Lagrangian 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.
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. Finally we show the existence of weak KAM
solutions for coveringsof M and explain the relationship between Fathi's results and Mañé's Critical Values and action potentials.
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.
We prove that it is possible to define the average
linking number (Hopf invariant) for every pair of invariant measures which
dont have a common periodic orbit of positive measure.
dvi file and ps file available.
We study the behavior of Mañé's action potential $\Phi_k$ associated to a
convex superlinear Lagrangian, for $k$ bigger than the critical value $c(L)$ .
We obtain growth estimates of the action potential as function of $k$.
We prove that the action potential can be written as $\Phi_k(x,y)=D_F(x,y)+f(y)-f(x)$
where $f$ is a smooth function and $D_F$ is the distance function associated to a
Finsler metric.
dvi file available.
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 construct a minimax selector for eventually quadratic hamiltonians
on cotangent bundles. We use it to give a relation between Hofer's
energy and Mather's action minimizing function. We also study the
local flatness of the set of twist maps.
dvi file available.
We give a new proof of the existence of the Green Bundles.
dvi file available.
The dynamics of the multidimensional randomly forced Burgers equation
is studied in the limit of vanishing viscosity. It is shown both theoretically
and numerically
that the shocks have a universal global structure
which is determined by the topology of the configuration space. This
structure is shown to be particularly rigid for the case of periodic
boundary conditions.
Let N be a 2n-dimensional manifold equipped with a symplectic
structurew and G(N)$ be the Lagrangian Grassmann bundle over N.
Consider a flow t^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.
dvi file available.
In this work we generalize to periodic Lagrangians several results
-originally stated for autonomous Lagrangians- including the existence
of a Ma\~n\'e's critical value , its characterization in terms of
smooth subsolutions of the Hamilton Jacobi equation as in
Lagrangian graphs... , and
the existence of Fathi's weak KAM solutions.
dvi file available.
We consider spatially periodic inviscid random forced Burgers
equation in
arbitrary dimension and the random time-dependent Lagrangian system
related to it. We construct a unique stationary distribution for
``viscosity" solutions of burgers equation. We also show that
with probability 1 there exists a unique minimizing trajectory
for the random Lagrangian system. These minimizing trajectories
generate a unique minimizing measure for the non-random skew-product
extension of the Lagrangian system.
dvi file available.