16,900 research outputs found
Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians
We prove the following rigidity result for the Tonelli Hamiltonians. Let T *
M be the cotangent bundle of a closed manifold M endowed with its usual
symplectic form. Let (F\_n) be a sequence of Tonelli Hamiltonians that C^0
converges on the compact subsets to a Tonelli Hamiltonian F. Let (G\_n) be a
sequence of Hamiltonians that that C^0 converges on the compact subsets to a
Hamiltonian G. We assume that the sequence of the Poisson brackets ({F\_n ,
G\_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F,
G}
Pseudographs and Lax-Oleinik semi-group: a geometric and dynamical interpretation
Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact
and connected manifold and let u be a semi-concave function defined on M. If E
(u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian
flow of H, we prove that for t > 0 small enough, \phi-t (E (u)) is an exact
Lagrangian Lipschitz graph. This provides a geometric
interpretation/explanation of a regularization tool that was introduced by
P.~Bernard to prove the existence of C 1,1 subsolutions
Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting
We consider locally minimizing measures for the conservative twist maps of
the -dimensional annulus or for the Tonelli Hamiltonian flows defined on a
cotangent bundle . For weakly hyperbolic such measures (i.e. measures
with no zero Lyapunov exponents), we prove that the mean distance/angle between
the stable and the unstable Oseledet's bundles gives an upper bound of the sum
of the positive Lyapunov exponents and a lower bound of the smallest positive
Lyapunov exponent. Some more precise results are proved too
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