  Finding special orbits (as periodic orbits) of dynamical systems by
variational methods and especially by minimization methods is an old method
(just think to the geodesic flow). More recently, new results concerning the
existence of minimizing sets and minimizing measures were proved in the setting
of conservative twisting dynamics. These twisting dynamics include geodesic
flows as well as the dynamics close to a completely elliptic periodic point of
a symplectic diffeomorphism where the torsion is positive definite . Two
aspects of this theory are called the Aubry-Mather theory and the weak KAM
theory. They were built by Aubry \& Mather in the '80s in the 2-dimensional
case and by Mather, Ma{\~n}{\'e} and Fathi in the '90s in higher dimension. We
will explain what are the conservative twisting dynamics and summarize the
existence results of minimizing measures. Then we will explain more recent
results concerning the link between different notions for minimizing measures
for twisting dynamics: their Lyapunov exponents; their Oseledet's splitting;
the shape of their support. The main question in which we are interested is:
given some minimizing measure of a conservative twisting dynamics, is there a
link between the geometric shape of its support and its Lyapunov exponents? Or
: can we deduce the Lyapunov exponents of the measure from the shape of the
support of this measure? Some proofs but not all of them will be provided. Some
questions are raised in the last section.
