Summary: COUNTING GEODESICS WHICH ARE OPTIMAL IN
The profusion of recent papers investigating the properties of optimal orbits of
dynamical systems, generalizing or contradicting some results specific to Lagrangian
systems, leaves no doubt as to the interest of such a study in a general setting.
However, it seems that a unified motivation remains to be found.
Optimal orbits and optimal measures appear most naturally in the Aubry-
Mather theory for Lagrangian systems : they are orbits, or measures, minimizing
globally the integral of the lagrangian. In this context they are classically called
action-minimizing orbits (and measures). More generally, given a dynamical system
and a potential f on the phase space, one may ask for a description of the orbits,
or of the invariant measures, minimizing the integral of f. Such a question appears
for instance in [YH] (from where we took the expression "optimal orbits") where
it is linked to the question of "controlling chaos"; in [CLT], as a variant of Ma~ne's
work on Lagrangian systems. In [J] and in [Bo] specific examples are treated for
their intrinsic interests.
In the papers mentioned above, the dynamical systems under study are ex-
panding or hyperbolic. It is a situation where the behaviour of arbitrary orbits is
very well known : the system is expansive, satisfies specification in fact, lots of