  We consider the problem of approximating numerically the moments and the
supports of measures which are invariant with respect to the dynamics of
continuous- and discrete-time polynomial systems, under semialgebraic set
constraints. First, we address the problem of approximating the density and
hence the support of an invariant measure which is absolutely continuous with
respect to the Lebesgue measure. Then, we focus on the approximation of the
support of an invariant measure which is singular with respect to the Lebesgue
measure.
  Each problem is handled through an appropriate reformulation into a linear
optimization problem over measures, solved in practice with two hierarchies of
finite-dimensional semidefinite moment-sum-of-square relaxations, also called
Lasserre hierarchies.
  Under specific assumptions, the first Lasserre hierarchy allows to
approximate the moments of an absolutely continuous invariant measure as close
as desired and to extract a sequence of polynomials converging weakly to the
density of this measure.
  The second Lasserre hierarchy allows to approximate as close as desired in
the Hausdorff metric the support of a singular invariant measure with the level
sets of the Christoffel polynomials associated to the moment matrices of this
measure.
  We also present some application examples together with numerical results for
several dynamical systems admitting either absolutely continuous or singular
invariant measures.
