  We introduce a new class of adelic heights on the projective line.
  We estimate their essential minimum and prove a result of equidistribution
(at every place) for points of small height with estimates on the speed of
convergence. To each rational function R in one variable and defined over a
number field K, is associated a normalized height on the algebraic closure of
K. We show that these dynamically defined heights are adelic in our sense, and
deduce from this equidistribution results for preimages of points under R at
every place of K. Our approach follows that of Bilu, and relies on potential
theory in the complex plane, as well as in the Berkovich space associated to
the projective line over C_p, for each prime p.
