  We investigate from a statistical perspective the arithmetic properties of
the dynamics of polynomials of fixed degree and defined over the field of
rational numbers. To start with, ordering their affine conjugacy classes by
height, we show that their average number of rational preperiodic points is
equal to zero, thereby proving a strong average version of the uniform
boundedness conjecture of Morton and Silverman. Next, inspired by the analogy
with the successive minima of a lattice we define the dynamical successive
minima of a polynomial. Noting that these quantities are invariant under the
action by conjugacy of the affine group we study their average behaviour using
the aforementioned ordering by height. In particular, we prove an optimal
statistical version of the dynamical Lang conjecture on the canonical height of
rational non-preperiodic points.
