  We study the asymptotics, box dimension, and Minkowski content of
trajectories of some discrete dynamical systems. We show that under very
general conditions, trajectories corresponding to parameters where saddle-node
bifurcation appears have box dimension equal to 1/2, while those corresponding
to period-doubling bifurcation parameter have box dimension equal to 2/3.
Furthermore, all these trajectories are Minkowski nondegenerate. The results
are illustrated in the case of logistic map.
