  We investigate the surjectivity of the word map defined by the n-th Engel
word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is
surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently
large. Moreover, we give an estimate for Q(n). We also present examples
demonstrating that this does not hold for all q.
  We conclude that the n-th Engel word map is surjective for the groups
PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show
that for any n<5, the corresponding map is surjective for all the groups
PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel
words in finite simple groups.
  In addition, we show that the n-th Engel word map is almost measure
preserving for the family of groups PSL(2,q), with q odd, answering another
question of Shalev.
  Our techniques are based on the method developed by Bandman, Grunewald and
Kunyavskii for verbal dynamical systems in the group SL(2,q).
