  In this paper, we construct a compactification of the space of Bridgeland
stability conditions on a smooth projective curve, as an analogue of Thurston
compactifications in Teichm\"uller theory.
  In the case of elliptic curves, we compare our results with the classical one
of the torus via homological mirror symmetry and give the Nielsen-Thurston
classification of autoequivalences using the compactification.
  Furthermore, we observe an interesting phenomenon in the case of the
projective line.
