Given a sequence of 3-manifolds M(n) converging strongly to a
simply degenerate one, obtained from a once-punctured torus group, we
study properties of limit sets. In particular we prove that the limit set
of the limit manifold contains a Jordan curve L with Hausorff dimension
bigger then one and L is the Hausdorff limit of chain of circles in the
limit sets of M(n). This is proved using Minsky model manifold and
properties of Cannon-Thurston map.