We consider a univariate two-scale difference equation,
which is studied in approximation theory, curve design and
wavelets theory. This paper analysis the correlation between the existence of
smooth compactly supported solutions of this equation and the convergence
of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion that
expresses this correlation in terms of mask of the equation.
It was shown that the convergence of subdivision scheme
depends on values that the mask takes at the points of its
generalized cycles. In this paper we show that the criterion is sharp in the
sense that an arbitrary generalized cycle causes the divergence
of a suitable subdivision scheme. To do this we construct
a general method to produce divergent subdivision schemes
having smooth refinable functions. The criterion therefore
establishes a complete classification of divergent subdivision schemes
