We study numerically the behaviour of the generalized dimensions D(q)
and the f(alpha) spectrum in dependence on a control parameter in the
parametrically driven, damped pendulum. We find a continuous transitio
n of the D(q) of a chaotic attractor near a boundary crisis to those c
haracterizing a chaotic saddle into which the attractor is converted w
hen the crisis occurs. At an interior crisis a chaotic saddle collides
with a small chaotic attractor and both chaotic sets merge to a large
chaotic attractor. In the vicinity of the crisis-value of the control
parameter the D(q) of the large attractor are close to the D(q) of th
e small attractor for positive q and near to those of the chaotic sadd
le for non-positive q. Correspondingly. the f(alpha) spectrum of the l
arge chaotic attractor near the crisis-value is very broad and has a t
ypical phase-transition-like shape.