  This paper generalizes the results of [13] and then provides an interesting
example. We construct a family of $W$-like maps $\{W_a\}$ with a turning fixed
point having slope $s_1$ on one side and $-s_2$ on the other. Each $W_a$ has an
absolutely continuous invariant measure $\mu_a$. Depending on whether
$\frac{1}{s_1}+\frac{1}{s_2}$ is larger, equal or smaller than 1, we show that
the limit of $\mu_a$ is a singular measure, a combination of singular and
absolutely continuous measure or an absolutely continuous measure,
respectively. It is known that the invariant density of a single piecewise
expanding map has a positive lower bound on its support. In Section 4 we give
an example showing that in general, for a family of piecewise expanding maps
with slopes larger than 2 in modulus and converging to a piecewise expanding
map, their invariant densities do not necessarily have a positive lower bound
on the support.
