  We consider a piecewise smooth expanding map of the interval possessing two
invariant subsets of positive Lebesgue measure and exactly two ergodic
absolutely continuous invariant probability measures (ACIMs). When this system
is perturbed slightly to make the invariant sets merge, we describe how the
unique ACIM of the perturbed map can be approximated by a convex combination of
the two initial ergodic ACIMs.
