  If a system mixes too slowly, putting a hole in it can completely destroy the
richness of the dynamics. Here we study this instability for a class of
intermittent maps with a family of slowly mixing measures. We show that there
are three regimes: 1) standard hyperbolic-like behavior where the rate of
mixing is faster than the rate of escape through the hole, there is a unique
limiting absolutely continuous conditionally invariant measure (accim) and
there is a complete thermodynamic description of the dynamics on the survivor
set; 2) an intermediate regime, where the rate of mixing and escape through the
hole coincide, limiting accims exist, but much of the thermodynamic picture
breaks down; 3) a subexponentially mixing regime where the slow mixing means
that mass simply accumulates on the parabolic fixed point. We give a complete
picture of the transitions and stability properties (in the size of the hole
and as we move through the family) in this class of open systems. In particular
we are able to recover a form of stability in the third regime above via the
dynamics on the survivor set, even when no limiting accim exists.
