  The discovery of transiting circumbinary planets by the Kepler mission
suggests that planets can form efficiently around binary stars. None of the
stellar binaries currently known to host planets has a period shorter than 7
days, despite the large number of eclipsing binaries found in the Kepler target
list with periods shorter than a few days. These compact binaries are believed
to have evolved from wider orbits into their current configurations via the
so-called Lidov-Kozai migration mechanism, in which gravitational perturbations
from a distant tertiary companion induce large-amplitude eccentricity
oscillations in the binary, followed by orbital decay and circularization due
to tidal dissipation in the stars. Here we explore the orbital evolution of
planets around binaries undergoing orbital decay by this mechanism. We show
that planets may survive and become misaligned from their host binary, or may
develop erratic behavior in eccentricity, resulting in their consumption by the
stars or ejection from the system as the binary decays. Our results suggest
that circumbinary planets around compact binaries could still exist, and we
offer predictions as to what their orbital configurations should be like.
