  We describe a correspondence between domain wall solutions of Einstein
gravity with a single scalar field and self-interaction potential. The
correspondence we call 'conformal scale factor inversion (CSFI)' is a map
comprising the inversion of the scale factor in conformal coordinates, and a
transformation of the field and its potential which preserves the form of the
Einstein equations for static and isotropic domain walls. By construction, CSFI
maps the asymptotic AdS boundary to the vicinity of a naked singularity in a
theory with a special Liouville (exponential) scalar field potential; it is
also a map in the parameter space of exponential potentials. The correspondence
can be extended to linear fluctuations, being akin to an S-duality, and can be
interpreted in terms of 'SUSY quantum mechanics' for the fluctuation modes. The
holographic implementation of CSFI relates the UV and IR regimes of a pair of
holographic renormalization group flows; in particular, it is a symmetry of the
GPPZ flow.
