  We study the influence of electron-phonon coupling on electron transport
through a Luttinger liquid with an embedded weak scatterer or weak link. We
derive the renormalization group (RG) equations which indicate that the
directions of RG flows can change upon varying either the relative strength of
the electron-electron and electron-phonon coupling or the ratio of Fermi to
sound velocities. This results in the rich phase diagram with up to three fixed
points: an unstable one with a finite value of conductance and two stable ones,
corresponding to an ideal metal or insulator.
