  The metallic surface states of a topological insulator support helical Dirac
fermions protected by topology with their spin locked perpendicular to their
momentum. They can acquire mass through magnetic doping or through
hybridization of states on opposite faces of a thin sample. In this case there
can be a component of electron spin oriented perpendicular to the surface
plane. The electron-phonon interaction renormalizes the dynamics of the charge
carriers through their spectral density. It also modifies the gap channel and a
second spectral function enters which, not only determines the out of plane
spin component, but also comes into in-plane properties. While the out of plane
spin component is decreased below the Fermi momentum ($k_F$), the in plane
component increases. There are also correlation tails extending well beyond
$k_F$. The angular resolved photo-emission line shapes aquire Holstein side
bands. The effective gap in the density of states is reduced and the optical
conductivity aquires distinct measurable phonon structure even for modest value
of the electron-phonon coupling.
