  We investigate the holographic renormalization of the
Einstein-Maxwell-dilaton theory which provides an asymptotic Lifshitz geometry
dual to a Lifshitz field theory. In this case, the existence of a field
combination with zero scaling dimension causes an ambiguity in fixing local
counter terms. Nevertheless, we show that all possible local counter terms give
rise to consistent thermodynamic quantities with the Lifshitz black brane
results. In addition, we also study the retarded Green functions of the current
and momentum operator of a non-relativistic Lifshitz field theory. In the
non-zero momentum regime, the results show intriguingly that there exists a
massive quasi-normal mode whose effective mass is linearly proportional to
temperature and that even at zero temperature there exists a quasi-normal mode
in the non-relativistic Lifshitz medium.
