We study the dynamical stability of the BTZ black string against fermonic and
gravitational perturbations. The BTZ black string is not always stable against
these perturbations. There exist threshold values for m2 related to the
compactification of the extra dimension for fermonic perturbation, scalar part
of the gravitational perturbation and the tensor perturbation, respectively.
Above the threshold values, perturbations are stable; while below these
thresholds, perturbations can be unstable. We find that this non-trivial
stability behavior qualitatively agrees with that predicted by a
thermodynamical argument, showing that the BTZ black string phase is not the
privileged stable phase.Comment: 9 pages, revised version to appear in Phys. Rev.
