  The phase structure of the layered sine-Gordon (LSG) model is investigated in
terms of symmetry considerations by means of a differential renormalization
group (RG) method, within the local potential approximation. The RG analysis of
the general N-layer model provides us with the possibility to consider the
dependence of the vortex dynamics on the number of layers. The Lagrangians are
distinguished according to the number of zero eigenvalues of their mass
matrices. The number of layers is found to be decisive with respect to the
phase structure of the N-layer models, with neighbouring layers being coupled
by terms quadratic in the field variables. It is shown that the LSG model with
N layers undergoes a Kosterlitz--Thouless type phase transition at the critical
value of the parameter \beta^2_{c} = 8 N \pi. In the limit of infinitely many
layers the LSG model can be considered as the discretized version of the
three-dimensional sine-Gordon model which has been shown to have a single phase
within the local potential approximation. The infinite critical value of the
parameter \beta_c^2 for the LSG model in the continuum limit (N -> \infty) is
consistent with the latter observation.
