  We analyze the effective action and the phase structure of N-layer
sine-Gordon type models, generalizing the results obtained for the two-layer
sine-Gordon model found in [I. Nandori, S. Nagy, K. Sailer and U. D.
Jentschura, Nucl. Phys. B725, 467-492 (2005)]. Besides the obvious field
theoretical interest, the layered sine-Gordon model has been used to describe
the vortex properties of high transition temperature superconductors, and the
extension of the previous analysis to a general N-layer model is necessary for
a description of the critical behaviour of vortices in realistic multi-layer
systems. The distinction of the Lagrangians in terms of mass eigenvalues is
found to be the decisive parameter with respect to the phase structure of the
N-layer models, with neighbouring layers being coupled by quadratic terms in
the field variables. By a suitable rotation of the field variables, we identify
the periodic modes (without explicit mass terms) in the N-layer structure,
calculate the effective action and determine their Kosterlitz-Thouless type
phase transitions to occur at a coupling parameter \beta^2_{c} = 8 N \pi, where
N is the number of layers (or flavours in terms of the multi-flavour Schwinger
model).
