  We consider the tensors generating matrix product states and density
operators in a spin chain. For pure states, we revise the renormalization
procedure introduced by F. Verstraete et al. in 2005 and characterize the
tensors corresponding to the fixed points. We relate them to the states
possessing zero correlation length, saturation of the area law, as well as to
those which generate ground states of local and commuting Hamiltonians. For
mixed states, we introduce the concept of renormalization fixed points and
characterize the corresponding tensors. We also relate them to concepts like
finite correlation length, saturation of the area law, as well as to those
which generate Gibbs states of local and commuting Hamiltonians. One of the
main result of this work is that the resulting fixed points can be associated
to the boundary theories of two-dimensional topological states, through the
bulk-boundary correspondence introduced by Cirac et al. in 2011.
