  The Perron-Frobenius theory for nonnegative matrices has been generalized to
order-preserving homogeneous mappings on a cone and more recently to
nonnegative multilinear forms.
  We unify both approaches by introducing the concept of order-preserving
multi-homogeneous mappings, their associated nonlinear spectral problems and
spectral radii. We show several Perron-Frobenius type results for these
mappings addressing existence, uniqueness and maximality of nonnegative and
positive eigenpairs. We prove a Collatz-Wielandt principle and other
characterizations of the spectral radius and analyze the convergence of
iterates of these mappings towards their unique positive eigenvectors. On top
of providing a new extension of the nonlinear Perron-Frobenius theory to the
multi-dimensional case, our contribution poses the basis for several
improvements and a deeper understanding of the current spectral theory for
nonnegative tensors. In fact, in recent years, important results have been
obtained by recasting certain spectral equations for multilinear forms in terms
of homogeneous maps, however as our approach is more adapted to such problems,
these results can be further refined and improved by employing our new
multi-homogeneous setting.
