Invariants for certain discrete dynamical systems given by rational mappings

We study the existence of invariants for the family of systems in an open domain D of R-n or C-n whose components are linear fractionals sharing denominator. Such systems can be written with the aid of homogeneous coordinates as the composition of a linear map in with a certain projection and their behaviour is strongly determined by the spectral properties of the corresponding linear map. The paper is committed to prove that if n >= 2 then every system of this kind admits an invariant, both in the real and in the complex case. In fact, for a sufficiently large n several functionally independent invariants can be obtained and, in many cases, the invariant can be chosen as the quotient of two quadratic polynomials.