Invariants for certain discrete dynamical systems given by rational mappings

Abstract

We study the existence of invariants for the family of systems in an open domain $\mathcal {D}$ of $\mathbb {R}^n$ or $\mathbb {C}^n$ 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 $\mathbb {K}^{n+1}$ 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 $n\ge 2$ 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.