  In [FJ07], Favre and Jonsson developed tools from valuative theory to study
the dynamics of a dominant endomorphism of the complex affine plane. We extend
this theory to the case of any affine surface, over any field. We give a new
method to construct an eigenvaluation of an endomorphism. We generalize the
result of Favre and Jonsson and show that the first dynamical degree of a
dominant endomorphism of anormal affine surface is an algebraic integer of
degree at most 2. Plus, we obtain a new result of rigidity. The set of first
dynamical degrees of loxodromic automorphisms of a given affine surface must be
contained in the set of integers or in the set of algebraic numbers of degree
2.
