Dynamics and Rational Maps a New Section for Chapter 7 of the Arithmetic of Dynamical Systems

N be a rational map, where f 0 ,. .. , f N are homogeneous polynomials of degree d with no common factors. Then φ is defined at all points not in the indeterminacy locus Z(φ) = { P ∈ P N : f 0 (P) = · · · = f N (P) = 0 }. The map φ is said to be dominant if the image φ (P N Z(φ)) is Zariski dense, i.e., does not lie in a proper algebraic subset of P N. Alternatively, the map φ is dominant if the polynomials f 0 ,. .. , f N do not satisfy a non-trivial polynomial relation. If φ : P N → P N is a morphism, then it is not hard to check that deg(φ n) = (deg φ) n for all n ≥ 1, but if φ is merely assumed to be a dominant rational map, then the degree of φ n may be strictly smaller than (deg φ) n. For example, φ : P 2 −→ P 2 , φ ([x, y, z]) = [xy, xz, z 2 ] satisfies φ 2 ([x, y, z]) = [x 2 yz, xyz 2 , z 4 ] = [x 2 y, xyz, z 3 ], so deg(φ) = 2 and deg(φ 2) = 3. More generally, we have the following elementary results.