Monomial maps on P2 and their arithmetic dynamics

We say that a rational map on Pn is a monomial map if it can be expressed in some coordinate system as [F0:⋯:Fn] where each Fi is a monomial. We consider arithmetic dynamics of monomial maps on P2. In particular, as Silverman (1993) explored for rational maps on P1, we determine when orbits contain only finitely many integral points. Our first result is that if some iterate of a monomial map on P2 is a polynomial, then the first such iterate is 1, 2, 3, 4, 6, 8, or 12. We then completely determine all monomial maps whose orbits always contain just finitely many integral points. Our condition is based on the exponents in the monomials. In cases when there are finitely many integral points in all orbits, we also show that the sizes of the numerators and the denominators are comparable. The main ingredients of the proofs are linear algebra, such as Perron–Frobenius theorem.