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Monomial maps on and their arithmetic dynamics. (English) Zbl 1242.37064

Authors’ abstract: We say that a rational map on \(\mathbb P^n\) is a monomial map if it can be expressed in some coordinate system as \([F_{0}: \cdots :F_n]\) where each \(F_i\) is a monomial. We consider arithmetic dynamics of monomial maps on \(\mathbb P^{2}\). In particular, as Silverman (1993) explored for rational maps on \(\mathbb P^{1}\), we determine when orbits contain only finitely many integral points. Our first result is that if some iterate of a monomial map on \(\mathbb P^{2}\) 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 the Perron-Frobenius theorem.

MSC:

37P55 Arithmetic dynamics on general algebraic varieties
15A42 Inequalities involving eigenvalues and eigenvectors
11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.)
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References:

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