Monomial maps on P 2 and their arithmetic dynamics

We say that a rational map on P n is a monomial map if it can be expressed in some coordinate system as [ F 0 : ⋯ : F n ] where each F i is a monomial. We consider arithmetic dynamics of monomial maps on P 2 . In particular, as Silverman (1993) explored for rational maps on P 1 , we determine when o...

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Bibliographic Details
Published in:Journal of number theory 2011-12, Vol.131 (12), p.2409-2425
Main Authors: Gregor, Aryeh, Yasufuku, Yu
Format: Article
Language:English
Subjects:
Algebraic dynamics
Cyclotomic theory
Integral points
Monomials
Orbits
Perronʼs theorem
Vojtaʼs conjecture
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Summary:We say that a rational map on P n is a monomial map if it can be expressed in some coordinate system as [ F 0 : ⋯ : F n ] where each F i is a monomial. We consider arithmetic dynamics of monomial maps on P 2 . In particular, as Silverman (1993) explored for rational maps on 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 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 Perron–Frobenius theorem.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2011.06.012