Integer points, diophantine approximation, and iteration of rational maps.

*(English)*Zbl 0811.11052Let \(\varphi(z)\in \mathbb{Q}(z)\) be a rational function with rational coefficients. Then \(\varphi\) determines an endomorphism of \(\mathbb{P}^ 1 (\mathbb{Q})\); this in turn gives rise to a dynamical system. Interest in such systems stems from \(\varphi\)-canonical heights, as studied by G. S. Call and the author [Compos. Math. 89, 163-205 (1993)].

This paper studies the diophantine properties of orbits under such systems. For example: Theorem. If \(\varphi(z)\) has degree at least 2 and if \(\varphi\circ \varphi\not\in \mathbb{Q}[z]\) then for any \(t\in \mathbb{P}^ 1 (\mathbb{Q})\) the orbit \(\{t, \varphi(t), \varphi(\varphi(t)), \varphi(\varphi (\varphi (t))), \dots\}\) contains only finitely many distinct integers.

This can often be made more quantitative, as follows. Theorem. If, furthermore, \(1/ (\varphi\circ \varphi)\not\in \mathbb{Q}[ 1/z]\) and the orbit is nonrepeating, then writing the \(n\)-th iterate of \(t\) as \(\varphi^ n(t)= a_ n/b_ n\) in lowest terms, we have \(\lim_{n\to\infty} (| a_ n|/ | b_ n|)=1\).

Finally, this paper proves that, in many cases, it is possible to show that orbits do not get extremely close to \(\infty\) in many cases (described precisely in the paper), in the sense that \[ \lim_{n\to\infty} {{\delta(A, \varphi^ n(t))} \over {(\deg \varphi)^ n}} =0 \] for fixed \(A\in \mathbb{P}^ 1 (\mathbb{C})\), where \(\delta\) denotes a distance function on \(\mathbb{P}^ 1 (\mathbb{C})\).

The methods of the paper consist of reducing to diophantine equations such as Thue equations or more generally those covered by Siegel’s theorem. Therefore some results are ineffective in the sense that some constants cannot be explicitly computed.

This paper studies the diophantine properties of orbits under such systems. For example: Theorem. If \(\varphi(z)\) has degree at least 2 and if \(\varphi\circ \varphi\not\in \mathbb{Q}[z]\) then for any \(t\in \mathbb{P}^ 1 (\mathbb{Q})\) the orbit \(\{t, \varphi(t), \varphi(\varphi(t)), \varphi(\varphi (\varphi (t))), \dots\}\) contains only finitely many distinct integers.

This can often be made more quantitative, as follows. Theorem. If, furthermore, \(1/ (\varphi\circ \varphi)\not\in \mathbb{Q}[ 1/z]\) and the orbit is nonrepeating, then writing the \(n\)-th iterate of \(t\) as \(\varphi^ n(t)= a_ n/b_ n\) in lowest terms, we have \(\lim_{n\to\infty} (| a_ n|/ | b_ n|)=1\).

Finally, this paper proves that, in many cases, it is possible to show that orbits do not get extremely close to \(\infty\) in many cases (described precisely in the paper), in the sense that \[ \lim_{n\to\infty} {{\delta(A, \varphi^ n(t))} \over {(\deg \varphi)^ n}} =0 \] for fixed \(A\in \mathbb{P}^ 1 (\mathbb{C})\), where \(\delta\) denotes a distance function on \(\mathbb{P}^ 1 (\mathbb{C})\).

The methods of the paper consist of reducing to diophantine equations such as Thue equations or more generally those covered by Siegel’s theorem. Therefore some results are ineffective in the sense that some constants cannot be explicitly computed.

Reviewer: P.Vojta (Berkeley)

##### MSC:

11J99 | Diophantine approximation, transcendental number theory |

14G25 | Global ground fields in algebraic geometry |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

37B99 | Topological dynamics |

##### Keywords:

iteration of rational maps; integral point; dynamical system; \(\varphi\)- canonical heights; diophantine properties of orbits; orbit; diophantine equations; Thue equations; Siegel’s theorem
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\textit{J. H. Silverman}, Duke Math. J. 71, No. 3, 793--829 (1993; Zbl 0811.11052)

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##### References:

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[2] | G. Call and J. Silverman, Canonical heights on varieties with morphisms , to appear in Compositio Math. · Zbl 0826.14015 · numdam:CM_1993__89_2_163_0 · eudml:90256 |

[3] | R. Devaney, An Introduction to Chaotic Dynamical Systems , Addison-Wesley Studies in Nonlinearity, Addison-Wesley, Redwood City, Calif., 1989. · Zbl 0695.58002 |

[4] | S. Lang, Elliptic Curves: Diophantine Analysis , Grund. Math. Wiss., vol. 231, Springer-Verlag, Berlin, 1978. · Zbl 0388.10001 |

[5] | S. Lang, Fundamentals of Diophantine Geometry , Springer-Verlag, New York, 1983. · Zbl 0528.14013 |

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[8] | J. H. Silverman, The Arithmetic of Elliptic Curves , Graduate Texts in Math., vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026 |

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