Integer points, diophantine approximation, and iteration of rational maps. (English) Zbl 0811.11052

Let \(\varphi(z)\in \mathbb{Q}(z)\) 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.
Reviewer: P.Vojta (Berkeley)


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
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