an:00558986
Zbl 0811.11052
Silverman, Joseph H.
Integer points, diophantine approximation, and iteration of rational maps
EN
Duke Math. J. 71, No. 3, 793-829 (1993).
00018135
1993
j
11J99 14G25 30D05 37B99
iteration of rational maps; integral point; dynamical system; \(\varphi\)- canonical heights; diophantine properties of orbits; orbit; diophantine equations; Thue equations; Siegel's theorem
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 \textit{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.
P.Vojta (Berkeley)