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A nondensity property of preperiodic points on Chebyshev dynamical systems. (English) Zbl 1214.11070
Summary: Let \(k\) be a number field with algebraic closure \(\overline{k}\), and let \(S\) be a finite set of primes of \(k\), containing all the infinite ones. Consider a Chebyshev dynamical system on \({\mathbb P}^2\). Fix the effective divisor \(D\) of \({\mathbb P}^2\) that is equal to a line nondegenerate on \([-2,2]^2\). Then we prove that the set of preperiodic points on \({\mathbb P}^2(\overline{k})\) which are \(S\)-integral relative to \(D\) is not Zariski dense in \({\mathbb P}^2 \).

MSC:
37P15 Dynamical systems over global ground fields
11G05 Elliptic curves over global fields
11G35 Varieties over global fields
11J71 Distribution modulo one
14G05 Rational points
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
37P35 Arithmetic properties of periodic points
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