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A nondensity property of preperiodic points on the projective plane. (English) Zbl 1244.37052
The main result of this paper concerns Zariski nondensity of \((D, S)\)-integral preperiodic points for a certain dynamical system on \(\mathbb{P}^2\). In particular, let \(k\) be a number field, let \(\rho\) be the squaring map on \(\mathbb{P}^2\), let \(S\) be any finite set of primes of \(k\) (including the archimedean ones), and let \(D\) be a nonzero effective divisor at least one of whose irreducible components is a line that intersects the torus \(\mathbb{S}^1 \times \mathbb{S}^1 \subseteq \mathbb{A}^2 \subseteq \mathbb{P}^2\) transversely. Then the set of \((D, S)\)-integral preperiodic points for \(\rho\) is not Zariski dense. Of course, in this case, preperiodic is the same as being expressible with all coordinates equal to zero or roots of unity. This result fits into a far-reaching conjecture stating that Zariski nondensity of \((D, S)\)-integral preperiodic points should be true for any dynamical system where \(D\) is a nonzero effective divisor, at least one of whose irreducible components contains no dense subsets of preperiodic points. The result in this paper is the first one toward the above conjecture in the case of a dynamical system in dimension greater than one.
The proof relies on an intricate calculation involving archimedean and non-archimedean absolute values. Let \(f\) be the equation of the line giving the irreducible component of \(D\) in the hypothesis, appropriately normalized. Given a generic sequence of preperiodic \((D, S)\)-integral points \(P_n\) (i.e., eventually leaving any proper Zariski closed subset of \(\mathbb{P}^2\)), and an absolute value \(|\cdot|_v\) of the number field \(k\), one can calculate \[ \lim_{n \to \infty} (|f(P_n)|_v)^{\text{avg}}, \] where \(|f(P_n)|_v^{\text{avg}}\) is the average absolute value at \(v\) of \(f\) evaluated at the Galois conjugates of \(P_n\). When one sums this over all values of \(v\), one obtains zero, by the product formula (interchanging the limit and the sum is shown to be possible because of the integrality hypothesis). However, when one considers each \(v\) separately, various equidistribution results due to Tate, Voloch, Scanlon, and Bilu show that one obtains, essentially, the logarithmic height of \(f\), which is known not to be zero (applying these results to the situation of the paper requires a great deal of care). This gives the contradiction that proves the main theorem.

MSC:
37P15 Dynamical systems over global ground fields
37P35 Arithmetic properties of periodic points
11G35 Varieties over global fields
11G50 Heights
11J71 Distribution modulo one
11J86 Linear forms in logarithms; Baker’s method
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
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