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Boundedness results for periodic points on algebraic varieties. (English) Zbl 1086.11029

Let \(X\) be an algebraic variety over a field \(K\) and let \(f : X \to X\) be a morphism. A point \(P \in X(K)\) is \(f\)-periodic if \(f^n(P)=P\) for some \(n>0\), and the smallest such \(n\) is called the period of \(P\).
In this article, the author gives short proofs of the following results:
Theorem: Let \(X\) be a proper variety over a field \(K\) which is finitely generated over the prime field and let \(f : X \to X\) be a morphism.
(1) if \(\text{char}(K)=0\) then the set of periods of all \(f\)-periodic points in \(X(K)\) is finite.
(2) if \(\text{char}(K)=p \neq 0\) then there exists \(n>0\) such that all the \(f^n\)-periodic points in \(X(K)\) have periods which are powers of \(p\).
It is not known whether the periods can really be unbounded if \(\text{char}(K)>0\).
The author gives examples where the theorem applies in the case where \(X\) is not proper. He also obtains sufficient conditions for the number of \(f\)-periodic points to be finite.
Proposition: Let \(X\) be a proper variety over a finitely generated field \(K\) of characteristic zero and \(f : X \to X\) a morphism. Suppose that there does not exist any positive dimensional subvariety \(Y\) of \(X\) such that \(f\) induces an automorphism of finite order of \(Y\). Then the number of \(f\)-periodic points in \(X(K)\) is finite. In particular, this is the case if \(X\) is a projective variety and \(L\) a line bundle on \(X\) such that \(f^*(L) \otimes L^{-1}\) is ample. He finally considers the case of \(p\)-adic fields.
Theorem: Let \(\mathcal{O}\) be the ring of integers in \(K\), a finite extension of \(\mathbb{Q}_p\), and let \(X\) be a proper scheme of finite type over \(\text{Spec}(\mathcal{O})\). Then there exists a constant \(M>0\) such that for any \(\mathcal{O}\)-morphism \(f : X \to X\), the periods of the \(f\)-periodic points in \(X(K)\) are all less than \(M\).

MSC:

11G35 Varieties over global fields
11G20 Curves over finite and local fields
14G20 Local ground fields in algebraic geometry
14G25 Global ground fields in algebraic geometry
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References:

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