Periods of orbits modulo primes
Let \(K\) be a global field and let \(V\) be a quasi-projective variety, embedded in \(\mathbb{P}^m_K\). Let \(d>1\) and let \(\Phi\) be an endomorphism of \(V\) which can be written using homogeneous polynomials of degree at most \(d\). Let \(\alpha \in V(K)\) and consider the orbit \(O(\alpha) := \{ \Phi^n(\alpha) : n \geq 0 \}\). For all but finitely many places \(v\) of \(K\) we may consider the reduced orbit \(O_v(\alpha) := \{ \Phi^n(\alpha \bmod v) : n \geq 0 \}\). Of course the latter is finite. The authors show a lower bound for the size of \(O_v(\alpha)\), for ``most \(v\)''. More specifically: assume that \(O(\alpha)\) is infinite. For a place \(v\) denote by \(N_v\) the cardinality of its residue field. The authors show that for every \(\epsilon < 1/\log d\) the set of \(v\) for which \(O_v(\alpha)\) has less than \(\varepsilon \log N_v\) elements has density zero. They also prove similar results for orbits under finitely generated monoids, and for orbits under \(\mathbb{F}_p[t]\) in Drinfeld modules.
scientific article; zbMATH DE number 5604007
scientific article