Periods of orbits modulo primes
type: Item
instance of: scholarly article
title: Periods of orbits modulo primes
zbMATH Open document ID: 1254.37065
author: Amir Akbary
author: Dragos Ghioca
published in: Journal of Number Theory
publication date: 2009-09-15T00:00:00Z
review text: 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.
reviewed by: Lenny Taelman
Mathematics Subject Classification ID: 37P55
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 5604007
zbMATH Keywords: algebraic dynamics
zbMATH Keywords: Drinfeld modules
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.03.007
OpenAlex ID: W2071550419
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DOI: 10.1016/J.JNT.2009.03.007
description: scientific article; zbMATH DE number 5604007