The size of semigroup orbits modulo primes

Wade Hindes and Joseph H. Silverman

Vol. 325 (2023), No. 2, 281–297

DOI: 10.2140/pjm.2023.325.281

Abstract

Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\in V(K)$. For each prime $\mathfrak{𝔭}$ of $K$, let $m_{\mathfrak{𝔭}}(S,P)$ denote the number of points in the orbit of $Pmod\mathfrak{𝔭}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\mathfrak{𝔭}}(S,P)$ and use it to show that the set of primes for which $m_{\mathfrak{𝔭}}(S,P)$ grows subexponentially as a function of $N_{K∕\mathbb{Q}}\mathfrak{𝔭}$ is a set of density zero. For $V={\mathbb{P}}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.

Keywords

arithmetic dynamics, finite field dynamics, semigroup dynamics

Mathematical Subject Classification

Primary: 37P25

Secondary: 37P05, 37P55

Milestones

Received: 31 March 2023

Revised: 1 September 2023

Accepted: 14 September 2023

Published: 3 November 2023