Variation of Periods Modulo p in Arithmetic Dynamics

TL;DR

This paper investigates the behavior of orbit sizes in arithmetic dynamics, showing that for most primes, the reduced orbit size exceeds a logarithmic threshold, revealing statistical properties of dynamical systems over number fields.

Contribution

It establishes a lower bound on orbit sizes modulo primes for a broad class of dynamical systems, extending understanding of orbit variation in arithmetic dynamics.

Findings

Orbit sizes are larger than a logarithmic function for almost all primes.

The result holds for a broad class of self-morphisms over number fields.

Provides a statistical density result for orbit sizes in arithmetic dynamics.

Abstract

Let F : V --> V be a self-morphism of a quasiprojective variety defined over a number field K and let P be a point in V(K) with infinite orbit under iteration of F. For each prime ideal p of good reduction, let m_p(F,P) be the size of the F-orbit of the reduction of P modulo p. Fix any e > 0. We show that for almost all primes p, in the sense of analytic density, the orbit size m_p(F,P) is larger than (log(N(p)))^(1-e), where N(p) is the norm of the ideal p.