On the Number of Periodic Points of Quadratic Dynamical Systems Modulo a Prime

TL;DR

This paper investigates the behavior of periodic points in quadratic dynamical systems modulo primes, revealing a consistent pattern across different maps and suggesting a Rayleigh distribution for their counts based on experimental data.

Contribution

It extends previous asymptotic results to a broader class of quadratic maps and uncovers a universal pattern in the distribution of periodic points through computational experiments.

Findings

Sum of periodic points behaves similarly across various quadratic maps.

Numerical evidence suggests the distribution of periodic points follows a Rayleigh distribution.

The observed patterns are consistent with earlier theoretical results for specific maps.

Abstract

In 2004 Vasiga and Shallit studied the number of periodic points of two particular discrete quadratic maps modulo prime numbers. They found the asymptotic behaviour of the sum of the number of periodic points for all primes less than some bound, assuming the Extended Riemann Hypothesis. Later that same year Chou and Shparlinski proved this asymptotic result without assuming any unproven hypotheses. Inspired by this we perform experiments and find a striking pattern in the behaviour of the sum of the number of periodic points for quadratic maps other than the two particular ones studied previously. From simulations it appears that the sum of the number of periodic points of all quadratic maps of this type behave the same. Finally we find that numerically the distribution of the amounts of periodic points seems to be Rayleigh.