Proportion of periodic points in reduction of polynomials
Volume 225 / 2026
Acta Arithmetica 225 (2026), 163-185
MSC: Primary 37C25; Secondary 37P25, 13A35, 11R45, 37F10
DOI: 10.4064/aa260330-1-8
Published online: 10 October 2026
Abstract
In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given $K$ a number field and $f$ a rational function with coefficients in $K$, if $f_\mathfrak {p}$ denotes the reduction of $f$ modulo a prime ideal $\mathfrak {p}$ in the ring of integers of $K$, what is the limit inferior of the proportion of periodic points of $f_\mathfrak {p}$ when the norm of $\mathfrak {p}$ goes to infinity?
Recent results of Fariña-Asategui and the author show that when $f$ is a polynomial of degree $d \geq 2$ that is not linearly conjugate over $\mathbb C$ to a Chebyshev polynomial then the limit inferior is zero.
In this article, we address the remaining cases to give a complete classification of the problem in the case of polynomials.
