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arXiv · 2603.21620

Proportion of periodic points in reduction of polynomials

Santiago Radi
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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 Fari\~na-Asategui and the author show that when $f$ is a polynomial of degree $d \geq 2$ non-linearly conjugate over $\mathbb{C}$ to a Chebyshev polynomial then the limit is zero. In this article, we address the remaining cases to give a complete classification of the problem in the case of polynomials.

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Santiago Radi. 2026-03-23. Proportion of periodic points in reduction of polynomials. https://arxiv.org/abs/2603.21620

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