Proportion of periodic points in reduction of polynomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908906728456192 |
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| author | Radi, Santiago |
| author_facet | Radi, Santiago |
| contents | 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 $d \geq 2$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21620 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proportion of periodic points in reduction of polynomials Radi, Santiago Number Theory 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 $d \geq 2$ 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. |
| title | Proportion of periodic points in reduction of polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.21620 |