Preperiodic points of polynomial dynamical systems over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916421147033600 |
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| author | Andersen, Aaron Garton, Derek |
| author_facet | Andersen, Aaron Garton, Derek |
| contents | For a prime $p$, positive integers $r,n$, and a polynomial $f$ with coefficients in $\mathbb{F}_{p^r}$, let $W_{p,r,n}(f)=f^n\left(\mathbb{F}_{p^r}\right)\setminus f^{n+1}\left(\mathbb{F}_{p^r}\right)$. As $n$ varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of $f$ on $\mathbb{F}_{p^r}$. In this paper we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of $\mathbb{F}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Preperiodic points of polynomial dynamical systems over finite fields Andersen, Aaron Garton, Derek Dynamical Systems Number Theory Primary 37P05, Secondary 37P25, 37P35, 11T06, 13B05 For a prime $p$, positive integers $r,n$, and a polynomial $f$ with coefficients in $\mathbb{F}_{p^r}$, let $W_{p,r,n}(f)=f^n\left(\mathbb{F}_{p^r}\right)\setminus f^{n+1}\left(\mathbb{F}_{p^r}\right)$. As $n$ varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of $f$ on $\mathbb{F}_{p^r}$. In this paper we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of $\mathbb{F}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages. |
| title | Preperiodic points of polynomial dynamical systems over finite fields |
| topic | Dynamical Systems Number Theory Primary 37P05, Secondary 37P25, 37P35, 11T06, 13B05 |
| url | https://arxiv.org/abs/2405.00605 |