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Preperiodic points of polynomial dynamical systems over finite fields
- Source :
- Int. J. Number Theory 20 (2024), no. 9, 2307-2316
- Publication Year :
- 2024
-
Abstract
- 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.<br />Comment: 9 pages
Details
- Database :
- arXiv
- Journal :
- Int. J. Number Theory 20 (2024), no. 9, 2307-2316
- Publication Type :
- Report
- Accession number :
- edsarx.2405.00605
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1142/S1793042124501124