1. Preperiodic points of polynomial dynamical systems over finite fields
- Author
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Andersen, Aaron and Garton, Derek
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Number Theory ,Primary 37P05, Secondary 37P25, 37P35, 11T06, 13B05 - 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., Comment: 9 pages
- Published
- 2024
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