Preperiodic points of polynomial dynamical systems over finite fields
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.00605
- arXiv:
- arXiv:2405.00605
- Bibcode:
- 2024arXiv240500605A
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Number Theory;
- Primary 37P05;
- Secondary 37P25;
- 37P35;
- 11T06;
- 13B05
- E-Print:
- 9 pages