Research PaperNo Access

Fariborz Maseeh Department of Mathematics, and Statistics, Portland State University, United States

and

Derek Garton

https://orcid.org/0000-0002-6030-8571

Fariborz Maseeh Department of Mathematics, and Statistics, Portland State University, United States

E-mail Address: gartondw@pdx.edu

Corresponding author.

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https://doi.org/10.1142/S1793042124501124Cited by:0 (Source: Crossref)

Abstract

For a prime p, positive integers $r, n$ , and a polynomial f with coefficients in $𝔽_{p^{r}}$ , let $W_{p, r, n} (f) = f^{n} (𝔽_{p^{r}}) \ f^{n + 1} (𝔽_{p^{r}})$ . 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 $𝔽_{p^{r}}$ . 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 $𝔽_{p^{r}}$ 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.

Keywords:

AMSC: 37P05, Secondary 37P25, 37P35, 11T06, 13B05

References

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