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Juliana Patrone

Abstract/Description

In this research, we observe the properties of orbits produced from dynamical systems of the form f(x) = xd +c defined over a finite field GF(q). We will investigate how varying the values of d, c, and q = pk affect the cycles created. We will perform numerous experiments and analyze the features of every directed graph created. We will analyze functions whose graphs have the same cyclic structure and when those occur. We will then prove some foundational theorems and special cases that lead up to a generalized theorem. Our specialized cases state that if d = p = k or if d = p and k is a prime number, then there are only two different structures for the graphs depending on the p and k values. These cases then lead up to a more general result that states that we still get two different structures for the graphs when d = p and k is a composite number that is not a multiple of p.

Note on the Author

Juliana Patrone graduated Summa Cum Laude from Bridgewater State University as a Departmental Honors student. She majored in Mathematics with a double concentration in Pure Mathematics and Statistics with a minor in Music. Mentored by Dr. Jacqueline Anderson (Department of Mathematics), Juliana was awarded an Adrian Tinsley Program (ATP) summer research grant to delve into the world of arithmetic dynamics in the summer of 2023. Juliana was also a member of Bridgewater State University’s Track and Field team where her main event was the high jump. Juliana will be going back to Bridgewater State University to get her master's in mathematics.

Rights Statement

Articles published in The Undergraduate Review are the property of the individual contributors and may not be reprinted, reformatted, repurposed or duplicated, without the contributor’s consent.

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