Graph and number theoretic properties of certain maps over finite field

Abstract

In this thesis, we study the graph over a finite field $\mathbb{F}_q$, where $q$ is a prime power, obtained by iterations of the map $g(x)=x^p$, where $p$ is a prime number. Some properties of the graphs such as a characterization of their vertices and the number of cycles with specific length are showed. Moreover, some statistical estimates about the tail and cycle lengths of the related graphs are established.