Polynomial endomorphisms over finite fields: experimental results

Given a finite field ${\text{F}}_q$ and $n\in {\mathbb{N}}^{\ast }$, one could try to compute all polynomial endomorphisms ${\text{F}}_q^n\text{lp}{\text{F}}_q^n$ up to a certain degree with a specific property. We consider the case $n=3$. If the degree is low (like 2,3, or 4) and the finite field is small ($q\le 7$) then some of the computations are still feasible. In this article we study the following properties of endomorphisms: being a bijection of ${\text{F}}_q^n\text{lp}{\text{F}}_q^n$, being a polynomial automorphism, being a {\em Mock automorphism}, and being a locally finite polynomial automorphism. In the resulting tables, we point out a few interesting objects, and pose some interesting conjectures which surfaced through our computations.