The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: t_1,t_2 \in \mathbb{F}_p \}$ be the $2$-dimensional vector space over $\mathbb{F}_p$. Denote by $\text{Sym}(\mathcal{V})$ the group of bijections on $\mathcal{V}$. Then any element of $\text{Sym}(\mathcal{V})$ is defined by a pair $ \langle f_1(x_1,x_2),f_2(x_1,x_2) \rangle $ of functions on $\mathcal{V}$. Since $\mathbb{F}_p$ is a finite field, we can assume that $f_1$ and $f_2$ are polynomials (of degree at most $p-1$ in each variable). Now, suppose we are given an arbitrary pair $$ \langle g_1(x_1,x_2),g_2(x_1,x_2) \rangle, $$ where $g_1,g_2 \in \mathbb{F}_p[x_1,x_2]$. Can we determine if the map $$ (x_1,x_2) \mapsto (g_1(x_1,x_2),g_2(x_1,x_2)) $$ induces:

*a***permutation**on $\mathcal{V}$?*a***$p^2$-cycle**on $\mathcal{V}$?

**Update:** There are plenty of types of pairs which are obvious bijections. For example, so called "triangular": $\alpha= \langle x_1+f_1,x_2+f_2(x_1) \rangle$, where $f_1 \in \mathbb{F}_p$ and $f_2(x_1) \in \mathbb{F}_p[x_1]$. Here, $\alpha$ is a $p^2$-cycle if and only if $f_1 \neq 0$ and $\text{deg}f_2 = p-1$.