On dynamical irreducible set of polynomials

In this article, a necessary and sufficient condition is proved for the dynamical irreducibility of a family of polynomials over a finite field. Using this result, an explicit construction of a dynamically irreducible set of polynomials is given over the finite field $\mathbb{F}_{p^p}$ where $p$ is a prime. Moreover, the existence of dynamically irreducible sets of size at least $p^2$ is also established over every finite field $\mathbb{F}_q$ where $q$ is a $p$ power. Finally, a bound on the cardinality of the set is given that needs to be tested for the dynamical irreducibility of a set of polynomials.