The inverse stability of Artin-Schreier polynomials over finite fields

Let $p$ be a prime number and $q$ a power of $p$. Let ${\mathbb{F}}_q$ be the finite field with $q$ elements. For a positive integer $n$ and a polynomial $\phi (X)\in {\mathbb{F}}_q[X]$, let $d_{n,\phi }(X)$ denote the denominator of the $n$th iterate of $\frac{1}{\phi (X)}$. The polynomial $\phi (X)$ is said to be inversely stable over ${\mathbb{F}}_q$ if all polynomials $d_{n,\phi }(X)$ are irreducible polynomial over ${\mathbb{F}}_q$ and distinct. In this paper, we characterize a class of inversely stable polynomials over ${\mathbb{F}}_q$. More precisely, for $\phi (X)=X^{p^t}+aX+b\in {\mathbb{F}}_q[X]$ with $t$ being a positive integer, we provide a sufficient and necessary condition for $\phi (X)$ to be inversely stable over ${\mathbb{F}}_q$.