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 \(\varphi (X)\in {\mathbb F}_{q}[X]\) , let \(d_{n,\varphi }(X)\) denote the denominator of the nth iterate of \(\frac{1}{\varphi (X)}\) . The polynomial \(\varphi (X)\) is said to be inversely stable over \({\mathbb F}_{q}\) if all polynomials \(d_{n,\varphi }(X)\) are irreducible polynomial over \({\mathbb F}_{q}\) and distinct. In this paper, we characterize a class of inversely stable polynomials over \({\mathbb F}_{q}\) . Actually, for \(\varphi (X)=X^p-X+\xi \in {\mathbb F}_{q}[X]\) , we provide a sufficient and necessary condition for \(\varphi (X)\) to be inversely stable over \({\mathbb F}_{q}\) .
