In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations
Theorem 1 seems to prove a discrete-time analog of Chetaev theorem. The proof is quite elementary, but I believe it is wrong. In particular, I do not think the quantity $\gamma$ (in the definition there is perhaps a typo, since the minimum should be over the closure of $\tilde{\Omega}$) is necessarily greater than zero, because the closure of $\tilde{\Omega}$ (which is merely the closure of the orbit, in my understanding) could contain $ x^* $, and $ \Delta V(x^*)=0 $. Thus the proof crumbles. Am I misunderstanding something?