For each smooth variety over a field with positive characteristic, let \(F\) denote its Frobenius. Let \(R\) be an integral domain of finite type over \(\mathbb {Z}\) and \(f: X \to \mathrm{Spec} R\) a smooth projective morphism of relative dimension \(1\). Let \(E\) be a vector bundle on \(X\). Assume that for infinitely many \(q_n\in\mathrm{Spec} R\) there are vector bundles \(A_{q_n}\) on the fiber \(X_{q_n}\) and integers \(e_n>0\) such that \(E_{q_n} \cong F^{\ast e_n}(A_{q_n})\). If \(e_n\) increases enough, they proved that the restriction of \(E\) to the generic fiber of \(f\) is semistable. This s a very strong result. They also proved (over a field with positive characteristic), several results on flat vector bundles or on vectors bundles \(A\) such that \(F^{\ast s}(A) \cong F^{\ast t}(A)\) for some \(s>t\). They give generalizations of several results of D. Gieseker [Ann. Sci École Norm. Sup. (4) 6, 95–101 (1973; Zbl 0281.14013)]. Stronger results are obtained if the smooth variety is defined over a finite field. If \(Y\) is a smooth projective variety defined over a finite field \(K\) with trivial fundamental group (over the algebraic closure), then every flat vector bundle on \(X\) is trivial.