Let R be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if i is a maximal Cohen-Macaulay R-module which begins an Auslander-Reiten sequence, then this sequence is produced by an endomorphism of m, which we call a Frobenius element. We observe that Frobenius elements can be easier to identify when R is a graded ring, instead of complete local. We give an example application, determining the shape of some components of Auslander-Reiten quivers. We also adapt results due to Zacharia and others, from the setting of Artin algebras. This allows us to list the potential shapes of the components of AR quivers in our setting. It also has an application to special cases of the Huneke-Wiegand Conjecture.