On the Ratliff-Rush closure of an ideal of a one-dimensional ring

Let $I$ be an ideal in a Noetherian ring $R$ and let $\tilde{I}$ be its Ratliff-Rush closure. In this paper we study the asymptotic Ratliff-Rush number, i.e. $h(I)=\min \{n\in {\mathbb{N}}_{+}\mid I^m=\widetilde{I^m},\mathrm{\forall }m\ge n\}$, in the one-dimensional case. Since $1\le h(I)\le r(I)$, where $r(I)$ is the reduction number of $I$, we look for conditions that determine the extremal values of $h(I)$.