In June 2000, two remarkably similar $M_s$ 6.6 earthquakes stroke southern Iceland in the space of 3 days. This doublet is here analyzed to test whether aftershock triggering is linear, i.e., if the triggering patterns of these two similar mainshocks are also similar. Methodologically, the key issue is to separate the two contributions to the overall seismicity patterns. For this purpose, we model the first three days (June $17^{th}$ to $21^{st}$) of this aftershock sequence with an Omori-Utsu law. We then extrapolate it beyond June $21^{st}$ in order to estimate the variations linked with the second shock. We correct for the large transient changes in catalogue completeness. Our results indicate that instantaneous short-lived (hours to days) dynamic triggering took place at the Hengill triple junction, on the Reykjanes Peninsula and around the \'{A}rnes fault plane that ruptured on the $17^{th}$ of June. We observe delayed, long-lasting (longer than weeks) episodes of quiescence at Hengill and on the \'{A}rnes fault plane. At Hengill this quiescence remains significant up to at least the end of year 2000. These quiescences are more consistent with Coulomb stress calculations than the preceeding episodes of triggering. Comparison between the number of triggered events by the two mainshocks reveals that the second one triggered only $2.1\%$ at Hengill and $0.3\%$ at Reykjanes of the number (respectively $3.6\%$ and $0.1\%$ of the seismic moment) that could be expected with a simple stress-threshold model of dynamic triggering. This clear non-linearity in patterns of dynamic triggering is consistent with a model in which the population of nucleating earthquakes is depleted by the first trigger.