Equal-mass stars in young open clusters and loose associations exhibit a wide spread of rotation periods, which likely originates from differences in the initial rotation periods and in the primordial disc lifetimes. We want to explore if the gravitational effects by nearby companions may play an additional role in producing the observed rotation period spread. We measure the photometric rotation periods of components of multiple stellar systems and look for correlations of the period differences among the components to their reciprocal distances. In this paper, we analysed the triple system AU Mic + AT Mic A&B in the 25-Myr beta Pictoris Association. We have retrieved from the literature the rotation period of AU Mic (P = 4.85d) and measured from photometric archival data the rotation periods of both components of AT Mic (P = 1.19d and P = 0.78d) for the first time. Moreover, we detected a high rate of flare events from AT Mic. Whereas the distant component AU Mic has evolved rotationally as a single star, the A and B components of AT Mic, separated by about 27 AU, exhibit a rotation rate a factor 5 larger than AU Mic. Moreover, the A and B components, despite have about equal mass, show a significant difference (about 40%) between their rotation periods. A possible explanation is that the gravitational forces between the A and B components of AT Mic (that are a factor about 7.3 x 10^6 more intense than those between AU Mic and AT Mic) have enhanced the dispersal of the AT Mic primordial disc, shortening its lifetime and the disc-locking phase duration, making the component A and B of AT Mic to rotate faster than the more distant AU Mic. We suspect that a different level of magnetic activity between the A and B components of AT Mic may be the additional parameter responsible for the difference between their rotation periods.
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