  We have observed Proxima Centauri and Barnard's Star with Hubble Space Telescope Fine Guidance Sensor 3. Proxima Centauri exhibits small-amplitude, periodic photometric variations. Once several sources of systematic photometric error are corrected, we obtain 2 milli-magnitude internal photometric precision. We identify two distinct behavior modes over the past four years: higher amplitude, longer period; smaller amplitude, shorter period. Within the errors one period (P ~ 83d) is twice the other. Barnard's Star shows very weak evidence for periodicity on a timescale of approximately 130 days. If we interpret these periodic phenomena as rotational modulation of star spots, we identify three discrete spots on Proxima Cen and possibly one spot on Barnard's Star. We find that the disturbances change significantly on time scales as short as one rotation period. 
Photometry of Proxima Centauri and Barnard's Star Using HST Fine   Guidance Sensor 3: A Search for Periodic Variations