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Tidally locked gas giants are typically in several-day orbits, implying a modest role for rotation in the atmospheric circulation. Nevertheless, there exist a class of gas-giant, highly irradiated objects---brown dwarfs orbiting white dwarfs in extremely tight orbits---whose orbital and hence rotation periods are as short as 1-2 hours. Phase curves and other observations have already been obtained for this class of objects, raising fundamental questions about the role of increasing planetary rotation rate in controlling the circulation. So far, most modeling studies have investigated rotation periods exceeding a day, as appropriate for typical hot Jupiters. Here we investigate atmospheric circulation of tidally locked atmospheres with decreasing rotation periods down to 2.5 hours. With decreasing rotation period, the width of the equatorial eastward jet decreases, consistent with the narrowing of the equatorial waveguide due to a decrease of the equatorial deformation radius. The eastward-shifted equatorial hot spot offset decreases accordingly, and the off-equatorial westward-shifted hot areas become increasingly distinctive. At high latitudes, winds become weaker and more rotationally dominated. The day-night temperature contrast becomes larger due to the stronger influence of rotation. Our simulated atmospheres exhibit variability, presumably caused by instabilities and wave interactions. Unlike typical hot Jupiter models, thermal phase curves of rapidly rotating models show a near alignment of peak flux to secondary eclipse. This result helps to explain why, unlike hot Jupiters, many brown dwarfs orbiting white dwarfs exhibit IR flux peaks aligned with secondary eclipse. Our results have important implications for understanding fast-rotating, tidally locked atmospheres.