(ABRIDGED) We use high resolution (~0.1) F814W ACS images from the HST ACS Treasury survey of the Coma cluster at z~0.02 to study bars in massive disk galaxies (S0s), and in dwarf galaxies in the Coma core. Our study helps constrain the evolution of bars and disks in dense environments and provides a comparison point for studies in lower density environments and at higher redshifts. (1) We characterize the fraction and properties of bars in a sample of 32 bright (M_V <= -18, M_* > 10^9.5 M_sun) S0 galaxies, which dominate the population of massive disk galaxies in the Coma core. Measuring the S0 bar fraction must be handled carefully, as the results depend on the method used: the bar fraction for bright S0s in the Coma core is 50%+/-11%, 65%+/-11%, and 60%+/-11% for three methods of bar detection: strict ellipse fit criteria, relaxed ellipse fit criteria, and visual classification. (2) We compare the S0 bar fraction across different environments (Coma core, A901/902, Virgo). We find that the bar fraction among bright S0 galaxies does not show a statistically significant variation across environments spanning two orders of magnitude in galaxy number density (n~300-10,000 gal/Mpc^3). We speculate that the S0 bar fraction is not significantly enhanced in rich clusters because S0s in rich clusters are less prone to bar instabilities as they are dynamically hot and gas poor due to ram pressure stripping and accelerated star formation. In addition, high-speed encounters in rich clusters may be less effective than slow, strong encounters in inducing bars. (3) We analyze a sample of 333 faint (M_V > -18) dwarf galaxies in the Coma core. Using unsharp-masking, we find only 13 galaxies with bar and/or spiral structure. The paucity of disk structures in Coma dwarfs suggests that either disks are not common in these galaxies, or that any disks present are too hot to develop instabilities.
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