Results are presented of a deep optical survey of the Ursa Major Cluster, a spiral-rich cluster of galaxies at a distance of 18.6 Mpc which contains about 30% of the light but only 5% of the mass of the nearby Virgo Cluster. Fields around known cluster members and a pattern of blind fields along the major and minor axes of the cluster were studied with mosaic CCD cameras on the Canada-France-Hawaii Telescope. The dynamical crossing time for the Ursa Major Cluster is only slightly less than a Hubble time. Most galaxies in the local Universe exist in similar moderate density environments. The Ursa Major Cluster is therefore a good place to study the statistical properties of dwarf galaxies since this structure is at an evolutionary stage representative of typical environments yet has enough galaxies that reasonable counting statistics can be accumulated. The main observational results of our survey are: (i) The galaxy luminosity function is flat, with a logarithmic slope alpha = -1.1 for -17 < M_R < -11, from a power-law fit. The error in alpha is likely to be less than 0.2 and is dominated by systematic errors, primarily associatedd with uncertainties in assigning membership to specific galaxies. This faint end slope is quite different to what was seen in the Virgo Cluster where alpha=-2.26. (ii) Dwarf galaxies are as frequently found to be blue dwarf irregulars as red dwarf spheroidals in the blind cluster fields. The density of red dwarfs is significantly higher in the fields around luminous members than in the blind fields. The most important result is the failure to detect many dwarfs. If the steep luminosity function claimed for the Virgo Cluster were valid for Ursa Major then in our blind fields we should have found about 1000 galaxies with -17 < M_R <-11 where we have found two dozen.
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