ISOTROPY, HOMOGENEITY, AND DIPOLE SATURATION

A distribution of points that satisifes the property of local isotropy is not necessarily homogeneous: homogeneity is implied by the conditi on of local isotropy together with the assumption of analyticity or re gularity. Here we show that the evidence of dipole saturation in catal ogs of galaxies (and clusters), together with a monotone growth of the monopole, is evidence of isotropy but not of homogeneity. This is ful ly compatible with a fractal structure which has the property of local isotropy, but it is nonanalytic and nonhomogeneous.