IS THERE ANY SCALING IN THE CLUSTER DISTRIBUTION

We apply fractal analysis methods to investigate the scaling propertie s in the Abell and ACO catalogs of rich galaxy clusters. The methods a re adapted to account for the incompleteness of the samples by appropr iately using the known selection functions in the partition sums. We a lso discuss different technical aspects of the method when applied to data sets with small number of points as the cluster catalogs. Results are compared with simulations based on the Zel'dovich approximation. We limit our analysis to scales less than 100 h(-1) Mpc. The cluster d istribution show a scale-invariant multifractal behavior in a limited scale range. For the Abell catalog this range is 15-60 h(-1) Mpc, whil e for the ACO sample it extends to smaller scales. Despite this differ ence in the extension of the scale-range where scale-invariant cluster ing takes place, both samples are characterized by remarkably similar multifractal spectra in the corresponding scaling regime. In particula r, the correlation dimension turns out to be D-2 similar or equal to 2 .2 for both Abell and ACO clusters. Although it is difficult to point out the scale at which homogeneity is reached with the present size of these redshift surveys, our results indicate that the cluster distrib ution shows a tendency to homogeneity at large scales, disproving the picture of a pure scale invariant fractal structure extending to arbit rarily large distances.