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The distribution of visible matter in the universe, such as galaxies andgalaxy clusters, has its origin in the week fluctuations of density thatexisted at the epoch of recombination. The hierarchical distribution of theuniverse, with its galaxies, clusters and super-clusters of galaxies indicatesthe absence of a natural length scale. In the Newtonian formulation, numericalsimulations of a one-dimensional system permit us to precisely follow theevolution of an ensemble of particles starting with an initial perturbation inthe Hubble flow. The limitation of the investigation to one dimension removesthe necessity to make approximations in calculating the gravitational fieldand, on the whole, the system dynamics. It is then possible to accuratelyfollow the trajectories of particles for a long time. The simulations show theemergence of a self-similar hierarchical structure in both the phase space andthe configuration space and invites the implementation of a multifractalanalysis. Here, after showing that symmetry considerations leads to theconstruction of a family of equations of motion of the one-dimensionalgravitational system, we apply four different methods for computing generalizeddimensions $D_q$ of the distribution of particles in configuration space. Wefirst employ the conventional box counting and correlation integral methodsbased on partitions of equal size and then the less familiar nearest-neighborand k-neighbor methods based on partitions of equal mass. We show that thelatter are superior for computing generalized dimensions for indices $q<-1$which characterize regions of low density.

Cosmology in one dimension: Symmetry role in dynamics, mass oriented approaches to fractal analysis