Cosmology in one dimension: Symmetry role in dynamics, mass oriented
  approaches to fractal analysis
  The distribution of visible matter in the universe, such as galaxies and
galaxy clusters, has its origin in the week fluctuations of density that
existed at the epoch of recombination. The hierarchical distribution of the
universe, with its galaxies, clusters and super-clusters of galaxies indicates
the absence of a natural length scale. In the Newtonian formulation, numerical
simulations of a one-dimensional system permit us to precisely follow the
evolution of an ensemble of particles starting with an initial perturbation in
the Hubble flow. The limitation of the investigation to one dimension removes
the necessity to make approximations in calculating the gravitational field
and, on the whole, the system dynamics. It is then possible to accurately
follow the trajectories of particles for a long time. The simulations show the
emergence of a self-similar hierarchical structure in both the phase space and
the configuration space and invites the implementation of a multifractal
analysis. Here, after showing that symmetry considerations leads to the
construction of a family of equations of motion of the one-dimensional
gravitational system, we apply four different methods for computing generalized
dimensions $D_q$ of the distribution of particles in configuration space. We
first employ the conventional box counting and correlation integral methods
based on partitions of equal size and then the less familiar nearest-neighbor
and k-neighbor methods based on partitions of equal mass. We show that the
latter are superior for computing generalized dimensions for indices $q<-1$
which characterize regions of low density.
