## 2. Generalized dimensionA fractal point distribution is usually characterized by its dimension and there exists a large variety of ways in which the dimension can be defined and measured. Of these possibilities two which are particularly simple and can be easily applied to a finite distribution of points are the box-counting dimension and the correlation dimension. In this section we discuss the "working definitions" of these two quantities that we have adopted for analyzing a distribution of a finite number of points. For more formal definitions of these dimensions the reader is referred to Borgani (1995) and references therein. The formal definitions usually involve the limit where the number of particles tends to infinity and they cannot be directly applied to galaxy distributions. We first consider the box-counting dimension. In calculating the
box-counting dimension for a distribution of points, the space is
divided into identical boxes and we count the number of boxes which
contain at least one point inside them. We then progressively reduce
the size of the boxes while counting the number of boxes with at least
one point inside them at every stage of this process. This gives the
number of non-empty boxes as a
function of the size of one edge of the box we then define To compute the correlation dimension for a point distribution with
N points we proceed by first labeling the points using an index j
which runs from 1 to For every point where is the position vector of
the point and
is the Heaviside function.
for
and
for
. We next divide
by the total number of points
If the probability exhibits a scaling relation we then define to be the correlation dimension. As with the box-counting dimension, the nature of the scaling behaviour may be different on different length scales and we may then get more than one value for the correlation dimension, each different value being valid over a range of scales. It is very clear that - which is
the probability of finding a point within a sphere of radius For a mono-fractal the box-counting dimension and the correlation dimension will be the same, and for a homogeneous, space filling point distribution they are both equal to the dimension of the ambient space in which the points are embedded. The box-counting dimension and the correlation dimension quantify
different aspects of the scaling behaviour of a point distribution and
they will have different values in a generic situation. The concept of
a generalized dimension connects these two definitions and provides a
continuous spectrum of dimensions
for a range of the parameter which is used to define the generalized dimension The quantity may exhibit different scaling behaviour over different ranges of length scales and we will then get more than one spectrum of generalized dimensions each being valid over a different range of length scales. From Eqs. (6) and (7) it is clear that the generalized dimension corresponds to the correlation dimension at . In addition corresponds to the box-counting dimension at . For a mono-fractal the generalized dimension is a constant i.e.
which reflects the fact that for a
mono-fractal the point distribution is characterized by a unique
scaling behaviour. For a generic multi-fractal the values of
will be different for different
values of Finally it should be pointed out that the Minkowski-Bouligand generalized dimension is one of the many possible definitions of a generalized dimension. The minimal spanning tree used by van der Weygaert & Jones (van der Weygaert & Jones, 1992) is another possible method which can be used. The Minkowski-Bouligand generalized dimension has the advantage of being easy to compute. In addition the various selection effects which have to be taken into account when analyzing redshift surveys can be easily accounted for when determining the Minkowski-Bouligand generalized dimension and hence we have chosen this particular method for the multi-fractal characterization of the galaxy distribution in LCRS, © European Southern Observatory (ESO) 1999 Online publication: November 3, 1999 |