**3.5. Multi-Fractals**

The power-law nature of the two-point and higher order correlation functions on small scales is suggestive of some kind of scaling behaviour, at least in the range of scales where the power law is observed. The simplest structure that has such scaling properties is the simple fractal, first studied in this context by Efstathiou, Fall and Hogan (1979). Martínez and Jones (1990) have since shown that even in this apparent scaling regime, the distribution of galaxies is significantly more complex.

If the structure is not that of a simple fractal, what might it be?
Jones et al (1988)
have suggested that it may be *Multifractal* - that
is, a distribution which over a certain range of scales can be
characterized by a set of dimensions rather than just one
dimension. There is scaling, albeit of a rather complex kind.

The multifractal description of the clustering process goes beyond
the two-point correlation function, encapsulating all high order
correlation functions in one function, *D*_{q}. The power
of the technique has been demonstrated by
Martínez et
al. (1990)
who examine the
scaling structure of a number of well known clustering models and
provide a variety of algorithms for calculating the dimensionality
function *D*_{q}. The simplest of these is from the
formulae for what is in fact the Renyi dimensions of the point set:

(59) | |

Here, *n*_{i}(*r*) is the count of particles in the
ith cell of size *r*,
and *N* is the total number of particles in all cells. There is a
technical problem involved in taking the limit in a discrete sample as
the cell size goes to 0.

In practise, box counting methods of determining dimensions are rather inefficient and tend to be dominated by shot noise. Van de Weygaert, Jones and Martínez (1991) have shown how to use the minimal spanning tree construct to calculate these dimensions with considerably fewer points than would be required by standard box counting methods.

The function *D*_{q} is related to the moment generating
function, *m*_{q}(*r*)
of the clustering distribution. The relationship is

(60) |

Hence each *D*_{q} encapsulates the information contained
in the statistical moments of the distribution of the point set. The fact
that in general one can in principle translate between the moments and
the dimensions means that they contain the same information, though
the information is presented in a different way. It is arguable that
*D* - *q* has a more immediate physical appeal.

The simplest application of the method is to compare the Hausdorf
dimension, *D*_{0}, with the Correlation Dimension,
*D*_{2}. Applying this to the ZCAT redshift catalogue gives
values

(61) |

This leads to the strong conclusion that the universe is not a simple
fractal characterized by one dimension. The value of
*D*_{0} indicates that
the characteristic structures are sheet-like rather than
filament-like. The value of *D*_{2} is just
3 - where
= 1.8 is
the slope of the two-point correlation function.