  Successive divisions of compact metric spaces appear in many different areas
of mathematics such as the construction of self-similar sets, Markov partitions
associated with hyperbolic dynamical systems, dyadic cubes associated with a
doubling metric space. The common feature in these is to divide a space into a
finite number of subsets, then divide each subset into finite pieces and repeat
this process again and again. In this paper we generalize such successive
divisions and call them partitions. Given a partition, we consider the notion
of a weight function assigning a "size" to each piece of the partition.
Intuitively we believe that a partition and a weight function should provide a
"geometry" and an "analysis" on the space of our interest. We are going to
pursue this idea in three parts. In the first part, the metrizability of a
weight function, i.e. the existence of a metric "adapted to" a given weight
function, is shown to be equivalent to the Gromov hyperbolicity of the graph
associated with the weight function. In the second part, the notions like
bi-Lipschitz equivalence, Ahlfors regularity, the volume doubling property and
quasisymmetry will be shown to be equivalent to certain properties of weight
functions. In particular, we find that quasisymmetry and the volume doubling
property are the same notion in the world of weight functions. In the third
part, a characterization of the Ahlfors regular conformal dimension of a
compact metric space is given as the critical index p of p-energies associated
with the partition and the weight function corresponding to the metric.

Metric Geometry