Multiscale substitution schemes and Kakutani sequences of partitions.

Substitution schemes provide a classical method for
constructing tilings of Euclidean space. Allowing multiple scales to
appear  in the substitution rule, multiscale substitution schemes are
introduced. In the talk we will consider some interesting new
geometric objects which are generated by such multiscale schemes.

We will focus on Kakutani sequences of partitions, in which every
element is defined by the substitution of all tiles of maximal measure
in the previous partition, and include the sequences of partitions of
the unit interval considered by Kakutani as a special case. Applying
new path counting results for directed weighted graphs, we will show
that such sequences of partitions are uniformly distributed, thus
extending Kakutani's result. Furthermore, we will describe certain
limiting frequencies associated with sequences of partitions, which
relate to the distribution of tiles of a given type and the volume
they occupy.