# 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.

- Last modified: 25/02/2019