### Multiscale substitution schemes and Kakutani sequences of partitions

Ergodic Theory and Dynamical Systems Seminar

22nd November 2018, 1:00 pm – 2:00 pm

Howard House, 4th Floor Seminar Room

Substitution schemes provide a classical method for constructing tilings

of Euclidean space. Allowing multiple scales in the scheme, we introduce

a rich family of sequences of tile partitions generated by the substitution

rule, which include the sequence of partitions of the unit interval

considered by Kakutani as a special case. In this talk we will use new path counting

results for directed weighted graphs to show that such sequences

of partitions are uniformly distributed, thus extending Kakutani's

original 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.

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