Trace splittings in $C^*$-algebras of tiling systems via colourings

Tiles of a hierarchical tiling system are coloured with given colours. The resulting system implements colour symmetries and prescribed frequencies and is itself a hierarchical system whose prototile types admit an elegant description. The frequencies of occurrence of colours is interpreted using the unique trace on the $C^*$-algebra of the given tiling system and the trace on the $C^*$-algebra of the coloured tiling system.