Complexity function and forcing in the 2D quasi-periodic rauzy tiling

The quantitative characteristics of the long-range translational order in the 2D quasi-periodic Rauzy tiling (complexity function and forcing depth) have been investigated. It is proved that the complexity function C(n) is equal to the number of figures in the n-corona grown from a seed composed of three figures of different types. The complexity function c(n) is found to be additive. A relationship between the jumps in the maximum forcing depth and large incomplete particular expansions in a chain fraction of irrational angles of rotation of a unit circle, which determine the growth of geodetic chains, is established.