Quasi-periods of layer-by-layer growth of Rauzy tiling

Abstract

The dynamic characteristics of shaping of a layer-by-layer growth polyhedron in the 2D quasi-periodic Rauzy tiling have been investigated. Quasi-periodicity of deviations ρi(n) of sectoral growth rates from their means has been found in a computer experiment and rigorously mathematically substantiated. It is shown that sectoral deviations ρi(n) have exact upper and low boundaries, which are determined by induced maps L′i, isomorphous to the rotation of a unit circle by an angle βi. Quasi-periods and amplitudes of deviations ρi(n) are calculated from partial quotients of the expansion of βi in a continued fraction as well as from the denominators of the partial convergents determined by these expansion.