Coboundary on colored tiling space as Rauzy fractal

Abstract

We consider the space Ω of colored tilings corresponding to a weighted substitution introduced by one of the authors in [1], which is a kind of natural extension of the f-expansion for a piecewise linar f. We give a characterization of adapted coboundaries, which are α- G-homogeneous on the space of integer points Ω 0 , where α is a complex number with a negative real part. The image of such a coboundary corresponding to a weighted substitution of cubic Pisot type, is a fractal set called Rauzy fractal. We also consider the Fibonacci tiling and the α- G-homogeneous, adapted coboundary on it. This coboundary together with the fractional part give a geometrical representation of the Fibonacci expansions that is more or less known.