Diophantine approximations, substitutions, and fractals

Abstract

In this chapter we shall show how to associate with a substitution σ a domain R _σ with fractal boundary, generalizing Rauzy’s famous construction of the Rauzy fractal [350] (see also Chap. 7). The substitutions are of Pisot type and unimodular, and our method is constructive. In this way, we obtain a geometric representation of the substitution as a domain exchange and, with a stronger hypothesis, we get a rotation on a torus. In fact, there are two quite different types of dynamics which act on the set R _σ: the first one (given by the shift) corresponds to the exchange of domains, while the dynamics of the substitution is given by a Markov endomorphism of the torus whose structure matrix is equal to the incidence matrix of the substitution. The first one has zero entropy, whereas the second one has positive entropy. The domain X _σ is interesting both from the viewpoint of fractal geometry, and of ergodic and number theory; see for instance [350], [353], [218], [220], [291], [290], [375] and Chap. 7.

This chapter has been written by S. Ito