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The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we(More)
Sturmian words are infinite words that have exactly n+ 1 factors of length n for every positive integer n. A Sturmian word sα,ρ is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x 7→ x+ α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main(More)
A generating method of self-affine tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in [AI01]. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is(More)
For a substitution satisfying the Pisot, irreducible, unimodular condition, a tiling substitution plays a key role in the construction of Rauzy fractals (see [15, 3]). To try extending techniques developed for substitutions to automorphisms, [6] gives the way to construct Rauzy fractals by using tiling substitutions for automorphisms related to hyperbolic(More)
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