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We i n troduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a Z 2-action by rotations on the circle. This gives a new geometric interpretation(More)
Sturmian words are infinite words that have exactly n + 1 factors of length n for every positive integer n. A Sturmian word s α,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 R α : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main(More)
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)
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)
Suppose A ∈ Gl d (Z) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called " good star " , and has A * ≥ 0, where A * is an oriented compound of A. A morphism θ of the free group on {1, 2,. .. , d} is called a non-abelianization of A if it has structure matrix A. We show that there is a tiling substitution Θ whose " boundary(More)
We give a construction that extends to the general hyperbolic case recent work on the geometric representation of Pisot substitutions and the explicit construction of Markov partitions for Pisot toral automorphisms. For the sake of simplicity, we consider a simple example of an automorphism of the free group on 4 generators, whose associated matrix has 4(More)
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