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Abstract. 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 object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric proof of Yasutomi's characterization of all pairs (α, ρ) such that s α,ρ is a fixed point of some non-trivial substitution.

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International audience 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 show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.

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