We work with shift spaces defined by infinite sequences of simple combinatorial rules called substitutions. To such a system we associate a renormalization cocycle, intimately related with a multidimensional continued fraction algorithm. Under suitable metric assumptions on the cocycle, namely the Pisot condition on the Lyapunov exponents, the shift space is measurably conjugate to an exchange of pieces on a fractal domain, known as Rauzy fractal. The study of the tiling theory by these fractals allows to deduce further informations on the spectrum of the symbolic subshift. When the spectrum is purely discrete we use certain suspensions of these fractals to construct explicit non-stationary Markov partitions for the sequence of toral automorphisms associated with the sequence of substitutions. We further analyze the action of the Weyl chamber flow (a higher-dimensional analogue of the Teichmüller flow) on these Markov partitions, recovering connections with continued fractions.
This is a joint work with P. Arnoux, V. Berthé, W. Steiner and J. Thuswaldner.