Anne Siegel: Coverings associated with a substitution and arithmetic conditions for tilings

A substitution is a morphism for the concatenation on the set of finite words over a finite alphabet. It naturally extends to the set of one-sided or two-sided sequences over this alphabet. Under a condition of primitivity, a substitution always admits a periodic point, that can be associated with the shift map to generate a minimal symbolic dynamical system. We will detail how this symbolic system can be represented by an exchange of domains on an Euclidean compact set called the Rauzy fractal of the substitution. This Rauzy fractal may generate several classes of tilings, among which a periodic tiling, a self-similar tiling and a Markov Partition tilings. These tilings should be seen as generalizations of Thurston's tilings associated to a $ \beta$-numeration system. We will explain how an arithmetical approach allows one to deduce an algorithmic condition to check whether these tilings exist or not.

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